From another angle, 0^0 = 1 for the x^0 function. n^0 = 1, where n is negative real. p^0 = 1, where p is positive real. The x^0 function is continuous at x = 0. Using the x^0 function makes 0^0 = 1, which makes 0^2 - 0^0 = 0 - 1 = -1. Earlier, I used the 0^x function that makes 0^0 undefined and 0^2 - 0^0 undefined. Both solutions contradict each other. I don't know which "convention" to use, but favor the answer is Undefined.
-1 If you're accumulating a sum, you initialize your accumulator sum to 0. Similarly, if you're accumulating a power, a number of multiplications of something, you initialize your "accumulator product" to 1. If you multiply it by something zero times, your accumulator is still 1. That's why logically 0^0 is 1. Added support for that is if you’re taking something to the 0th power, you’re multipling a starting value by it that number of times. “0” times means you’re multiplying by that number not even once, so that “0” doesn’t make it any different from any number to the 0th power. So you should get the same result as any other number to the 0th power is, namely “1”.
And that is wrong. 0⁰ is undefined. You could argue that 0⁰ = 1 because x⁰ = 1, but you could also argue that 0⁰ = 0, because 0^x = 0. 0⁰ is undefined, the same way as x/0, log₀ and log₁ are undefined.
@@cricri593 No. By convention, 0⁰ is ambiguous, and therefore undefined. You can say x⁰ = 1, but you can also say 0^x = 0. But here ist the reason 0⁰ is undefined: 0⁰ = 0² ⋅ 0⁻² = 0² ⋅ 1/0² = 0²/0² = 0/0. And devision by zero is undefined.
Since you get contradictory results for x^y as x approaches zero, from what you get as y approaches zero, 0^0 is undefined, and not necessarily equal to 1. It is considered an indeterminate form, like 0 divided by 0.
Please be informed that there is a condition in general, a^0=1 when "a" is not equal to "zero". So, the power of anything (here "a") is zero except a=0. Therefore, in this argument the result will be undefined.
I agree that Zero to Zero is 1. Here is proof. If x ^ 0 = 1 you cannot have 1 = nothing (since there is no x to multiply with). Therefore the form x ^ 0 = 1 is because x ^ 1 = 1 * (no x's). So 0 ^ 0 = 1 * (no zeroes) = 1
As Don Benson put it, whether to define 0^0 is a matter of convenience not correctness. Many calculations get unnecessarily complex if we don't assign a value. And Don Knuth noted the real problem was we were comparing apples and oranges. 0^0 as a value is 1; 0^0 as a limiting form is undetermined.
The strange thing is...I do 0^0 on my windows calculator I get 1. BUT if I do this equation a different way - 0 divided by 0 - my calculator THEN gives me "result is undefined"
Orange is known as Applesin (son of apple) in some languages, both are fruits, both have comparable qualities - very easy to compare. Given enough intelligence one can compare anything 😁
The reason any number raised to the 0 power is 1 is because x^m/x^n = x^m-n. When you divide a number by itself, i.e., X^m / x^m that equals X^m-m or x^0, which is 1 because anything divided by itself is 1. That doesn't work for 0 though because 0/0 is undefined.
You need to put parentheses around your powers when they have multiple terms when using the ^ to indicate exponentiation, since there is no physically visible superscript. That is, for example, w^a-b = (w^a) - b by order of operations. To make the entire a - b be clearly the power, use w^(a-b).
Woohoo! Working on almost 50 here. I loved math in school but I didn’t enjoy exponents. I currently feel like a genius for remembering and had the answer in just a few seconds before opening the video to see if I was right. 😂😂 Thanks for the self confidence boost!
An interesting thing to do is to use a good graphing calculator and graph the following: x^0, 0^x, and x^x and look at the graphed results. x^0 results in a line at 1 as you would expect. 0^x results in a line at 0 when x is greater than 0. x^x results in a very interesting shape.
You could make 0^0 equal to any number between 0 and 1 if you try hard enough. For example, turn (0^0)/2 + (0^0)/2 into (\lim_{x\to 0} 0^x)/2 + (\lim_{x\to 0} x^x)/2 = 0/2 + 1/2 = 1/2.
"0^0" is, by definition, not a number. It is a disallowed operation similar to division by 0, so it's use in an equation is disallowed. The equation is the equivalent of saying "What does 0^0 minus blue equal?"
Who decides? You? I don’t think so, unless you have a Field medal in your pocket. Mathematicians decide. And mathematicians disagree on this, and they are divided into two groups . One group finds the value 0 more appropriate. Another group thinks it is better to assign the value one. Tao and Bourbaki are in this second group. But the point is that no one, as far as I know, says that 0^0 is not a number.
@@ZannaZabriskie In my experience, the two sides of the "debate" are 0^0 = 1 and 0^0 is undefined. I don't think anyone seriously suggests 0^0 = 0 should be the definition.
@@MuffinsAPlenty I remember some text with 0^0=0 position, (essentially to maintain the continuity over R+ of the function 0^x) but now I don't feel like finding them again. And, look, I could be wrong! Maybe you’re right and the other more common position is: 0^0 undefined. But in the end I just don’t care at all. I don’t care, basically because I am not passionate about this diatribe. Don’t get me wrong: I always read this kind of posts, and I'm always very amused to see these guys get passionate, and argue their positions vehemently, as if they were reciting the gospel. But the issue itself I am not passionate about at all. The main reason is that - correct me if I'm wrong - choice is irrelevant. Math doesn’t change whichever choice you make. None of three choices (0,1,undef) leads to an antinomy. None of the three opens up new mathematical worlds. at most, you will be able to write a summation that starts at zero instead of one plus an additional term, or something like that. As convenient as defining 0!=1. But the mathematical castle would certainly not collapse by setting 0!=ndef: you would write less elegant expressions, but nothing would change. With sympathy
So, out of two HPs and two TIs, all 4 give ERR (undefined). The only calculator I have the does otherwise is the Calculator program that comes with Windows!
At least the Windows calculator receives updates here and then, those calculators were stuck in the old era they were originally made with no updates ever since, so it couldn't keep up with the new defined numbers.
Assume that n is a positive real number. The reason that b^0 = 1 for b not 0 is that 1 = b^n/b^n = b^(n-n) = b^0, so therefore b^0 = 1 for b a nonzero real number. However, if we allow b to equal 0, then we obtain 0/0 = 0^n/0^n = 0^(n-n) = 0^0, so therefore 0^0 = 0/0 which is undefined, ergo 0^0 is undefined.
Why is 0^0 left undefined? Because you can find functions f(x) and g(x) such that f(x) and g(x) both approach 0 but f(x)^g(x) does not approach 1. For example, let f(x) = sin(x) and g(x) = 1/ln(x), where ln(x) denotes the natural logarithm of x. As x approaches 0 from numbers greater than 0, the given functions for f(x) and g(x) both approach 0. However, using techniques from Calculus II, we see that f(x)^g(x) is approaching e, which is approximately 2.718, rather than approaching 1. You can use your graphing calculator to help visualize the fact that, as x approaches 0 from numbers greater than 0, f(x)^g(x) is approaching e in this case.
This is, indeed, the reason why mathematicians un-defined 0^0 in the early 19th century (prior to that point, mathematicians considered 0^0 to be equal to 1). But this argument _shouldn't_ be seen as convincing, at least when you think of 0^0 as an arithmetic expression, instead of as a limiting form. What I mean by this is: when we say 0^0 is an indeterminate form, we are not talking about the _number_ 0 raised to the power of the _number_ 0. Instead, we are talking about _a function approaching 0_ raised to the power of _a function approaching 0._ To me, an arithmetic expression is when we have operations between _actual numbers,_ and a limiting form is when we replace _functions with their limits_ in an attempt to avoid computing the actual limit directly. The next thing to note is this: it is perfectly consistent for the _arithmetic expression_ 0^0 to have a value of 1 while simultaneously the _limiting form_ 0^0 is indeterminate. This is because, if we go back to the meaning of limiting forms, 0^0 being an indeterminate limiting form simply means: "knowing f(x)→0 and g(x)→0 is insufficient information to determine the limit of f(x)^g(x)". And one of the key points about limits is that *_the limit of a function can be different from the value of a function._* Said in more symbolic terms, f(a) can be different from lim(x→a) f(x). *_If lim(x→a) f(x) always had to match f(a), then there would be no purpose to limits whatsoever._* But the argument that 0^0 is an indeterminate limiting form hence must be undefined as an arithmetic expression is essentially saying: "If f(a) is defined, then f(a) might not match lim(x→a) f(x), so we cannot have f(a) defined." Do you see the problem with this reasoning? It undermines the entire concept of limits. But even worse than that, I have never seen a natural example of a discontinuity _caused by_ 0^0 being defined as 1 (except for the situation where we have 0^f(x)). Even the example you gave is not a discontinuity caused by 0^0, but rather is a discontinuity caused by ln(0) being undefined and/or −∞ not being a number. Letting f(x) = sin(x) and g(x) = 1/ln(x), we do indeed get a limiting form 0^0 for f(x)^g(x) as x approaches 0 from the right. However, this is not the same thing as _plugging in_ x = 0. If we plug in x = 0, we get g(0) = 1/ln(0), which is undefined since ln(0) is undefined. If you _ignore_ the fact that ln(0) is undefined and pretend that ln(0) = −∞, then you still have the issue of g(0) = 1/−∞, which is undefined since −∞ isn't a real or complex number. If you _ignore_ the fact that −∞ isn't a real or complex number and pretend that 1/−∞ = 0, then you can conclude that g(0) = 0. But in order to conclude that g(0) = 0, you have to ignore the fact that ln(0) is undefined and ignore the fact that −∞ isn't a real or complex number. So the discontinuity of sin(x)^(1/ln(x)) at x = 0 is not the fault of 0^0, but rather the fault of 1/ln(x) being undefined at x = 0. Even if we defined 0^0 as e, sin(0)^(1/ln(0)) would still be undefined, by virtue of ln(0) being undefined. To reiterate, even if we defined 0^0 = e, sin(x)^(1/ln(x)) would still be discontinuous at x = 0, because sin(x)^(1/ln(x)) would _still be undefined at x = 0._ Therefore, we can see that 0^0 is not to blame for the discontinuity in sin(x)^(1/ln(x)) at x = 0, so claiming that sin(x)^(1/ln(x)) is an example of why 0^0 must be undefined as an arithmetic expression is just bad reasoning.
But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@MuffinsAPlenty But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
May be it helps to think about where a^0=1 comes! For example (a^5)/(a^5)=a^(5-5)=a^0=1 Here you have to write 0^n / 0^n = ? But before applying the formula n-n=0 and therefore 0^0 is 1, you must see that 0^n is 0 and that a division to 0 is undefined! 👍
Another thing: 0! is undefined as the factorial function is defined to be multiplication, and that (-1)! is infinity, and 0! = 0 * infinity... uh, oh...
-1 I suppose one might verify that any number ^0 = 1 by exploring n^x as x->0 or limit as x ->0 i.e. x=0.1 =0.01 =0.001 =0.0001 say 5^x 5^0.1=1.175 5^0.01=1.016 5^0.001=1.002 5^0.0001=1.0001✔️✔️✔️ etc.... 0^0.1=0 0^0.01=0 0^0.001=0 0^1e-20=0 ???❌️ 0.1^0.1=0.794 0.01^0.01=0.955 0.001^0.001=0.993 0.0001^0.0001=0.999✔️✔️✔️ for n=0 revise the limits to n^x as n->0 and x->0 = 1
First a word about 3^0, for example. We have 3^4=3^3×3, 3^3=3^2×3 and 3^2=3^1×3. If we want also to get 3^1=3^0×3, we have to take 3^0=1. This a choice, not a demonstration, but it’s often useful. We can’t do the same with 0^0, because to get 0^1=0^0×0, we can chose any value for 0^0. But what can we do? It’s not so easy, because it needs the functions exp et ln. For any real numbers x et n, with x>0, x^n can be defined by x^n=exp(nln(x). For example 3^2=exp(2ln(3))=9 et 3^2,1=exp(2,1ln(3))=10,045....... (we can’t do this with x0). Then we can observe that 0,1^0,1=0,794.... and 0,01^0,01=0,954.....and 0,001^0,001=0,993......For this reason, it seem’s to be a good idea to take 0^0=1. It’s a choice, often taken and which can be useful, not the result of a demonstration.
0^0 is generally undefined. The most common is to set it to 1 so that x^k evaluates to 1 for x=0 and k=0 in a sum or series. But one may set it to anything suitable for the case where it's used. So all of the three alternatives, as well as any other answer, is actually correct.
0⁰ in my calculator give "ERROR"... as it should. If you calculate lim Xˣ for x->0⁺, you'll get 1. But if you calculate lim 0ˣ for x->0⁺, you'll get 0. This ambiguity leads to the statement, that 0⁰ is undefined, as it depends on the situation.
"Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@bobh6728 But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@AlbertTheGamer-gk7sn there is philosophy and there is just plain dumb. This is in the latter category. "undefined" in this context simply means, that there is no unique answer for this question, but math requires a unique answer. Therefore it is "undefined", as - in the rules and axioms of math - this is unsolvable.
@@m.h.6470 Well, we humans find new ways to solve problems, as negative numbers, irrational numbers, transcendental numbers, and imaginary numbers are ways to make unsolvable problems solvable.
To get a number to the zero power start with that number to any power and divide it by the same number to the same power. The answer is always 1. e.g. a to the n divided by a to the n = 1 ( how many a to the nth in a to the nth?). The problem is that when applying this to zero we are dividing by 0 which is a no no. I have a "proof" that 1=2. Guess where the mistake is hidden.
I LOVE my HP 11C. I had one since 1974, lost it around 2002 and thank God found one in pristine condition on E-Bay. I guess it is the RPN, that spoils us. When forced to use any Non-RPN Calculator, it is a disaster. Before I watch further, I'm guessing he will use The Calculus, specifically the Limit of x^x as x->0 to prove 0^0 is undefined
@@robertakerman3570 - Reverse Polish Notation. It’s a way of entering calculations without needing parenthesis. You enter the operands first, then the operator, so instead of (2+4)*(3+5)=, you’d key 2 ENTER 4 + 3 ENTER 5 + * So you get a partial result 6 when you keyed 2 ENTER 4 + and a partial result 8 when you keyed 3 ENTER 5 + and then the final result 48 when you keyed *
There is still discussion between the math scholars if 0^0 = 1. Some say it is simply 0, others say it is 1. I myself think it is 0, since 0^1 also is 0. For all other numbers this is not the case.
I still don't understand why 0 to the 0 power isn't 0. If I have 0 apples, no matter how many times I multipy no apples, it still comes out to no apples and you can't make a 3.14 with no apples.. And I don't need a calculater to figure this out. This is why I am 71 years old and never understood math - plus I hated it. Otherwise, I was mostly an A student. Math is just not rational, and in fact, all numbers just seem wholey irrational to me. That is why I choose to bake apples instead of counting them. Also, most of the people I know who do understand this are men. That is also why I don't understand men.......................I probably won't be back - I'm going to go bake an apple pie without counting how many it takes. Thanks anyway. P.S. I decided to come back just to give you a like, a 100%, a smiley face, an A++ and a few stars for trying since you seem to think it all makes sense and who am I to burst your bubble?
It is actually undetermined. 0 to the 0 power could be anything because you are essentially dividing 0 by 0. In essence you asking what number times 0 equals 0, which could be any number.
0^0 can be interpreted as the limit of x^x where x -> 0, which can be proved to be 1. This video has missed golden opportunity to even mention, let alone explain, it. But then, again, this guy is well known for talking a lot while saying little …
The value of zero to the power zero is a subject of debate and depends on the context in which it is used. In different areas of mathematics, it can be assigned different values: 1. In combinatorics and set theory: is often defined as 1. This is because there is exactly one way to choose zero elements from a set of zero elements. When we interpret exponentiation in set theory, it refers to the number of functions that can be defined from a set of size (the exponent) to a set of size (the base). For example, 3 to the power 2 represents the number of functions from a set of 2 elements (e.g., ) to a set of 3 elements (e.g., ), which is because each element in the domain (2 elements) can map to any of the 3 elements in the codomain (3 elements), i.e., . Now, consider the case : If you have two sets, both of size 0 (i.e., the empty set), how many functions can be defined from the empty set to the empty set? In set theory, there is exactly one function from the empty set to the empty set. This function is called the empty function, which is a vacuously defined function because there are no elements to map. This is a subtle but important point: even though there are no elements in the domain, there is still exactly one way to define a function, and that is to define nothing. 2. In calculus and analysis: it is considered an indeterminate form because defining it could lead to inconsistencies in certain scenarios, such as limits in calculus. 3. In general arithmetic: It is sometimes left undefined due to potential contradictions, especially when both the base and the exponent are zero. 4. In computer science: it is typically defined as 1 to simplify algorithms and calculations. So, the value depends on the mathematical context and the conventions being used. The above equation is in simple maths, so should be considered as "undefined".
EXCELLENT! Advanced mathematical theory, presented in a manner that obfuscates & bamboozles! You'd make a great accountant! CEO: "How much tax do we owe?" Accountant" "How much do you WANT to owe?":
Undefined. 0^p=0, where p is positive real. 0^n is undefined, n is negative real. 0^0 is undefined because the limit 0^0+ is 0, and the limit of 0^0- is undefined. 0^x is not continuous at x=0, so 0^0 cannot be determined. 0^2 - 0^0 = 0 - undefined = Undefined.
Zero to the power of zero, denoted by 0⁰, is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines 0⁰ = 1. In mathematical analysis, the expression is sometimes left undefined. -- wikipedia
"Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
Consider 0^5/0^5. Applying the rule ‘when you divide like bases you subtract their exponents’ you would get 0^(5-5) or 0^0. Since division by 0 is undefined 0^0 is also undefined. Not 1.
5:15 in advanced mathematics 0^0 -can be interpreted as- *is* undefined. Let's emphasize that. I wonder whether the author of this video conducted any research beyond putting "0 ^ 0" in his calculator. Any sources? Is there any school book claiming the result is 1? *EDIT:* My claim was wrong. The result depends on the subject. For example, in discrete mathematics 1 is a sensible result. Thx MuffinsAPlenty.
But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
"in advanced mathematics 0^0 -can be interpreted as- is undefined." Depends on which subject you're dealing with. In basically all of discrete mathematics, 0^0 = 1 is correct. And this is because the empty product (a product with no factors) is 1. And the reason that the empty product is 1 is because that's the _only possible value_ a product with no factors could have if we expect a product with no factors to abide by the associative property of multiplication. In discrete mathematics, exponentiation represents repeated multiplication: x^n means a product with n factors, all factors being x. So, in discrete mathematics, x^0 means a product with 0 factors, all factors being x. In this context, regardless of the value of x, this is the empty product (the fact that all factors are x is _vacuously_ true, regardless of the value of x), and therefore, must have a value of 1. So pretty much any discrete setting, 0^0 = 1 is correct. And it actually shows up in formulas too (though you often wouldn't think to _use_ those formulas in those cases, which is why many non-discrete mathematicians don't notice this), and in every formula involving discrete exponentiation where 0^0 shows up, you _only_ get the right answer when 0^0 is evaluated as 1. And this makes sense because _everything_ about discrete exponentiation is based on the associative property of multiplication. Feel free to look up "empty product", although it belongs to a more general idea of "empty operation" which can be found in things like universal algebra and category theory. I recommend the article "too simple to be simple" on nLab, particularly the subsection on biased definitions. Now, in analysis things get a bit more sticky, and I'm happy to talk about that if you want, but I wanted to make the discrete case clear first.
@@MuffinsAPlenty Thanks for the detailed and informative reply. To be honest, after wrting the comment I noticed that, for example, the binomial theorem (x + y)^n = x^0 * y^n + ... requires 0^0 = 1 to not break in the trivial case. Funnily enough, this didn't cross my mind during my studies. So, by now I would agree that 1 is "more correct" than 0 so to speak. What I'm wondering now: are there any areas (analysis) where people agree on 0^0 = 0? [The answer in this video ("the calculator says so") still feels a bit unsatisfying, but it's understandable he doesn't want to dive into advanced math to provide an example.]
@@michaelmuller5856 "What I'm wondering now: are there any areas (analysis) where people agree on 0^0 = 0?" Not that I'm aware of. One time someone referred me to the concept of Munchausen numbers (or Münchhausen numbers - there ). A Munchausen number is a number where, if you take all of the digits in the representation of the number and raise them to themselves and then take the sum, you get the number back. So for example, 3435 is a Munchausen number since 3^3 + 4^4 + 3^3 + 5^5 = 3435. Obviously, Munchausen numbers are _super_ base dependent. Like 3435 is a Munchausen number in base ten but wouldn't be in other bases. *_Some_* people who have done work on Munchausen numbers adopt the convention that 0^0 = 0. For example, if one takes 0^0 = 0, then 438579088 is an additional Munchausen number in base ten. It's been proven that 0*, 1, 3435, and 438579088* are the only Munchausen numbers in base ten (* if one takes 0^0 = 0). Now, as you can probably guess, working on Munchausen numbers is a very fringe topic in mathematics, partially because it is not much more than a curiosity, it's super base-dependent, and then we get results which tell us that there really are only a few in any given base (so there isn't even much more that can be said about them anyway). But even in this fringe topic, taking 0^0 = 0 *isn't* agreed upon. For example, the person who named them "Munchausen numbers" to begin with says that the only two reasonable approaches are taking 0^0 = 1 or having 0^0 be undefined. (And in some bases, you get more Munchausen numbers when taking 0^0 = 1, as opposed to 0^0 = 0). But this is pretty much the _only_ situation I have seen where some people say taking 0^0 = 0 is what should be done. It's a fringe topic that seems uninteresting and arbitrary to me, and even those who have done work on this don't agree with that stance. In terms of analysis, typically analysts just take 0^0 to be undefined, and that's pretty much because of limits and continuity. The two-variable function f(x,y) = x^y cannot be made continuous at (0,0) even if one were to redefine 0^0. So that means some functions which have a limiting form of 0^0 will have a discontinuity associated with 0^0. And since analysts often want to deal with continuous functions, they will often take 0^0 to be undefined so that they don't have to deal with functions discontinuous at some point in their domains. Of course, there are _still_ plenty of situations in which they have to take 0^0 = 1, such as using power series/dealing with analytic functions. But still, I have _never_ seen an analyst suggest that 0^0 = 0 is "correct".
Hi, Some devices and software, but not all, will give you 0^0==1. My old TI-85 and Realcalc for Android failed. ❌❗ I tried the free version of WolframAlpha, and some functions worked if called a certain way. The output notes warn that some intermediate steps may run through Real and Complex routines that can affect some results (making it hard to tell simplify or solve to work over just the domain of Real numbers). I did find that the Limit function was consistently correct. And the notes do not list 0^0 as one of the 8 undefined or indeterminate numbers like 0/0. Anyway, the expression below is as simple as I could make it. One call to Limit passing two limit variables to approach 0: X and Y for Y^2 and X^X expressions. I add so the result will be positive. 0^2 + 0^0 == 1 or 0 + 1 == 1 ✅ Good old limits. ❤ Here's the expression: lim(x, y)->(0, 0) (x^x)-(y^2) Regards, Eric
To me, it's pointless, since I don't think there is any application for it. IOW, my answer would be "meaningless". It's like, "Where's the center of the surface of a sphere?" Or "What's farther north than the North Pole?" Or "What's infinity divided by 6?"
My answer was -1. My scientific calculator on my mobile says: underterminate. If I remember it correctly anything raised to power 0 is defined to be 1, because that conforms to the rules of addition of exponents. I watched the video just to be reminded of what I learned in math 50 years ago. But it's easy to see why 0^0 has to be defined to 1.. How anyone can think it's undefined baffles me, though. That would create some havoc!
@@thomass2451 Except that 0^0 isn't undefined in pretty much any branch of discrete mathematics. When it comes to discrete mathematics, 0^0 = 1, and doing anything else (including calling it "undefined") just makes things unnecessarily complicated. (Depending on your definition of exponentiation, this is even sometimes _provable_ from the definition!) The decision by mathematicians in the early 19th century to un-define 0^0 (which was previously taken to be equal to 1, including by Leonhard Euler) is being recognized by more and more mathematicians as a mistake based on not properly understanding the concepts of limits and continuity. Because yes, the _only_ reason any mathematicians consider 0^0 undefined today is because of limits/continuity arguments.
@@thomass2451 Bourbaki write 0^0=1 - after 1970. So all Bourbaki-mathematicians thinked that was the appropriate choice. They were not a few, and they were important mathematicians. Nowadays, Terence Tao in his Analysis write the same thing. Not me: Terence Tao.
lim_(x->0^+) x^x = 1 and lim_(x->0^+) 0^x = 0 are two different ways of looking at this problem. Understand both limits assume positive diminishing x approaching 0. The former limit raises positive diminishing x to x (itself) and gives a result of 1, while the latter raises 0 to diminishing x and gives a result of 0. This is a conflict. The two interpretations do not match. Depending on your problem domain, one of the two interpretations will be correct. That is why C, now has 3 power functions; powr(), which is the former interpretation and is explicit; pow() this also the former and is implied; pown() which is the latter and is explicit. If you are a C programmer and use pow(), start analysing which version you need and use powr()/pown() where appropriate. This will disambiguate between the two versions and, make your choice, (and it is a choice,) explicit for maintainers/other users of your code. As an aside, WolframAlpha explicitly chooses undefined. It does not allow the use of the C pow() function, you must choose powr() or pown() and, use the appropriate limit syntax to tell it what you want to use I hope this helps.
But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@abhimanyubhattacharyya2403 But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@trannhatlong1968 But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
I was average or worse at maths when at school. Now, in my 60s I seem to have improved significantly, having answered -1 within two seconds. However, I have never heard of undef as a result
Un numero real o complejo al ser dividido por cero, el resultado se convierte en indeterminado o indefinido, como el caso del gringo bestia, que lo complica innecesariamente.
Ok, Ok. What I want to know is which Gray Beard was able to sell this definition? When will he die, and when will the new Gray Beard change the definition? What will A.I. say?
0 to the zeroth power is undefined no matter what, there's no opinions here, it's not humanitarian studies, math is precise . The property of x^0=1 is derived from the property a^m/a^n = a^(m-n), where we put m=n, but it is valid only if a≠0, because it is in the denominator of the LHS
-1 Anything to the Zero power is 1 / undefined. Zero to any power above zero is zero. Since there is a minus in front of the Zero to the Zero power, the answer is -1.
Thanks for helping some of us overcome those feelings of shame, failure and dread we grew up with in math class all those decades ago. Truly a traumatic experience. "Heads, I win; tails, you lose." Nothing ever explained, nothing translated into English. Just a dismissive sneer and a lot of humiliating red ink. A game you couldn't win but had to keep playing. Every once in a while, I gather the courage to try to comprehend this bizarre and opaque system of thought. Why does 0 to the 0 power equal 1? "Because I SAY SO!" I'm hoping someday to understand what a "quadratic equation" is.
(Amateur musings here, happy for anyone to correct me). I feel a bit funny about saying 0^0 is 1. I have always thought (maybe wrongly) that x^0 = 1 because it's really x/x, which is 1. Why? Because with negative exponents, x^-1 is x/x/x, x^-2 is x/x/x/x and so on. It's just the 'getting smaller' version of exponents getting bigger - except instead of multiplying itself more and more times, it's dividing itself more and more times. The problem I have is 0/0 isn't really definable. What's 'nothing' divided by 'nothing'? 1? I mean, does that really make sense? Can you have 'one' nothing? That would imply you could have numbers of nothing, which again, makes no sense at all. If you have a million nothings and I have one nothing, we both still have nothing. It seems better to say it's actually not definable because it doesn't make sense. Edit to say... well now that I think about it, if x/x = 1, which it should be, why should that still not be true if x=0? Still feels odd, since dividing by zero really makes no sense in the context of division.
This is (I hope) just a case of remembering that Zero to the power Zero is 1 Zero to any other power is Zero. So, we have Zero squared - Zero to the power Zero 0 - 1 = - 1 EDIT (after watching your video) I still don't quite get how an equation can have an answer and be 'undefined' as well.
4:59: This is utterly wrong. You can argue that 0⁰ = 1, since x⁰ = 1. But you could also argue that 0⁰ = 0, as 0^x = 0. Due to this arbitrary approach, 0⁰ is undefined, and therefore undefined is the only correct answer of this problem!
This sort of reasoning may seem convincing on its face, but when you think more carefully about it, it becomes unconvincing. 0^x = 0 is only true for _positive_ x, whereas x^0 = 1 is true for all x (except possibly 0). So the "rule" that 0^x = 0 _necessarily_ breaks at x = 0 no matter what we do, since 0^x is undefined for all negative numbers x (0^x is also undefined for the left half of the complex plane, including the imaginary axis). But x^0 = 1 is true for _all_ complex numbers (and even elements of an abstract algebraic structure, too!) So the 0^x = 0 rule will _always_ break at x = 0, _no matter what we do,_ even if we chose to define 0^0 = 0. This rule would _still_ break at x = 0. But the rule x^0 = 1 doesn't have to break _at all_ if we take 0^0 = 1. Indeed, any choice other than 0^0 = 1 (including having 0^0 be undefined) causes the rule x^0 = 1 to break at x = 0 and _only_ at x = 0.
@@KarlWork-n3i The product of no factors, known as the empty product, is almost universally taken to have a value of 1, because that's the only possible way for the product of no factors to be consistent with the associative property of multiplication. If one changes the meaning of exponentiation away from being repeated multiplication, then sure, the empty product reasoning breaks down. Nevertheless, even when one isn't using repeated multiplication, 1 is still almost always the "correct" value to assign 0^0 in those contexts. There is pretty much _never_ a situation where 0^0 should be assigned the value of 0. And this is why the two approaches commonly taken to 0^0 by professional mathematicians are "1 or undefined" (which doesn't include 0).
I said 0^0 is undefined but COULD be given values of zero or 1 if necessary. I was not implying that it has ever been given value of Zero. But if in the future and if a reason arose for the product of zero zeros to equal zero then that's ok but don't worry I would ask you if it's ok first.
Anyway with limits you can have situation where 0^0 = 0 Limit (F) = Limit (G) = zero as x approaches zero. Where F(x) = exp(-1/(x^2) and G(x) = x Then Limit (F^G) = zero as x approaches zero. Remember all tho for F, zero not in domain of F, it's Limit (F) still equals zero. So there are situations where 0^0 = 0
I got the answer -1 without any confusion. However, even though I have a degree in mathematics, I was confused when you said that undefined was also correct. 😅 I guess I’d better grab my encyclopedia of mathematics and brush up a little bit. 😊
If 0^0=1 because a^0=1 for any other a, then why is 0/0 undefined even though a/a=1 for any a other than 0? Also, 0^(1/n) = 0 for all n=1,2,3... , converging to 0^0=0. It may make sense to set 0^0=1 in many instances, but there are exceptions. Thus, undefined. btw, using the usual exponential law x^(a+b) = x^a * x^b, we could consider: 0^0 = 0^(0+0) = 0^0 * 0^0 thus, 0^0 must be some a that satisfies a=a^2, leading to exactly two solutions, 0^0=0, or 0^0=1.
Re why is 0/0 undefined even though a/a=1 for any a other than 0: this is good question. Remember that division by a number is the operation that "undoes" multiplication by that number. For example, 3×2=6, so 6÷2=3. Written in the opposite order, 6÷2=3, so 3×2=6. In general, a/b means the number you must multiply by b to get a. In symbols, a/b=c means that c solves c×b=a. To work out a/a, we put b=a to get: a/a=c, where c solves c×a=a. Now if a≠0, we can divide both sides of the equation by a to get c=1, and we're done. If a=0, the equation becomes c×0=0, and this is satisfied by any real value c. So in a sense 0/0 can take any real value. Because 0/0 doesn't have a definite value, we say that it is undefined.
You could define it as 1 because the limit of a^0 for a towards 0 is 1. Then again, you could define it as 0 because the limit of 0^x for positive x towards 0 is 0. At the end of the day, 0^0 is just as undefined as 0/0 or 0*infinity.
Undefined to me means that there is no answer or it is irrelevant because it can't be answered so its return is equal to 0. I still see the answer of 0 being valid.
Dave needed to add more context to show why 0 isn't an answer. Think of why x^0 is always one. For example, 2^2 divided by itself. Which is 4/4 which is 1. Using power laws, you subtract exponents when dividing, so you get 2^0. But we already know the answer is one, therefore 2^0 has to be equal. When it comes to a 0 base, and you do the same process, you get 0/0, and we know division by 0 can't be done, therefore it cannot be an answer. Now, if you are approaching your comment from a logic standpoint, with 0 as a return value for anything that isn't true, I can understand your thought. But this isn't a logic problem, and 0 in math isn't a replacement for 'undefined'.
Undefined is basically saying we humans don't have the intelligence to comprehend this. Nature however does it every time a black hole is created. Nature also has no problem with irrational or complex numbers. A perfect circle should be impossible because it is a ratio of its circumference and radius, i. e. it is a product of pi . Pi is irrational though and cannot be described as a ratio. So what gives. We do, and we know it. That's why concepts like undefined exist. We don't have the knowledge to describe it yet. I hope this helps.
@@anthonywarfield7348 You don't give humankind nearly enough credit for our understanding of certain basic mathematical concepts. Re - Perfect circle. Pi is indeed an irrational number which means, as you point out, it can't be expressed in the form of a ratio of integers. Basically it can't be expressed as a fraction. But the value of Pi is not in any way required to express the equation for a perfect circle. The value of a circle (simplified to locate the center at origin) is x^2 + y^2 = r^2. Pi is not required. You could, however, support an argument that the numerical value of the area of circle (or volume of a sphere) cannot be "perfectly" calculated due the irrational value of Pi. Re - "Undefined". We should come up with another term for that because it sounds like it means "geez, I dunno and might never know!". We understand the concept, or concepts. There are several ways an expression can result in an "undefined" solution. That's important - numbers can't be "undefined", only expressions can be "undefined". So what's the problem with division by zero? Let's use an example- 10 ÷ 5 = 2. We can check that by recognizing 2 × 5 = 10. Let's try that with division by zero. 10 ÷ 0 = y therefore y × 0 must equal 10. There is no solution for that expression. There's no number which can be multiplied by zero to equal 10. The answer is "undefined". We fully understand that. There's no mystery.
You show this as an algebraic expression. So, the rule is "any non-zero number to the power of zero is 1." Zero to the power of zero, denoted by 0^0, is a mathematical expression that is either defined as "1" or left "undefined," depending on context. In algebra and combinatorics, one typically defines 0^0 = 1...
@@pedinurse1 Hmmm. I used a computer in 1961 to plot a graph of x^x as x approached 0 from the positive side. The problem existed then as it still does.
The answer is UNDEFINED because of the reason for "anything to the zero power equals 1." We know anything divided by itself, except for zero, equals one; anything divided by zero is undefined. So 7^2 / 7^2 equals one - and the rules of exponents tells us 7^2 / 7^2 = 7^(2-2) = 7^0. Therefore 7^0 = 1. One could rewrite 0^0 as, say, 0^7 / 0^7, since by the rules of exponents this would be 0^(7-7) which is equal to 0^0. But we know anything divided by zero is UNDEFINED, so the term 0^7 in the denominator makes that expression UNDEFINED.
I could not believe that a calculator would return 1 for 0^0. This is the same as 0 divided by 0 and you cannot divide by zero. The only proper answer is undefined. So, I put it into my calculator and it gave an error, which is what it gives for any other division by zero.
I just for fun tried 0^0 on the rp calculator on my desktop, and it gave 1, which is just what I thought at first. However, I can also guess that it might be represented by a transfinite number, just as x/0 if I'm not getting totally confused of course.... Anybody wanna step in and enlighten me?
He ist right: 0^0 = 0^(1-1) = 0^(1)*0^(-1) = 0^1 / 0^1 = 0/0 Division by 0 i not forbidden, just simply UNDEFINED. (As long as you are not using 'limits'.) The video ist just wrong!!
This video is somewhat long-winded in getting to the crux of the issue. 0^2 is just a distraction. So the problem is simply what is 0^0? Again it's very easy to summarise this in a few words: x^0 = 1 for all values of x. But also 0^x = 0 for all values of x. So clearly 0^0 is a special case where two correct mathematical equations conflict as the answer could be 0 or 1 and is thus declared as undefined. My MacBook calculator app clearly displays the answer as "Not a number". Case closed - thank you!
This is all wrong from the start. The answer -1 is wrong. 0^0 is undefined. When you say x^0=1 this is true for any x different than 0. Proof: x^2 = x^(2+0) = x^2 * x^0. So if x^2 is non zero divide both sides by x^2 and obtain x^0=1. Now x^2 is different than zero as long as x is different than zero. So we proved that x^0=1 if x different than 0. If x=0 funny things can happen. This will depend on how fast you go close to 0. The result can be 0, infinity or a finite number. Second point: my calculator gives 0^0 as an error. I used the scientific calculator on my iPhone. Thirdly, please do nit encourage people to use the calculator blindly. This is so bad practice. Learn the concepts of mathematics. A calculator is just a machine, your brain can think.
But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
0 as it cannot be undefined or - 1. 0 is the substitute of nothing. It is impossible to have nothing multiples into nothing and subtract nothing. Therefore the answer is nothing or 0. This breaks the law of 0 and it’s definition.
Yeah I don't fully trust everything with a calculator. Sometimes you have to think about the problem. Undefined mean we don't know what this situation is yet (you don't know what you don't know). We can't prove or disprove so we leave it undefined. This means we cannot say 0^0=1 Thus the answers has to be undefined not (-1).
Not necessarily. It has long been taught that any number ‘x’ raised to the power ‘0’ (so x^0) is 1 algebraically. Therefore, when x=0’ 0^0 = 1. Remember, this is just standard convention. There are often special use cases that go against standard convention, which is why PEMDAS does not hold true for algebraic division. In such cases, everything after the division goes under the division dividing everything before the division. It is for this reason that undefined can also be correct under specific circumstances.
@@TheloniousCube Zero to the power of zero, denoted by 0^0, is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines 0^0 = 1. In mathematical analysis, the expression is sometimes left undefined. Computer programming languages and software also have differing ways of handling this expression. And that's why some missions to Mars CAN CRASH ... :) Beacuse of different DEFINITIONS ...
No, algebraically x^0 is defined as 1. Therefore 0^0 is 1. However, this is not the only convention that algebra varies. Algebra doesn’t follow PEMDAS. When applying division, algebra divides everything that comes before the division by everything that comes after the division. As with a lot of math, it depends. Which is why both 1 and undefined can both be correct leading to -1 in this case.
@@anwaraisling You have to realize that raising to the zero power means dividing the number by itself. 0/0 is undefined. 10/10 = 1 so 10 raised to the zero power equals 1.
x^x, let x-> 0 from the negative side or from the positive side. You will get closer to -1 and 1. see also f(y)^f(x)) -- how you approach 0 matters. Some people have argued that 0 does not exist, it is just a very small number, maybe so small that if you square it you would get to nothing. 🙂 or 😲 -- the zero daemon.
You got it wrong too. The definition of a^b is "take 0 and add a multiplied with itself for b times". So, it's obvious that the result should be 0, not -1, not undefined...
0 to the power 0 shall be left undefined. There is a reason why great mathematicians never touched the subject. If the calculator says the answer is 1 there is something wrong with the algorithm it’s using. Some calculators give an error message. A simple test would be to numerically determine the limit of the function f(x)=x^x as x ->0. Just substitute very small positive and negative values to demonstrate. On positive side the limit can actually be determined analytically using logarithms. You will realize the limit is 1 as x->0+ and it’s 0 as x->0- This means the function is discontinuous at x=0 and thus undefined.
I never cease to be amazed that people can take a simple idea and twist it to something complicated. If you take a calculator and substitute a number very close to 1, like x=1.0001 into the expression (x-1)/(x-1) you get a number very close to 1. If by hand you substitute x EXACTLY equal to 1 you get 0/0. Therefore 0/0=1 BUT if you take a calculator and substitute a number very close to 1 into the expression ((x^2)-1)/(x-1) you get a number very close to 2. If by hand you substitute x EXACTLY equal to 1 you get 0/0. Therefore 0/0=2 BUT if you do the same thing with the expression ((x^3)-1)/(x-1) you get 0/0=3 If you do the same thing with the expression ((x-4)-1)/(x-1) you get 0/0=4 If you use the right expression you can get 0/0 to equal any number at all. That is why 0/0 is meaningless.
Those expressions in x if plotted as functions not valid for x = 1 they have discontinuity there, your argument is totally wrong. Please try not to involve division by zero in any reasoning 0^0 is undefined but can equal 1 or zero if mathematically necessary. Nothing complicated about that.
What you are thinking about is the subject of Limits and the idea of continuity. Limit F(x) as x approaches "a" in the domain is NOT F(a) Sometimes it is and sometimes it's not.
Perhaps I didn't explain well. For a real function to be continuous at a point A in it's domain left and right hand Limits have to be equal and equal to f(A). For the last graph you mentioned both left and right hand Limits are the same and equal 4 but f(1) does NOT equal 4. So the graph has a discontinuity at x = 1. What that means is as you walk up the graph approaching f(1) from the left and right you will be approaching 4 but at x = 1 graph undefined and has discontinuity there. Same idea for the other graphs. So argument does not work.
@@KarlWork-n3i But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
ChatGPT4o says it is UNDEFINED or -1 it depends on the system we choose: The expression 0 2 − 0 0 0 2 −0 0 is undefined due to the ambiguity around 0 0 0 0 . Here's the breakdown: 0 2 = 0 0 2 =0, since any non-negative power of 0 results in 0. 0 0 0 0 is indeterminate or undefined in most contexts. Some mathematical conventions define 0 0 = 1 0 0 =1 in specific cases (such as combinatorics), but it generally remains undefined due to the lack of a consistent interpretation across all areas of math. Therefore, 0 2 − 0 0 0 2 −0 0 is undefined unless you adopt the convention that 0 0 = 1 0 0 =1, in which case: 0 2 − 0 0 = 0 − 1 = − 1. 0 2 −0 0 =0−1=−1. So, the result depends on how 0 0 0 0 is treated. If 0 0 = 1 0 0 =1, the answer is − 1 −1; otherwise, the expression is undefined
my gut instinct was to say the result is zero but then i thought about 0^0 for a bit and ended up at undefined. I couldn't see how an equation with only zeros resolves to a number that is not zero because that just didn't make any sense in my head. So undefined was my answer in the end. I bet calculators simply have the rule that ^0 is always 1 with no exception for 0^0.
If you set the equation up as a unknown with x being the unknown then the equation is a bit more clearer and algebraic . Substitututing x for the question mark will maintain the equality of the equation. 0^2 - 0^0 = X and solve for X . ✌🏿 Good stuff
"0 to the anything" (0^x; x0) is 0. "Anything to the 0" (x^0; x0) is 1. 0 to the 0 can't be both 0 and 1, unless it's Schroedinger's Zero, so you're left with an undefined. Best you can do is take the limit as either base or exponent *approaches* 0.
But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
Ok the fact is that zero to the power of zero is sometimes defined as 1 (good arguments see Donald Knuth) to catch some special cases in polynoms or eg the geometric sum formula or the binomic formula and in calcus it is undefined because the limit is not defined see the statement from french math. Cauchy. So sometimes it is practical to define it ans sometimes it is not, it is the decision of the programmer of a cas or calculator program and there is absolutly nothing wrong with it
There is another reason why I say it’s undetermined. Let’s assume 0^0=1. If it’s defined like that I will perform legal mathematical operations on it to create a contradiction. 0^(1-1)=1 (0^1)(0^(-1)=1 (0^1)/(0^1)=1 0/0=1 and that’s a contradiction. Assuming 0^0=1 all operations I performed are valid but they led to an undefined result. That means the original assumption is incorrect.
How about "Nobody knows?" "Undefined" normally means "Increases without bound," ie, "infinity," though we try to avoid using the "I" word. Then there's "indeterminate." It could have a value, or be undefined, we don't know. Sometimes, it can be determined, like the limit of the ratio of two functions, with the "Hospital Rule" (Calculus instructors are pulling their hair out!). But I can't conceive of WHAT it could POSSIBLY be! Actually, this would make a very good "Zen Koan." 🤗
Zero does not exist in algrebra. Only mean median mode and range does. A lack of something is negative.. i call that a negative state. If you have 0 it means its a mean median mode and range of some quanity. Basically a baseline. Algebra has symbols for that less than, equal too and so on.
I think is because nothing to the power of 2 must be nothing. But 0 at the power of 0 is regarded as the starting point of a process, not physical objects, therefore 1. (but still thinking…)
At 5:03 you say that the calculator gives 1 if you enter 0 to the power 0 Please try that on the iphone calculator. You will see that your statement is not true. The result depends on the type of calculator you use.
The problem with 0^0 is that it may theoretically = 1 but it that does not work all of the time. For example, log(0^0) should = 0*log(0). But, there is no logarithm (exponent) I know of that can be applied to any base that results in 0. In that sense 0^0 is undefined.
The problem is that you're applying a rule in a situation where it doesn't apply. The rule that log(a^b) = b*log(a) is only valid if a > 0. You applied it in the context that a = 0, and therefore, the contradiction you're arriving at follows from applying a "rule" outside of the context where it is valid. Another example of this faulty reasoning: (-1)^2 can't be equal to 1. If it were, then we would have 0 = log(1) = log((-1)^2) = 2log(-1). But regardless of any branch cut you may take, log(-1) is never equal to 0 (which it would need to be in order for 0 = 2log(-1) to be true). Therefore, we can safely conclude (-1)^2 isn't equal to 1, right? Well, no, we can't conclude that. The issue is that I applied a rule outside of the situation in which it is valid.
@@MuffinsAPlenty First of all it is accepted in imaginary math that SWRT(-1)^2 can = -1 by special rules, and -1^2 = 1 of course, however we still say that SQRT(-1) or i or j is undefined,; it simply helps us handle some very important math issues in real life such as reactive power (vector) equaling real power (vector) plus imaginary power (vector) . Still, imaginary power is not actual power though it exists as temporarily stored power. In the case of 0^0 you are trying to apply classical math rules and assume the answer 1 is correct. I would call it undefined, not even imaginary as it has no useful application, because it fails a basic application of math i.e. applying a log conversion.
Contrary to your information you have provided , 0^2 - 0^0 does not equal -1. This expression actually equals 0. Here's why: 0 raised to any power, including 0 and 2, equals 0. This is because anything multiplied by 0 is 0. Therefore, 0^2 equals 0 and 0^0 also equals 0. Subtracting two identical numbers (in this case, both 0) results in 0. So, the correct answer is: 0^2 - 0^0 = 0
From another angle, 0^0 = 1 for the x^0 function. n^0 = 1, where n is negative real. p^0 = 1, where p is positive real. The x^0 function is continuous at x = 0. Using the x^0 function makes 0^0 = 1, which makes 0^2 - 0^0 = 0 - 1 = -1. Earlier, I used the 0^x function that makes 0^0 undefined and 0^2 - 0^0 undefined. Both solutions contradict each other. I don't know which "convention" to use, but favor the answer is Undefined.
-1 If you're accumulating a sum, you initialize your accumulator sum to 0.
Similarly, if you're accumulating a power, a number of multiplications of something, you initialize your "accumulator product" to 1.
If you multiply it by something zero times, your accumulator is still 1. That's why logically 0^0 is 1.
Added support for that is if you’re taking something to the 0th power, you’re multipling a starting value by it that number of times. “0” times means you’re multiplying by that number not even once, so that “0” doesn’t make it any different from any number to the 0th power. So you should get the same result as any other number to the 0th power is, namely “1”.
I agree with this.
And that is wrong. 0⁰ is undefined. You could argue that 0⁰ = 1 because x⁰ = 1, but you could also argue that 0⁰ = 0, because 0^x = 0. 0⁰ is undefined, the same way as x/0, log₀ and log₁ are undefined.
I did see it defined somewhere and before that it seemed strange to me too. Something about counting-principles or so...@@Nikioko
by convention it makes 1 for continuity
@@cricri593 No. By convention, 0⁰ is ambiguous, and therefore undefined. You can say x⁰ = 1, but you can also say 0^x = 0.
But here ist the reason 0⁰ is undefined: 0⁰ = 0² ⋅ 0⁻² = 0² ⋅ 1/0² = 0²/0² = 0/0. And devision by zero is undefined.
-1?
I know anything to the zero power is 1. Zero to the second power is 0. And zero to the zero power is 1. So 0-1 = -1.
Since you get contradictory results for x^y as x approaches zero, from what you get as y approaches zero, 0^0 is undefined, and not necessarily equal to 1. It is considered an indeterminate form, like 0 divided by 0.
Correct. Any other answer is bullsh*t.
That was my initial answer as well.
Please be informed that there is a condition in general, a^0=1 when "a" is not equal to "zero". So, the power of anything (here "a") is zero except a=0. Therefore, in this argument the result will be undefined.
I agree that Zero to Zero is 1.
Here is proof. If x ^ 0 = 1 you cannot have 1 = nothing (since there is no x to multiply with). Therefore the form x ^ 0 = 1 is because x ^ 1 = 1 * (no x's). So 0 ^ 0 = 1 * (no zeroes) = 1
As Don Benson put it, whether to define 0^0 is a matter of convenience not correctness. Many calculations get unnecessarily complex if we don't assign a value. And Don Knuth noted the real problem was we were comparing apples and oranges. 0^0 as a value is 1; 0^0 as a limiting form is undetermined.
The strange thing is...I do 0^0 on my windows calculator I get 1. BUT if I do this equation a different way - 0 divided by 0 - my calculator THEN gives me "result is undefined"
Orange is known as Applesin (son of apple) in some languages, both are fruits, both have comparable qualities - very easy to compare.
Given enough intelligence one can compare anything 😁
@MiklosKoncsek, in what sense is 0/0 another way of 0^0?
my calculator said zero
Cool. It doesn't change the answer, but cool.
The reason any number raised to the 0 power is 1 is because
x^m/x^n = x^m-n.
When you divide a number by itself, i.e.,
X^m / x^m that equals
X^m-m or x^0, which is 1 because anything divided by itself is 1.
That doesn't work for 0 though because 0/0 is undefined.
0
You need to put parentheses around your powers when they have multiple terms when using the ^ to indicate exponentiation, since there is no physically visible superscript. That is, for example, w^a-b = (w^a) - b by order of operations. To make the entire a - b be clearly the power, use w^(a-b).
I’ve been helping my daughter with College Algebra. I got it right!
I’ve been using your channel to help me help her. Thank you!
-1
0 squared is 0*0 + 0 to 0 power which is (1)=-1 it can also be undefined -
Why is 0 to the 0 power 1?
Not all calculators are giving the same answer: Android - "undefined or +1" , Google and Windows - "-1"
HP48 … 1
Woohoo! Working on almost 50 here. I loved math in school but I didn’t enjoy exponents. I currently feel like a genius for remembering and had the answer in just a few seconds before opening the video to see if I was right. 😂😂 Thanks for the self confidence boost!
An interesting thing to do is to use a good graphing calculator and graph the following: x^0, 0^x, and x^x and look at the graphed results. x^0 results in a line at 1 as you would expect. 0^x results in a line at 0 when x is greater than 0. x^x results in a very interesting shape.
I dont see that at all
x^x is non elementary
You could make 0^0 equal to any number between 0 and 1 if you try hard enough. For example, turn (0^0)/2 + (0^0)/2 into (\lim_{x\to 0} 0^x)/2 + (\lim_{x\to 0} x^x)/2 = 0/2 + 1/2 = 1/2.
"0^0" is, by definition, not a number. It is a disallowed operation similar to division by 0, so it's use in an equation is disallowed. The equation is the equivalent of saying "What does 0^0 minus blue equal?"
I strongly agree with you
Who decides? You? I don’t think so, unless you have a Field medal in your pocket.
Mathematicians decide.
And mathematicians disagree on this, and they are divided into two groups . One group finds the value 0 more appropriate. Another group thinks it is better to assign the value one. Tao and Bourbaki are in this second group.
But the point is that no one, as far as I know, says that 0^0 is not a number.
@@ZannaZabriskie In my experience, the two sides of the "debate" are 0^0 = 1 and 0^0 is undefined. I don't think anyone seriously suggests 0^0 = 0 should be the definition.
@@MuffinsAPlenty I remember some text with 0^0=0 position, (essentially to maintain the continuity over R+ of the function 0^x) but now I don't feel like finding them again. And, look, I could be wrong! Maybe you’re right and the other more common position is: 0^0 undefined. But in the end I just don’t care at all.
I don’t care, basically because I am not passionate about this diatribe.
Don’t get me wrong: I always read this kind of posts, and I'm always very amused to see these guys get passionate, and argue their positions vehemently, as if they were reciting the gospel.
But the issue itself I am not passionate about at all.
The main reason is that - correct me if I'm wrong - choice is irrelevant. Math doesn’t change whichever choice you make. None of three choices (0,1,undef) leads to an antinomy. None of the three opens up new mathematical worlds.
at most, you will be able to write a summation that starts at zero instead of one plus an additional term, or something like that.
As convenient as defining 0!=1.
But the mathematical castle would certainly not collapse by setting 0!=ndef: you would write less elegant expressions, but nothing would change.
With sympathy
So, out of two HPs and two TIs, all 4 give ERR (undefined). The only calculator I have the does otherwise is the Calculator program that comes with Windows!
At least the Windows calculator receives updates here and then, those calculators were stuck in the old era they were originally made with no updates ever since, so it couldn't keep up with the new defined numbers.
Assume that n is a positive real number. The reason that b^0 = 1 for b not 0 is that 1 = b^n/b^n = b^(n-n) = b^0, so therefore b^0 = 1 for b a nonzero real number. However, if we allow b to equal 0, then we obtain 0/0 = 0^n/0^n = 0^(n-n) = 0^0, so therefore 0^0 = 0/0 which is undefined, ergo 0^0 is undefined.
Why is 0^0 left undefined? Because you can find functions f(x) and g(x) such that f(x) and g(x) both approach 0 but f(x)^g(x) does not approach 1. For example, let f(x) = sin(x) and g(x) = 1/ln(x), where ln(x) denotes the natural logarithm of x. As x approaches 0 from numbers greater than 0, the given functions for f(x) and g(x) both approach 0. However, using techniques from Calculus II, we see that f(x)^g(x) is approaching e, which is approximately 2.718, rather than approaching 1. You can use your graphing calculator to help visualize the fact that, as x approaches 0 from numbers greater than 0, f(x)^g(x) is approaching e in this case.
This is, indeed, the reason why mathematicians un-defined 0^0 in the early 19th century (prior to that point, mathematicians considered 0^0 to be equal to 1).
But this argument _shouldn't_ be seen as convincing, at least when you think of 0^0 as an arithmetic expression, instead of as a limiting form. What I mean by this is: when we say 0^0 is an indeterminate form, we are not talking about the _number_ 0 raised to the power of the _number_ 0. Instead, we are talking about _a function approaching 0_ raised to the power of _a function approaching 0._ To me, an arithmetic expression is when we have operations between _actual numbers,_ and a limiting form is when we replace _functions with their limits_ in an attempt to avoid computing the actual limit directly.
The next thing to note is this: it is perfectly consistent for the _arithmetic expression_ 0^0 to have a value of 1 while simultaneously the _limiting form_ 0^0 is indeterminate. This is because, if we go back to the meaning of limiting forms, 0^0 being an indeterminate limiting form simply means: "knowing f(x)→0 and g(x)→0 is insufficient information to determine the limit of f(x)^g(x)". And one of the key points about limits is that *_the limit of a function can be different from the value of a function._* Said in more symbolic terms, f(a) can be different from lim(x→a) f(x). *_If lim(x→a) f(x) always had to match f(a), then there would be no purpose to limits whatsoever._* But the argument that 0^0 is an indeterminate limiting form hence must be undefined as an arithmetic expression is essentially saying: "If f(a) is defined, then f(a) might not match lim(x→a) f(x), so we cannot have f(a) defined." Do you see the problem with this reasoning? It undermines the entire concept of limits.
But even worse than that, I have never seen a natural example of a discontinuity _caused by_ 0^0 being defined as 1 (except for the situation where we have 0^f(x)). Even the example you gave is not a discontinuity caused by 0^0, but rather is a discontinuity caused by ln(0) being undefined and/or −∞ not being a number.
Letting f(x) = sin(x) and g(x) = 1/ln(x), we do indeed get a limiting form 0^0 for f(x)^g(x) as x approaches 0 from the right. However, this is not the same thing as _plugging in_ x = 0. If we plug in x = 0, we get g(0) = 1/ln(0), which is undefined since ln(0) is undefined. If you _ignore_ the fact that ln(0) is undefined and pretend that ln(0) = −∞, then you still have the issue of g(0) = 1/−∞, which is undefined since −∞ isn't a real or complex number. If you _ignore_ the fact that −∞ isn't a real or complex number and pretend that 1/−∞ = 0, then you can conclude that g(0) = 0. But in order to conclude that g(0) = 0, you have to ignore the fact that ln(0) is undefined and ignore the fact that −∞ isn't a real or complex number.
So the discontinuity of sin(x)^(1/ln(x)) at x = 0 is not the fault of 0^0, but rather the fault of 1/ln(x) being undefined at x = 0. Even if we defined 0^0 as e, sin(0)^(1/ln(0)) would still be undefined, by virtue of ln(0) being undefined. To reiterate, even if we defined 0^0 = e, sin(x)^(1/ln(x)) would still be discontinuous at x = 0, because sin(x)^(1/ln(x)) would _still be undefined at x = 0._ Therefore, we can see that 0^0 is not to blame for the discontinuity in sin(x)^(1/ln(x)) at x = 0, so claiming that sin(x)^(1/ln(x)) is an example of why 0^0 must be undefined as an arithmetic expression is just bad reasoning.
But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@MuffinsAPlenty But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@AlbertTheGamer-gk7sn Please stop spamming this everywhere. You don't even read what you're replying to. You simply see something and decide to spam.
@@MuffinsAPlenty I'm just telling us that we should define the undefined, as that allows us to boost technological growth.
May be it helps to think about where a^0=1 comes!
For example (a^5)/(a^5)=a^(5-5)=a^0=1
Here you have to write
0^n / 0^n = ?
But before applying the formula n-n=0 and therefore 0^0 is 1, you must see that 0^n is 0 and that a division to 0 is undefined! 👍
You mean a division BY 0 is undefined.
@@trwent
Of course! 👍
Another thing: 0! is undefined as the factorial function is defined to be multiplication, and that (-1)! is infinity, and 0! = 0 * infinity... uh, oh...
-1
I suppose one might verify that any number ^0 = 1 by exploring
n^x as x->0 or limit as x ->0
i.e.
x=0.1
=0.01
=0.001
=0.0001
say
5^x
5^0.1=1.175
5^0.01=1.016
5^0.001=1.002
5^0.0001=1.0001✔️✔️✔️
etc....
0^0.1=0
0^0.01=0
0^0.001=0
0^1e-20=0 ???❌️
0.1^0.1=0.794
0.01^0.01=0.955
0.001^0.001=0.993
0.0001^0.0001=0.999✔️✔️✔️
for n=0 revise the limits to
n^x as n->0 and x->0
= 1
But not ^zero or less. Zero to any power
First a word about 3^0, for example. We have 3^4=3^3×3, 3^3=3^2×3 and 3^2=3^1×3. If we want also to get 3^1=3^0×3, we have to take 3^0=1. This a choice, not a demonstration, but it’s often useful. We can’t do the same with 0^0, because to get 0^1=0^0×0, we can chose any value for 0^0. But what can we do? It’s not so easy, because it needs the functions exp et ln.
For any real numbers x et n, with x>0, x^n can be defined by x^n=exp(nln(x). For example 3^2=exp(2ln(3))=9 et 3^2,1=exp(2,1ln(3))=10,045....... (we can’t do this with x0). Then we can observe that 0,1^0,1=0,794.... and 0,01^0,01=0,954.....and 0,001^0,001=0,993......For this reason, it seem’s to be a good idea to take 0^0=1. It’s a choice, often taken and which can be useful, not the result of a demonstration.
0^0 is generally undefined. The most common is to set it to 1 so that x^k evaluates to 1 for x=0 and k=0 in a sum or series. But one may set it to anything suitable for the case where it's used. So all of the three alternatives, as well as any other answer, is actually correct.
0⁰ in my calculator give "ERROR"... as it should.
If you calculate lim Xˣ for x->0⁺, you'll get 1.
But if you calculate lim 0ˣ for x->0⁺, you'll get 0.
This ambiguity leads to the statement, that 0⁰ is undefined, as it depends on the situation.
Yes, it is undefined. A limit is not the same as a value you compute. The limit approaches 1 but it never reaches 1, so you can’t say the value is 1.
"Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@bobh6728 But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@AlbertTheGamer-gk7sn there is philosophy and there is just plain dumb. This is in the latter category. "undefined" in this context simply means, that there is no unique answer for this question, but math requires a unique answer. Therefore it is "undefined", as - in the rules and axioms of math - this is unsolvable.
@@m.h.6470 Well, we humans find new ways to solve problems, as negative numbers, irrational numbers, transcendental numbers, and imaginary numbers are ways to make unsolvable problems solvable.
To get a number to the zero power start with that number to any power and divide it by the same number to the same power. The answer is always 1. e.g. a to the n divided by a to the n = 1 ( how many a to the nth in a to the nth?). The problem is that when applying this to zero we are dividing by 0 which is a no no.
I have a "proof" that 1=2. Guess where the mistake is hidden.
My HP 11C (yes, it's older than you are, and still working fine) evaluates 0^0 as "ERR 0." So, I guess it subscribes to the "undefined" concept.
I've a TI 30. It began to melt(ha ha).
I LOVE my HP 11C. I had one since 1974, lost it around 2002 and thank God found one in pristine condition on E-Bay. I guess it is the RPN, that spoils us. When forced to use any Non-RPN Calculator, it is a disaster. Before I watch further, I'm guessing he will use The Calculus, specifically the Limit of x^x as x->0 to prove 0^0 is undefined
@@coldlogiccrusader365 Define RPN. TYSM
@@robertakerman3570 - Reverse Polish Notation. It’s a way of entering calculations without needing parenthesis. You enter the operands first, then the operator, so instead of (2+4)*(3+5)=, you’d key 2 ENTER 4 + 3 ENTER 5 + *
So you get a partial result 6 when you keyed 2 ENTER 4 +
and a partial result 8 when you keyed 3 ENTER 5 +
and then the final result 48 when you keyed *
@@markgearing Europeans gave Us so much!
There is still discussion between the math scholars if 0^0 = 1. Some say it is simply 0, others say it is 1. I myself think it is 0, since 0^1 also is 0. For all other numbers this is not the case.
For all other numbers, it is not the case that 0^0 = 0 ?
It is indeterminate
@@trwent Don't be silly.
@Kleermaker1000 Well, your statement was unclear. What "other numbers" are there room for in 0^0 = 1?
@@trwent I meant all numbers with the exponent 0.
I still don't understand why 0 to the 0 power isn't 0. If I have 0 apples, no matter how many times I multipy no apples, it still comes out to no apples and you can't make a 3.14 with no apples.. And I don't need a calculater to figure this out. This is why I am 71 years old and never understood math - plus I hated it. Otherwise, I was mostly an A student. Math is just not rational, and in fact, all numbers just seem wholey irrational to me. That is why I choose to bake apples instead of counting them. Also, most of the people I know who do understand this are men. That is also why I don't understand men.......................I probably won't be back - I'm going to go bake an apple pie without counting how many it takes. Thanks anyway. P.S. I decided to come back just to give you a like, a 100%, a smiley face, an A++ and a few stars for trying since you seem to think it all makes sense and who am I to burst your bubble?
Das rührt von einer Definition her, die besagt, dass jede beliebige reelle Zahl in der 0-ten Potenz immer 1 ergibt
It is actually undetermined. 0 to the 0 power could be anything because you are essentially dividing 0 by 0. In essence you asking what number times 0 equals 0, which could be any number.
It's 1 or it can be classed as undefined. It's optional dependant on context.
My answer was -1.
@@Wandjina104LOL...he thinks she wants an answer...you're the 👌
0^0 can be interpreted as the limit of x^x where x -> 0, which can be proved to be 1. This video has missed golden opportunity to even mention, let alone explain, it. But then, again, this guy is well known for talking a lot while saying little …
The value of zero to the power zero is a subject of debate and depends on the context in which it is used. In different areas of mathematics, it can be assigned different values:
1. In combinatorics and set theory: is often defined as 1. This is because there is exactly one way to choose zero elements from a set of zero elements.
When we interpret exponentiation in set theory, it refers to the number of functions that can be defined from a set of size (the exponent) to a set of size (the base).
For example, 3 to the power 2 represents the number of functions from a set of 2 elements (e.g., ) to a set of 3 elements (e.g., ), which is because each element in the domain (2 elements) can map to any of the 3 elements in the codomain (3 elements), i.e., .
Now, consider the case :
If you have two sets, both of size 0 (i.e., the empty set), how many functions can be defined from the empty set to the empty set?
In set theory, there is exactly one function from the empty set to the empty set. This function is called the empty function, which is a vacuously defined function because there are no elements to map. This is a subtle but important point: even though there are no elements in the domain, there is still exactly one way to define a function, and that is to define nothing.
2. In calculus and analysis: it is considered an indeterminate form because defining it could lead to inconsistencies in certain scenarios, such as limits in calculus.
3. In general arithmetic: It is sometimes left undefined due to potential contradictions, especially when both the base and the exponent are zero.
4. In computer science: it is typically defined as 1 to simplify algorithms and calculations.
So, the value depends on the mathematical context and the conventions being used.
The above equation is in simple maths, so should be considered as "undefined".
EXCELLENT! Advanced mathematical theory, presented in a manner that obfuscates & bamboozles! You'd make a great accountant!
CEO: "How much tax do we owe?"
Accountant" "How much do you WANT to owe?":
Zero to the zero power is undefined therefore it’s an illegal operation
Undefined. 0^p=0, where p is positive real. 0^n is undefined, n is negative real. 0^0 is undefined because the limit 0^0+ is 0, and the limit of 0^0- is undefined. 0^x is not continuous at x=0, so 0^0 cannot be determined. 0^2 - 0^0 = 0 - undefined = Undefined.
For any nonzero number a, a^0 = 1. The power 0^0 is undefined.
Zero to the power of zero, denoted by 0⁰, is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines 0⁰ = 1. In mathematical analysis, the expression is sometimes left undefined. -- wikipedia
"Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
Who decide that? You?
@@ZannaZabriskieuse imaginary numbers and you can now approach from four directions and the results will no agree.
Consider 0^5/0^5. Applying the rule ‘when you divide like bases you subtract their exponents’ you would get 0^(5-5) or 0^0. Since division by 0 is undefined 0^0 is also undefined. Not 1.
Training/ practicing for the IBEW aptitude test. your videos have been lots of help.
Good luck on the test!
Best of luck
x^0 comes from x^a/x^a => x^(a-a) -> x^0
BUT!
0^0 would then come from 0^a/0^a -> 0^a/0 -> division by zero -> undefined
5:15 in advanced mathematics 0^0 -can be interpreted as- *is* undefined. Let's emphasize that.
I wonder whether the author of this video conducted any research beyond putting "0 ^ 0" in his calculator.
Any sources? Is there any school book claiming the result is 1?
*EDIT:* My claim was wrong. The result depends on the subject. For example, in discrete mathematics 1 is a sensible result. Thx MuffinsAPlenty.
But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
"in advanced mathematics 0^0 -can be interpreted as- is undefined."
Depends on which subject you're dealing with. In basically all of discrete mathematics, 0^0 = 1 is correct. And this is because the empty product (a product with no factors) is 1. And the reason that the empty product is 1 is because that's the _only possible value_ a product with no factors could have if we expect a product with no factors to abide by the associative property of multiplication. In discrete mathematics, exponentiation represents repeated multiplication: x^n means a product with n factors, all factors being x. So, in discrete mathematics, x^0 means a product with 0 factors, all factors being x. In this context, regardless of the value of x, this is the empty product (the fact that all factors are x is _vacuously_ true, regardless of the value of x), and therefore, must have a value of 1.
So pretty much any discrete setting, 0^0 = 1 is correct. And it actually shows up in formulas too (though you often wouldn't think to _use_ those formulas in those cases, which is why many non-discrete mathematicians don't notice this), and in every formula involving discrete exponentiation where 0^0 shows up, you _only_ get the right answer when 0^0 is evaluated as 1. And this makes sense because _everything_ about discrete exponentiation is based on the associative property of multiplication.
Feel free to look up "empty product", although it belongs to a more general idea of "empty operation" which can be found in things like universal algebra and category theory. I recommend the article "too simple to be simple" on nLab, particularly the subsection on biased definitions.
Now, in analysis things get a bit more sticky, and I'm happy to talk about that if you want, but I wanted to make the discrete case clear first.
@@MuffinsAPlenty Thanks for the detailed and informative reply. To be honest, after wrting the comment I noticed that, for example, the binomial theorem (x + y)^n = x^0 * y^n + ... requires 0^0 = 1 to not break in the trivial case. Funnily enough, this didn't cross my mind during my studies.
So, by now I would agree that 1 is "more correct" than 0 so to speak. What I'm wondering now: are there any areas (analysis) where people agree on 0^0 = 0?
[The answer in this video ("the calculator says so") still feels a bit unsatisfying, but it's understandable he doesn't want to dive into advanced math to provide an example.]
@@michaelmuller5856 "What I'm wondering now: are there any areas (analysis) where people agree on 0^0 = 0?"
Not that I'm aware of. One time someone referred me to the concept of Munchausen numbers (or Münchhausen numbers - there ). A Munchausen number is a number where, if you take all of the digits in the representation of the number and raise them to themselves and then take the sum, you get the number back. So for example, 3435 is a Munchausen number since 3^3 + 4^4 + 3^3 + 5^5 = 3435.
Obviously, Munchausen numbers are _super_ base dependent. Like 3435 is a Munchausen number in base ten but wouldn't be in other bases. *_Some_* people who have done work on Munchausen numbers adopt the convention that 0^0 = 0. For example, if one takes 0^0 = 0, then 438579088 is an additional Munchausen number in base ten. It's been proven that 0*, 1, 3435, and 438579088* are the only Munchausen numbers in base ten (* if one takes 0^0 = 0).
Now, as you can probably guess, working on Munchausen numbers is a very fringe topic in mathematics, partially because it is not much more than a curiosity, it's super base-dependent, and then we get results which tell us that there really are only a few in any given base (so there isn't even much more that can be said about them anyway). But even in this fringe topic, taking 0^0 = 0 *isn't* agreed upon. For example, the person who named them "Munchausen numbers" to begin with says that the only two reasonable approaches are taking 0^0 = 1 or having 0^0 be undefined. (And in some bases, you get more Munchausen numbers when taking 0^0 = 1, as opposed to 0^0 = 0).
But this is pretty much the _only_ situation I have seen where some people say taking 0^0 = 0 is what should be done. It's a fringe topic that seems uninteresting and arbitrary to me, and even those who have done work on this don't agree with that stance.
In terms of analysis, typically analysts just take 0^0 to be undefined, and that's pretty much because of limits and continuity. The two-variable function f(x,y) = x^y cannot be made continuous at (0,0) even if one were to redefine 0^0. So that means some functions which have a limiting form of 0^0 will have a discontinuity associated with 0^0. And since analysts often want to deal with continuous functions, they will often take 0^0 to be undefined so that they don't have to deal with functions discontinuous at some point in their domains. Of course, there are _still_ plenty of situations in which they have to take 0^0 = 1, such as using power series/dealing with analytic functions. But still, I have _never_ seen an analyst suggest that 0^0 = 0 is "correct".
0^0=0 ( actually undefined but keep a chart going and you get help-=1^O-1 Then 0^0 = 1. Teferefore 0- 1 =-1
Loved that one just finished watching a criminal court case. Math and law are pretty similar. Numbers of great mathematicians were also lawyers.
That's because it's all arbitrary, arcane and handed down by others. The essence of authority.
Both sets are embezzlers also 😆
Hi,
Some devices and software, but not all, will give you 0^0==1. My old TI-85 and Realcalc for Android failed. ❌❗
I tried the free version of WolframAlpha, and some functions worked if called a certain way. The output notes warn that some intermediate steps may run through Real and Complex routines that can affect some results (making it hard to tell simplify or solve to work over just the domain of Real numbers).
I did find that the Limit function was consistently correct. And the notes do not list 0^0 as one of the 8 undefined or indeterminate numbers like 0/0.
Anyway, the expression below is as simple as I could make it. One call to Limit passing two limit variables to approach 0: X and Y for Y^2 and X^X expressions. I add so the result will be positive.
0^2 + 0^0 == 1 or
0 + 1 == 1 ✅
Good old limits. ❤ Here's the expression:
lim(x, y)->(0, 0) (x^x)-(y^2)
Regards, Eric
My old button calculator when using 0 X/y 0 it outputs 0. My phone using the same method outputs 1. To me, zero to the power of zero is zero.
Any number including 0 to the power of 0 is 1
To me, it's pointless, since I don't think there is any application for it. IOW, my answer would be "meaningless". It's like, "Where's the center of the surface of a sphere?" Or "What's farther north than the North Pole?" Or "What's infinity divided by 6?"
My answer was -1.
My scientific calculator on my mobile says: underterminate.
If I remember it correctly anything raised to power 0 is defined to be 1, because that conforms to the rules of addition of exponents.
I watched the video just to be reminded of what I learned in math 50 years ago. But it's easy to see why 0^0 has to be defined to 1..
How anyone can think it's undefined baffles me, though. That would create some havoc!
I was taught in the 1970s that 0 ^ 0 is undefined. You can't have nothing raised to the power of nothing, went the teacher's logic.
0^0 was undefined in the 1970s and is still undefined today. It will also be undefined in the year 3970.
@@thomass2451 Except that 0^0 isn't undefined in pretty much any branch of discrete mathematics. When it comes to discrete mathematics, 0^0 = 1, and doing anything else (including calling it "undefined") just makes things unnecessarily complicated. (Depending on your definition of exponentiation, this is even sometimes _provable_ from the definition!)
The decision by mathematicians in the early 19th century to un-define 0^0 (which was previously taken to be equal to 1, including by Leonhard Euler) is being recognized by more and more mathematicians as a mistake based on not properly understanding the concepts of limits and continuity. Because yes, the _only_ reason any mathematicians consider 0^0 undefined today is because of limits/continuity arguments.
@@MuffinsAPlentyOuch... :D
Well … Euler in 1752 defined 0^0 = 1. The man was a lot smarter at Maths than 99.99999999% of population.
@@thomass2451 Bourbaki write 0^0=1 - after 1970. So all Bourbaki-mathematicians thinked that was the appropriate choice. They were not a few, and they were important mathematicians.
Nowadays, Terence Tao in his Analysis write the same thing. Not me: Terence Tao.
In arithmetics 0 means nothing. In mathematics 0 means a very negligible quantity.
lim_(x->0^+) x^x = 1 and lim_(x->0^+) 0^x = 0 are two different ways of looking at this problem. Understand both limits assume positive diminishing x approaching 0. The former limit raises positive diminishing x to x (itself) and gives a result of 1, while the latter raises 0 to diminishing x and gives a result of 0.
This is a conflict. The two interpretations do not match.
Depending on your problem domain, one of the two interpretations will be correct.
That is why C, now has 3 power functions; powr(), which is the former interpretation and is explicit; pow() this also the former and is implied; pown() which is the latter and is explicit.
If you are a C programmer and use pow(), start analysing which version you need and use powr()/pown() where appropriate.
This will disambiguate between the two versions and, make your choice, (and it is a choice,) explicit for maintainers/other users of your code.
As an aside, WolframAlpha explicitly chooses undefined. It does not allow the use of the C pow() function, you must choose powr() or pown() and, use the appropriate limit syntax to tell it what you want to use
I hope this helps.
0^0 is undefined therefore the answer is undefined.
Right
O^2=0 minus 0^0= undefined ,therefore for me the answer should be (0-- undefined)=undefined.
But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@abhimanyubhattacharyya2403 But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@trannhatlong1968 But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
Any number to the 0 exponent is one.
I was average or worse at maths when at school. Now, in my 60s I seem to have improved significantly, having answered -1 within two seconds. However, I have never heard of undef as a result
It takes patience to be good at math, so maybe you have more patience now.
The more common undef is anything divided by zero
Un numero real o complejo al ser dividido por cero, el resultado se convierte en indeterminado o indefinido, como el caso del gringo bestia, que lo complica innecesariamente.
Ok, Ok. What I want to know is which Gray Beard was able to sell this definition? When will he die, and when will the new Gray Beard change the definition? What will A.I. say?
0 to the zeroth power is undefined no matter what, there's no opinions here, it's not humanitarian studies, math is precise . The property of x^0=1 is derived from the property a^m/a^n = a^(m-n), where we put m=n, but it is valid only if a≠0, because it is in the denominator of the LHS
-1 Anything to the Zero power is 1 / undefined. Zero to any power above zero is zero. Since there is a minus in front of the Zero to the Zero power, the answer is -1.
Thanks for helping some of us overcome those feelings of shame, failure and dread we grew up with in math class all those decades ago. Truly a traumatic experience. "Heads, I win; tails, you lose." Nothing ever explained, nothing translated into English. Just a dismissive sneer and a lot of humiliating red ink. A game you couldn't win but had to keep playing. Every once in a while, I gather the courage to try to comprehend this bizarre and opaque system of thought. Why does 0 to the 0 power equal 1? "Because I SAY SO!" I'm hoping someday to understand what a "quadratic equation" is.
(Amateur musings here, happy for anyone to correct me).
I feel a bit funny about saying 0^0 is 1.
I have always thought (maybe wrongly) that x^0 = 1 because it's really x/x, which is 1. Why? Because with negative exponents, x^-1 is x/x/x, x^-2 is x/x/x/x and so on. It's just the 'getting smaller' version of exponents getting bigger - except instead of multiplying itself more and more times, it's dividing itself more and more times.
The problem I have is 0/0 isn't really definable. What's 'nothing' divided by 'nothing'? 1? I mean, does that really make sense? Can you have 'one' nothing? That would imply you could have numbers of nothing, which again, makes no sense at all. If you have a million nothings and I have one nothing, we both still have nothing. It seems better to say it's actually not definable because it doesn't make sense.
Edit to say... well now that I think about it, if x/x = 1, which it should be, why should that still not be true if x=0? Still feels odd, since dividing by zero really makes no sense in the context of division.
This is (I hope) just a case of remembering that Zero to the power Zero is 1
Zero to any other power is Zero.
So, we have Zero squared - Zero to the power Zero 0 - 1 = - 1
EDIT (after watching your video) I still don't quite get how an equation can have an answer
and be 'undefined' as well.
👍
4:59: This is utterly wrong. You can argue that 0⁰ = 1, since x⁰ = 1. But you could also argue that 0⁰ = 0, as 0^x = 0. Due to this arbitrary approach, 0⁰ is undefined, and therefore undefined is the only correct answer of this problem!
This sort of reasoning may seem convincing on its face, but when you think more carefully about it, it becomes unconvincing.
0^x = 0 is only true for _positive_ x, whereas x^0 = 1 is true for all x (except possibly 0). So the "rule" that 0^x = 0 _necessarily_ breaks at x = 0 no matter what we do, since 0^x is undefined for all negative numbers x (0^x is also undefined for the left half of the complex plane, including the imaginary axis). But x^0 = 1 is true for _all_ complex numbers (and even elements of an abstract algebraic structure, too!)
So the 0^x = 0 rule will _always_ break at x = 0, _no matter what we do,_ even if we chose to define 0^0 = 0. This rule would _still_ break at x = 0.
But the rule x^0 = 1 doesn't have to break _at all_ if we take 0^0 = 1. Indeed, any choice other than 0^0 = 1 (including having 0^0 be undefined) causes the rule x^0 = 1 to break at x = 0 and _only_ at x = 0.
@@MuffinsAPlenty
The product of zero zeros is undefined but can equal 1 or zero if mathematically necessary
@@KarlWork-n3i The product of no factors, known as the empty product, is almost universally taken to have a value of 1, because that's the only possible way for the product of no factors to be consistent with the associative property of multiplication.
If one changes the meaning of exponentiation away from being repeated multiplication, then sure, the empty product reasoning breaks down. Nevertheless, even when one isn't using repeated multiplication, 1 is still almost always the "correct" value to assign 0^0 in those contexts.
There is pretty much _never_ a situation where 0^0 should be assigned the value of 0. And this is why the two approaches commonly taken to 0^0 by professional mathematicians are "1 or undefined" (which doesn't include 0).
I said 0^0 is undefined but COULD be given values of zero or 1 if necessary.
I was not implying that it has ever been given value of Zero. But if in the future and if a reason arose for the product of zero zeros to equal zero then that's ok but don't worry I would ask you if it's ok first.
Anyway with limits you can have situation where 0^0 = 0
Limit (F) = Limit (G) = zero as x approaches zero.
Where F(x) = exp(-1/(x^2)
and G(x) = x
Then Limit (F^G) = zero as x approaches zero.
Remember all tho for F, zero not in domain of F, it's Limit (F) still equals zero.
So there are situations where
0^0 = 0
You can’t subtract by something that is undefined.
The RULE is that any base elevated to the zero power = 1. Therefore, it’s NOT undefined because it’s already defined by the rule.
@@awcabot1
False. The rule says that any NONZERO base raised to the zero power is one. A power of zero is not defined for a base of zero.
X⁰ = 1, (X # 0.)
but 0^0 IS defined
I agree with you in principle but since they are defining 0 exponent as 1. Maybe he should have put that in the question.
On the use of the carat or the x to the y button, you forgot to mention that you need to press = after entering the exponent!
I got the answer -1 without any confusion. However, even though I have a degree in mathematics, I was confused when you said that undefined was also correct. 😅 I guess I’d better grab my encyclopedia of mathematics and brush up a little bit. 😊
The answer -1 is incorrect.
@@Nikioko I see that now, but I was still surprised by it. 😳
I jumped to the chase without thinking and said 0, but when I saw your post I thought ... yeah.
I have a degree in mathematics also and 0 to the 0 power is usually considered undefined.
@@eemmeennddeell I believe it now. I’m not sure what I was thinking, but I confused myself. 🤨
If 0^0=1 because a^0=1 for any other a,
then why is 0/0 undefined even though a/a=1 for any a other than 0?
Also, 0^(1/n) = 0 for all n=1,2,3... , converging to 0^0=0.
It may make sense to set 0^0=1 in many instances, but there are exceptions. Thus, undefined.
btw, using the usual exponential law x^(a+b) = x^a * x^b, we could consider:
0^0 = 0^(0+0) = 0^0 * 0^0
thus, 0^0 must be some a that satisfies a=a^2, leading to exactly two solutions, 0^0=0, or 0^0=1.
Re why is 0/0 undefined even though a/a=1 for any a other than 0: this is good question.
Remember that division by a number is the operation that "undoes" multiplication by that number.
For example, 3×2=6, so 6÷2=3.
Written in the opposite order, 6÷2=3, so 3×2=6.
In general, a/b means the number you must multiply by b to get a.
In symbols, a/b=c means that c solves c×b=a.
To work out a/a, we put b=a to get:
a/a=c, where c solves c×a=a.
Now if a≠0, we can divide both sides of the equation by a to get c=1, and we're done.
If a=0, the equation becomes c×0=0, and this is satisfied by any real value c.
So in a sense 0/0 can take any real value.
Because 0/0 doesn't have a definite value, we say that it is undefined.
Just a thought. I know 0^0=1 but I’d love to see the mathematical proof of this. That be interesting.
You could define it as 1 because the limit of a^0 for a towards 0 is 1. Then again, you could define it as 0 because the limit of 0^x for positive x towards 0 is 0.
At the end of the day, 0^0 is just as undefined as 0/0 or 0*infinity.
@@arthur_p_dent At the end of the day it's defined as 1
@@TheloniousCube at the end of the day, this is a matter of convention. You can define it as 1, or not.
@@arthur_p_dent In the context of algebra mathematicians define it as 1
It is a definition, not a proof. As such it is somewhat useful, sort of like i=✔️-1
Had a good chuckle with do not use a calculator. If ya don't know it's -1, not sure a calculator will help.
Undefined to me means that there is no answer or it is irrelevant because it can't be answered so its return is equal to 0. I still see the answer of 0 being valid.
Here's an experiment - take a pencil and do the following division ---- 7 ÷0. See what you get.
Dave needed to add more context to show why 0 isn't an answer. Think of why x^0 is always one. For example, 2^2 divided by itself. Which is 4/4 which is 1. Using power laws, you subtract exponents when dividing, so you get 2^0. But we already know the answer is one, therefore 2^0 has to be equal. When it comes to a 0 base, and you do the same process, you get 0/0, and we know division by 0 can't be done, therefore it cannot be an answer.
Now, if you are approaching your comment from a logic standpoint, with 0 as a return value for anything that isn't true, I can understand your thought. But this isn't a logic problem, and 0 in math isn't a replacement for 'undefined'.
Undefined is basically saying we humans don't have the intelligence to comprehend this. Nature however does it every time a black hole is created. Nature also has no problem with irrational or complex numbers. A perfect circle should be impossible because it is a ratio of its circumference and radius, i. e. it is a product of pi . Pi is irrational though and cannot be described as a ratio. So what gives. We do, and we know it. That's why concepts like undefined exist. We don't have the knowledge to describe it yet. I hope this helps.
@@anthonywarfield7348 You don't give humankind nearly enough credit for our understanding of certain basic mathematical concepts.
Re - Perfect circle. Pi is indeed an irrational number which means, as you point out, it can't be expressed in the form of a ratio of integers. Basically it can't be expressed as a fraction. But the value of Pi is not in any way required to express the equation for a perfect circle. The value of a circle (simplified to locate the center at origin) is x^2 + y^2 = r^2. Pi is not required. You could, however, support an argument that the numerical value of the area of circle (or volume of a sphere) cannot be "perfectly" calculated due the irrational value of Pi.
Re - "Undefined". We should come up with another term for that because it sounds like it means "geez, I dunno and might never know!". We understand the concept, or concepts. There are several ways an expression can result in an "undefined" solution. That's important - numbers can't be "undefined", only expressions can be "undefined".
So what's the problem with division by zero? Let's use an example- 10 ÷ 5 = 2. We can check that by recognizing 2 × 5 = 10. Let's try that with division by zero.
10 ÷ 0 = y therefore y × 0 must equal 10. There is no solution for that expression. There's no number which can be multiplied by zero to equal 10. The answer is "undefined". We fully understand that. There's no mystery.
It doesn't say thr equation is undefined. The andwer is -1
You show this as an algebraic expression. So, the rule is "any non-zero number to the power of zero is 1." Zero to the power of zero, denoted by 0^0, is a mathematical expression that is either defined as "1" or left "undefined," depending on context. In algebra and combinatorics, one typically defines 0^0 = 1...
One reason I was behind in math, two answers to the same question that are totally different
How about taking the square root of +1? There are two answers (+1, -1) that are totally different.
@@thomasw.eggers4303 come again?
How about describing someone who is fat and ugly. Both fat or ugly would be correct.
math has been twisted as they made up new rules
@@pedinurse1 Hmmm. I used a computer in 1961 to plot a graph of x^x as x approached 0 from the positive side. The problem existed then as it still does.
My calculator is showing "0⁰is ambiguous"
-1. Zero to the power of anything except zero is zero. Anything, even zero, to the power of zero, is 1. Zero minus one is -1.
"Anything, even zero, to the power of zero, is 1."
That's wrong.
The answer is UNDEFINED because of the reason for "anything to the zero power equals 1." We know anything divided by itself, except for zero, equals one; anything divided by zero is undefined. So 7^2 / 7^2 equals one - and the rules of exponents tells us 7^2 / 7^2 = 7^(2-2) = 7^0. Therefore 7^0 = 1. One could rewrite 0^0 as, say, 0^7 / 0^7, since by the rules of exponents this would be 0^(7-7) which is equal to 0^0. But we know anything divided by zero is UNDEFINED, so the term 0^7 in the denominator makes that expression UNDEFINED.
I could not believe that a calculator would return 1 for 0^0. This is the same as 0 divided by 0 and you cannot divide by zero. The only proper answer is undefined. So, I put it into my calculator and it gave an error, which is what it gives for any other division by zero.
In what universe is..x-0 the same thing as x÷0??? -1 is absolutely a correct answer.
I just for fun tried 0^0 on the rp calculator on my desktop, and it gave 1, which is just what I thought at first. However, I can also guess that it might be represented by a transfinite number, just as x/0 if I'm not getting totally confused of course.... Anybody wanna step in and enlighten me?
No, 0^0 is not equivalent to 0/0. Who taught you that? I’d like to remove their license.
He ist right:
0^0 = 0^(1-1) = 0^(1)*0^(-1) = 0^1 / 0^1 = 0/0
Division by 0 i not forbidden, just simply UNDEFINED. (As long as you are not using 'limits'.)
The video ist just wrong!!
0^0 is defined as 1
Please have Dr Terrance Howard on the channel. His new discoveries, specifically 1X1=2
how can 0 become -1 when if you have no money and empty pockets than logic says you have nothing
Debt?
totally agree
-1. Y^Y based on limits as Y approaches infinity. 0-1. I am guessing.
-1... Yay!... I'm in the 30%....
This video is somewhat long-winded in getting to the crux of the issue. 0^2 is just a distraction. So the problem is simply what is 0^0? Again it's very easy to summarise this in a few words: x^0 = 1 for all values of x. But also 0^x = 0 for all values of x. So clearly 0^0 is a special case where two correct mathematical equations conflict as the answer could be 0 or 1 and is thus declared as undefined. My MacBook calculator app clearly displays the answer as "Not a number". Case closed - thank you!
This is all wrong from the start. The answer -1 is wrong. 0^0 is undefined. When you say x^0=1 this is true for any x different than 0. Proof: x^2 = x^(2+0) = x^2 * x^0. So if x^2 is non zero divide both sides by x^2 and obtain x^0=1. Now x^2 is different than zero as long as x is different than zero. So we proved that x^0=1 if x different than 0. If x=0 funny things can happen. This will depend on how fast you go close to 0. The result can be 0, infinity or a finite number. Second point: my calculator gives 0^0 as an error. I used the scientific calculator on my iPhone. Thirdly, please do nit encourage people to use the calculator blindly. This is so bad practice. Learn the concepts of mathematics. A calculator is just a machine, your brain can think.
0^0 is defined as 1
But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
The binomial theorem: (a + b)^n=... would collaps for the case where b = 0 if 0^0 wasn't DEFINED to be = 1.❤
PEMDAS is your friend
It seems to me that zero is an easy answer, right?😢 I wish someone could explain logic behind nothing with no power can be something.
exactly
Yeah, try doing it with apples and see how many apples you end up with, zero.
It deals with sets of numbers. If you had 0 apple trees that produce 0 apples this year there is only 1 way you get this result.
0!: Am I a joke to you?
0 as it cannot be undefined or - 1. 0 is the substitute of nothing. It is impossible to have nothing multiples into nothing and subtract nothing. Therefore the answer is nothing or 0. This breaks the law of 0 and it’s definition.
69 y.o. mechanical engineer. -1 all day long.
Ajab khan khattak.What about night sir?
@@ajabkhan9320 It's sleeping.
Yeah I don't fully trust everything with a calculator. Sometimes you have to think about the problem. Undefined mean we don't know what this situation is yet (you don't know what you don't know). We can't prove or disprove so we leave it undefined. This means we cannot say 0^0=1 Thus the answers has to be undefined not (-1).
0 to the power of 0 is undefined
Not necessarily. It has long been taught that any number ‘x’ raised to the power ‘0’ (so x^0) is 1 algebraically. Therefore, when x=0’ 0^0 = 1. Remember, this is just standard convention. There are often special use cases that go against standard convention, which is why PEMDAS does not hold true for algebraic division. In such cases, everything after the division goes under the division dividing everything before the division. It is for this reason that undefined can also be correct under specific circumstances.
@@anwaraisling No. It is undefined, because it leads to the expression 0 / 0.
@@Ivan-fc9tp4fh4d No, it is defined as such. "It leads to..." is not a valid argument
@@TheloniousCube 0 . 0 - 0^0 = 0 - 1 = -1 ?
@@TheloniousCube Zero to the power of zero, denoted by 0^0, is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines 0^0 = 1. In mathematical analysis, the expression is sometimes left undefined. Computer programming languages and software also have differing ways of handling this expression.
And that's why some missions to Mars CAN CRASH ... :) Beacuse of different DEFINITIONS ...
It is always taught anything raised to zero. What is zero? It is 1 devided by infinity. So ( 1÷infinity) raised to zero is 1
Zero to power zero is undefined therefore the answer would be 0
Yah, I'd go w/that.
No.. base on ur logic 0 to power of 0 is undefine then ur ans will be undefine.
For me 0^0 is 1 according to the rule
No, algebraically x^0 is defined as 1. Therefore 0^0 is 1. However, this is not the only convention that algebra varies. Algebra doesn’t follow PEMDAS. When applying division, algebra divides everything that comes before the division by everything that comes after the division. As with a lot of math, it depends. Which is why both 1 and undefined can both be correct leading to -1 in this case.
@@anwaraisling Where do you get the idea that "Algebra doesn't follow PEMDAS"?
@@anwaraisling You have to realize that raising to the zero power means dividing the number by itself. 0/0 is undefined. 10/10 = 1 so 10 raised to the zero power equals 1.
x^x, let x-> 0 from the negative side or from the positive side. You will get closer to -1 and 1.
see also f(y)^f(x)) -- how you approach 0 matters. Some people have argued that 0 does not exist, it is just a very small number, maybe so small that if you square it you would get to nothing. 🙂 or 😲 -- the zero daemon.
The limit as x->0- x^x =0 and from positive side it’s 1. So f(x)=x^x is discontinuous at x=0 and thus undefined.
First comment!!! 🎉
WooooHoooo!
You got it wrong too. The definition of a^b is "take 0 and add a multiplied with itself for b times". So, it's obvious that the result should be 0, not -1, not undefined...
I think it more correct to call it indeterminate than undefined.
0 to the power 0 shall be left undefined. There is a reason why great mathematicians never touched the subject. If the calculator says the answer is 1 there is something wrong with the algorithm it’s using. Some calculators give an error message.
A simple test would be to numerically determine the limit of the function f(x)=x^x as x ->0. Just substitute very small positive and negative values to demonstrate. On positive side the limit can actually be determined analytically using logarithms.
You will realize the limit is 1 as x->0+ and it’s 0 as x->0-
This means the function is discontinuous at x=0 and thus undefined.
X high 0 IS per definition for every natural number 1, afaik. So the result should bei -1. Written before watching the Video.
Correct,real number to the power 0 is 1
I never cease to be amazed that people can take a simple idea and twist it to something complicated. If you take a calculator and substitute a number very close to 1, like x=1.0001 into the expression (x-1)/(x-1) you get a number very close to 1. If by hand you substitute x EXACTLY equal to 1 you get 0/0. Therefore 0/0=1
BUT if you take a calculator and substitute a number very close to 1 into the expression ((x^2)-1)/(x-1) you get a number very close to 2. If by hand you substitute x EXACTLY equal to 1 you get 0/0. Therefore 0/0=2
BUT if you do the same thing with the expression ((x^3)-1)/(x-1) you get 0/0=3
If you do the same thing with the expression ((x-4)-1)/(x-1) you get 0/0=4
If you use the right expression you can get 0/0 to equal any number at all. That is why 0/0 is meaningless.
Those expressions in x if plotted as functions not valid for x = 1 they have discontinuity there, your argument is totally wrong. Please try not to involve division by zero in any reasoning
0^0 is undefined but can equal 1 or zero if mathematically necessary. Nothing complicated about that.
What you are thinking about is the subject of Limits and the idea of continuity. Limit F(x) as x approaches "a" in the domain is NOT F(a)
Sometimes it is and sometimes it's not.
Perhaps I didn't explain well. For a real function to be continuous at a point A in it's domain left and right hand Limits have to be equal and equal to f(A).
For the last graph you mentioned both left and right hand Limits are the same and equal 4 but f(1) does NOT equal 4. So the graph has a discontinuity at x = 1.
What that means is as you walk up the graph approaching f(1) from the left and right you will be approaching 4 but at x = 1 graph undefined and has discontinuity there. Same idea for the other graphs.
So argument does not work.
@@KarlWork-n3i But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
ChatGPT4o says it is UNDEFINED or -1 it depends on the system we choose:
The expression
0
2
−
0
0
0
2
−0
0
is undefined due to the ambiguity around
0
0
0
0
.
Here's the breakdown:
0
2
=
0
0
2
=0, since any non-negative power of 0 results in 0.
0
0
0
0
is indeterminate or undefined in most contexts. Some mathematical conventions define
0
0
=
1
0
0
=1 in specific cases (such as combinatorics), but it generally remains undefined due to the lack of a consistent interpretation across all areas of math.
Therefore,
0
2
−
0
0
0
2
−0
0
is undefined unless you adopt the convention that
0
0
=
1
0
0
=1, in which case:
0
2
−
0
0
=
0
−
1
=
−
1.
0
2
−0
0
=0−1=−1.
So, the result depends on how
0
0
0
0
is treated. If
0
0
=
1
0
0
=1, the answer is
−
1
−1; otherwise, the expression is undefined
my gut instinct was to say the result is zero but then i thought about 0^0 for a bit and ended up at undefined. I couldn't see how an equation with only zeros resolves to a number that is not zero because that just didn't make any sense in my head. So undefined was my answer in the end. I bet calculators simply have the rule that ^0 is always 1 with no exception for 0^0.
You are correct about calculators.
I remember that any number to the 0 power is 1. The question is whether this applies to zero. If so. then answer is -1. Otherwise equation is invalid.
If you set the equation up as a unknown with x being the unknown then the equation is a bit more clearer and algebraic . Substitututing x for the question mark will maintain the equality of the equation. 0^2 - 0^0 = X and solve for X . ✌🏿 Good stuff
"0 to the anything" (0^x; x0) is 0.
"Anything to the 0" (x^0; x0) is 1.
0 to the 0 can't be both 0 and 1, unless it's Schroedinger's Zero, so you're left with an undefined.
Best you can do is take the limit as either base or exponent *approaches* 0.
But we need to define it. "Undefined" is just a code word of saying, "Screw this challenge. I'm turning back". This is very bad as it states that you are fearful and afraid of challenges. This is the exact opposite goal of humanity. Humans are meant to break away from nature using self-awareness, conscience, willpower, and imagination. This is why mankind managed to establish such civilization that sets them apart from all animals. We 21st-century humans must thank our long-gone ancestors by breaking away even more to make them proud. Einstein left in his will saying the first person that uses his theory of relativity to invent time travel must travel back to April 17th, 1955, to make him proud. "Undefined" is basically stating we are not used to those numbers, so let's just don't use them. It all depends on context. If we were living in Minecraft, a world without circles, and all of a sudden, a circle randomly appeared out of the blue, we would call it "undefined", but since in our world, we have polar coordinates, the premium package with the spherical bundle, we are accustomed to seeing circles, and we won't call them "undefined". Also, a long time ago, people worshipped the moon like a god at an "undefined" distance away from us, and they believed the sky's the limit, and everything they see in the night sky are basically pure celestial spheres of light at an "undefined" distance away from us, and the Earth was the point where those "undefined" distances converged to, but we managed to reach the moon and even send space probes outside our solar system, even attempting to reach the end of a universe, making such distances not "undefined" anymore. Finally, infinities are everywhere. Without it, the Big Bang wouldn't have happened, and every time you move, infinities are required to make it happen. Infinities created us, don't disrespect them by calling it "undefined" Divide by 0, spread your wings, learn how to fly, and do the impossible. We need infinities to make our dreams of time travel and superpowers come true.
@@AlbertTheGamer-gk7sn Give it to Spock.
Ok the fact is that zero to the power of zero is sometimes defined as 1 (good arguments see Donald Knuth) to catch some special cases in polynoms or eg the geometric sum formula or the binomic formula and in calcus it is undefined because the limit is not defined see the statement from french math. Cauchy. So sometimes it is practical to define it ans sometimes it is not, it is the decision of the programmer of a cas or calculator program and there is absolutly nothing wrong with it
There is another reason why I say it’s undetermined.
Let’s assume 0^0=1. If it’s defined like that I will perform legal mathematical operations on it to create a contradiction.
0^(1-1)=1
(0^1)(0^(-1)=1
(0^1)/(0^1)=1
0/0=1 and that’s a contradiction. Assuming 0^0=1 all operations I performed are valid but they led to an undefined result. That means the original assumption is incorrect.
Did this in my head and came up with -1. I then checked using my HP35 calculator and got undefined. Now I'm totally confused.
0:02 undefined. 0^0 is basically dividing 0 by 0, which isn’t valid.
How about "Nobody knows?" "Undefined" normally means "Increases without bound," ie, "infinity," though we try to avoid using the "I" word. Then there's "indeterminate." It could have a value, or be undefined, we don't know. Sometimes, it can be determined, like the limit of the ratio of two functions, with the "Hospital Rule" (Calculus instructors are pulling their hair out!). But I can't conceive of WHAT it could POSSIBLY be! Actually, this would make a very good "Zen Koan." 🤗
Zero does not exist in algrebra. Only mean median mode and range does. A lack of something is negative.. i call that a negative state. If you have 0 it means its a mean median mode and range of some quanity. Basically a baseline. Algebra has symbols for that less than, equal too and so on.
I think is because nothing to the power of 2 must be nothing.
But 0 at the power of 0 is regarded as the starting point of a process, not physical objects, therefore 1. (but still thinking…)
At 5:03 you say that the calculator gives 1 if you enter 0 to the power 0
Please try that on the iphone calculator. You will see that your statement is not true. The result depends on the type of calculator you use.
A calculator should give ERROR or something like that for 0^0
The problem with 0^0 is that it may theoretically = 1 but it that does not work all of the time. For example, log(0^0) should = 0*log(0). But, there is no logarithm (exponent) I know of that can be applied to any base that results in 0. In that sense 0^0 is undefined.
The problem is that you're applying a rule in a situation where it doesn't apply. The rule that log(a^b) = b*log(a) is only valid if a > 0. You applied it in the context that a = 0, and therefore, the contradiction you're arriving at follows from applying a "rule" outside of the context where it is valid.
Another example of this faulty reasoning: (-1)^2 can't be equal to 1. If it were, then we would have
0 = log(1) = log((-1)^2) = 2log(-1).
But regardless of any branch cut you may take, log(-1) is never equal to 0 (which it would need to be in order for 0 = 2log(-1) to be true).
Therefore, we can safely conclude (-1)^2 isn't equal to 1, right? Well, no, we can't conclude that. The issue is that I applied a rule outside of the situation in which it is valid.
@@MuffinsAPlenty First of all it is accepted in imaginary math that SWRT(-1)^2 can = -1 by special rules, and -1^2 = 1 of course, however we still say that SQRT(-1) or i or j is undefined,; it simply helps us handle some very important math issues in real life such as reactive power (vector) equaling real power (vector) plus imaginary power (vector) . Still, imaginary power is not actual power though it exists as temporarily stored power. In the case of 0^0 you are trying to apply classical math rules and assume the answer 1 is correct. I would call it undefined, not even imaginary as it has no useful application, because it fails a basic application of math i.e. applying a log conversion.
Contrary to your information you have provided , 0^2 - 0^0 does not equal -1. This expression actually equals 0.
Here's why: 0 raised to any power, including 0 and 2, equals 0. This is because anything multiplied by 0 is 0.
Therefore, 0^2 equals 0 and 0^0 also equals 0. Subtracting two identical numbers (in this case, both 0) results in 0.
So, the correct answer is: 0^2 - 0^0 = 0