But What Is ∞ ^ 0
ฝัง
- เผยแพร่เมื่อ 21 ส.ค. 2024
- In this video, we will discuss the concept of infinity to the power of 0. Contrary to popular belief, this mathematical expression is undefined, not equal to 1. While some may argue that it equals 1, the general consensus is that it cannot be defined due to the nature of infinity and its properties in exponentiation. Join me as we delve into the intricacies of this concept.
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How did u comment 3 days ago
Hey, can you please talk about the logarithm of 0? It always appears as undefined and I don’t get it.
@@krimzon69 private vedio earlier
This is 0, cuz anything * 0 = 0, why? cuz any number, everyone you chose to ^0=0, this will be 0infinites, which is 0.
plz do me a number is not really -
As Vsauce said, infinity is not a number, but a kind of number. Under certain axioms, there are infinitely many infinities.
its a kind of number but not a number, what a stupid statement to make
@@uggupuggu Why?
@@BenjaminMellor
Because being a kind of number means that it's a number.
Perhaps a better statement to make would be that infinities are simply not numbers.
... and more nonsensical statements that arise from only watching a vsauce video without doing the actual proper learning about it
what type of infinity is the amount of the infinities
I think a slightly modified approach that I like would be to define the symbol ∞ to be shorthand for "the limit as x grows to infinity". That fits the informal definition at the end of the video (∞⁰ = lim x⁰ = 1). The indeterminate forms would be more like ∞ ^(1/∞) or something similar with the exponent is some function that includes a limit approaching infinity in it as a parameter.
Actually you should also define 0 to be "the limist as x grows to 0"
@@_simobr 0 doesn’t need to be redefined, it’s already well defined as the cardinality of the empty set.
@@Bodyknock yes, but you said that inf^0 = inf^(1/inf)
and that, formally it isn't true
it is true only if you define 0 as the limit as x approaches 0 of x
@@_simobr no I didn’t say that.
Why are you assuming that the notion(which, by the way, means little to nothing) ∞⁰ has 0 being exact and ∞ being approached to? This is fun and games but this has the same sense in it as 0/0 = 0, because the zero at the top is exact and zero at the bottom is being approached to
This reminds me of Bertrand Russell writing that numbers are actually variables themselves. Infinity is a variable too and how it is to be used simply depends on what it actually stands for.
The calculator: *domain error*
What it actually did:
100E10^100=infinity in Google calculator
2:17 wtf is wrong with that calculator?!
guess what i didnt understand a single thing
Get brilliant! :D
🗿🍷👁️🕳️👁️
I take this figure and expand it. Intuitively, let infinity be a number divided by zero. Now using properties of exponent, 1^0/0^0 is what we get. As you said at the beginning of the video, 1^0 is 1, and you also said 0^0 is undefined. This means that the form whose value we try to find is undefined. We can settle by saying infinity^0 is an indeterminate form. This is my intuition. Please correct me if my intuition needs to be updated.
0^0 is 1, so it would be 1/1, or just 1
0:09 it's also true for negative real numbers as well.
Then there’s 0.
@@gallium-gonzollium We don't talk about 0^0. At least not in this video.
@@NOT_A_ROBOT it's either 1 or infinity, cuz there is one zero in zero, but on second thought, there can be infinite zeros in zero
@@__rogalik__7936 there are x zeros in zero
0^0 is 1
We should have more videos with completely black backgrounds. It looks so cool and futuristic like a hologram on OLED screens.
also saves your eyes (and battery)
You could use any positive number, set its power to 0, and voila! 1. Because infinity is every positive number, then it itself cannot have an exponent. However the infinite things inside of it can. As a result, it is safe to say, it's 1.
If everything you're working with is a set, then whatever set to the power of 0 is 1, because there is only one function that goes from whatever set to the empty set
Isn't it flipped, there's exactly one function from the empty set to any other set, while there is no function from any other set (except empty set) to empty set
@@selayarlaut2746 you're right, I did get that flipped
I really appreciate your marriage of Manim with talking head Brian that uses markers on a mirror or glass. You have your own style and I love it! Keep it up Bri!
This channel just makes videos on my thoughts which like his thoughts.
2:16 that is an unusual-looking calculator, to say the least.
Also, if you take the limit as x approaches infinity at x^(1/x) you get infinity^0, and it results also at this function as 1.
I think my biggest issue with this limit is that even with infinity being plugged for x, 1/infinity is still greater than 0
1:57 or inconsistent
Hey, any number you can write like:
N = 1 × M×M×...×M
And the count of the "M" is the power.
Thats reason why any number in 0 power is 1:
N=1×M repeted 0 times=> N=1
What is inf×1? Thats inf. Thats means that you con write that:
N¹=1×inf; N²=1×inf×inf ect. =>
Inf⁰=1×inf 0 times = 1
x^0 = x^-1 * x^1 = x / x
infinity^0 = infinity / infinity
im not taking about the type of number infinity think of it like this
sum from n=0 to infinity of 1 = infinity. Now lets divide sum from n=0 to infinity of 1 by itself that it essentially infinity / infinity
and if you think about it youre dividing something by itself for every term in the top infinity there is that same term in the bottom infinity so it will always be 1 no matter how far out the sum is written out, even to infinity. The exact same type of infinity divided by itself is 1. If they were different types of infinity it would come out to be either 0 or infinity.
However, when you take x^0 = x / x
x is the only variable in the equation. if x = infinity, the infinities youre dividing are exactly the same, so infinity^0 should be 1.
theres something wrong with that feel free to tell me what it is
00:08 is it not true for all negative real numbers?
you can go other way around:
lim ∞^x ≈ 0
x->0
At first glance, that approaches 1 as well since once x is less than 1 you're taking roots of infinity (the square root, the cube root, the 4th root, etc.). But the square root of infinity is likely undefined so the limit is probably undefined here as well.
1:37 the answer to ∞ to the power of 0 is actually true for that statement because 1/x=1/∞ and then you cross the symbol out and in the brackets and then you will get 1 +1 which equals 2
infinity^0 = e^ln(infinity^0) = e^0*ln(infinity) = e^0*infinity = e^(0/1)*(1/0) = e^(0*1)/(1*0) = e^(0/0) = e^nullity = nullity
I feel like there is a difference between something approaching infinity to the zero and actually infinity to the zero. For me it is sort of a "cheat" to say that something approaches that Expression but also it's probably not possible to just solve it.
I dunno - I am a bit confused but I just don't like how you always take an Expression like that and then talk about limits approaching that. It just doesn't feel right...
Rigorously, infinity only “exists” in the context of limits. It doesn’t really make sense to “plug it in”
(dealing with the real numbers and not some other system where infinity is defined to be a number)
The answer is definitely one
In set theory the k^p between any two sizes k and p (weather they are infinate or finite) is the amount of functions that can be defined from a set the size p to a set the size k. In the case of infinity^0 the problem isn't defind well becuase there are many infinate sizes. However thqt doesn't realy matter becuase no mater what size you put there there will alwaus be one function from the null set ø to an infinate group and that would be the null function.
I think you could just say that Infinity is a count of numbers and not a number itself
That is only true for cardinal infinity Aleph0. Ordinal infinity ω is just the next number after forever.
infinity^0 is just eight^0 but sideways
0 x Pizza = ?
0 pizza for you
nope its sadness
03:00 He looks like he's crying while solving maths 😂
2:54 or L’Hôpital’s rule
how many infinity and 0 relation (Not what you think) videos are going to come next?
Tsk, clickbait title! Probably, most maths enthusiasts do 'think' infinity to the zeroth power is undefined.
[Edited to add: Since I posted this comment, the title has been changed (without acknowledgment, sadly), removing the words _It's not what you think._ ]
If you have learned calculus then no
How is "But what is infinity ^ 0" clickbait
@@osmarfreitas8646
It isn't; the title has been changed; it used to include the words, _It's not what you think._
Thank you for drawing my attention to this unacknowledged change.
@@rogerkearns8094 ohh alright then 👌 i think that part meant that people who aren't math enthusiasts would often get the wrong answer, but yea, the old title still seems weirdly confident that no one who stumbles across the video knows it. You're welcome :)
Seems to me this "anythng to the 0 is 1" is something we learn in 9th grade as a rule, instead of something you compute. So, it's a definition of sort. Therefore you would be correct in saying any super large number, approaching infinity is still 1 to fit the definition. Infinity is a CONCEPT, not a NUMBER, it's pretty easy to grasp that idea.
Divide infinity by itself, and you get one. It’s one
Nah, infinity over infinity is indeterminate
Infinity over infinity is any positive number which when multiplied by infinity is infinity
Inf^0 = (0^-1)^0 = 0^(-1*0) = 0^0
Since 0^0 is defined by BriTheMathGuy as =1, Inf^0 must be as well
# input
s = float("inf")
print(s ** 0)
# output
'1.0'
poshel nahuy
If you saw my comment from the 0/0 video i have my own number system so lets define inf^0
Ω^0
Limit so: lim(x to inf) x^0 = 1
inf^0 = 1
the construction explanations are the circumference line such that the isometric paperwork is the answer
I wonder for how long Brilliant is going to be the sponsor of this guy's videos
So at least with regards to programming, I'd imagine that many pow(x,n) algorithms special case the situation where n is 0, instead immediately returning 1 without even looking at what x is. With such an algorithm, even stuff like pow(❤,0) would = 1
3:10 Looks like what floating point numbers do.
Nullity seems to behave a bit like None in Maybe types in languages like Haskell. Whenever None appears in your result anything further down the chain will also be None 🤔
On our perquusail
i tried out (on Python) infinity to the power of 0, and i got 1
According to c# it is 1, just a fun fact :)
Infinity is like a doorway to a whole other hotel of values. And in that hotel, each door leads to another infinite hotel.
Consider a modified Collatz where s2 = (3*s1 + 1)/2^n where n is that integer that yields a negative integer value for s2. What values of s1 lead to s2 = 1? Well, 1, 5, 21, 85, ... is an infinite list of values that reach s2 = 1. (This is basically (2^(2p) - 1)/3 for all integer p 1 to infinity.)
Now consider what values for s0 that lead to s1 = {1,5,21,85,...}. Each of these yields an infinite list of odd integers, each distinct from each other with no shared elements, AND each of THOSE does the same. It's like playing Plinko or following a binary tree, except rather than 2 options you have infinite options at each step.
So I say infinity isn't even a kind of number. It's another thing entirely.
for positive numbers? from the looks of it negative works too.
unless you mean how saying -1^0 is just -(1^0) so it's -1, but with -1 as the number, (-1)^0 is 1
Lol infinity is a number it’s just not a number on the number line
Number
Definition :
1. an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification. ✅. You can make cauclations with ♾ and I think you can order series or for identification
2. a quantity or amount. Absolutely yes ✅
Verbs
1. amount to (a specified figure or quantity); ✅ basically the same as the last one
Infinity definition
1. a number greater than any assignable quantity or countable number ✅ sure it’s still a value just very big
2. limitless or endless in space, extent, or size; impossible to measure or calculate. ✅ endless has a value
3.
Around 0:09, it's said "In fact, it (n^0 = 1) is true for all positive real numbers as well." Wouldn't this be true for negative numbers as well? And for 0, since 0^0 = 1?
first of all, 0^0 is not defined and of course it's possible for every real number (e.g. -1^0)
its true for any number in which a^b is defined by exp(b*ln(a)):
a^0 = exp(0 * ln(a)) = exp(0) = 1
this includes all complex numbers, real numbers, and pretty much every other number system
@@wyboo2019 So this relies on ln(x) being defined, right? So we can define it analytically over negative numbers (and at all complex inputs, actually)?
@@miraser5187 I believe the convention for 0^0 is 1, based on the limit of f(x) = x^x as it approaches 0. What is your reasoning for it being defined for every real number? Unfortunately, "of course" doesn't cut it for me :)
I love this video so much!!❤❤❤
I think it's 4
Infinity^0 = Infinity/Infinity which is undefined.
To define Infinity^0 we need to define infinity/infinity
If infinity were bound by the laws of regular numbers, then infinity/infinity would be 1, since x/x = 1; however, since infinity is not a number in the traditional sense, the laws of regular numbers don't apply. However, if we use limits to find x/x as x becomes infinitely large, then the answer would still be 1 as x/x still equals 1.
can you tell me some thing about this pls :
∞ + 1 = ∞
1 = ∞ - ∞
1 = 0
Imagine if Ramanujan is alive
this is the easy way to understand it:
∞ is not a 'single' number in mathematics, it references a direction on a coordinate grid or it is a 'unbounded' set of numbers.
so it doesnt always 'mean' the same thing every time you see it!
you can only perform a 'normal' mathematics operation (and an exponent is a 'normal' operation) on a single number!
ergo, "∞ ^0" is an invalid expression.
so attempting a 'normal' mathematical operation on infinity is akin to applying a screwdriver when the tool you really need is a crescent wrench... its not going to work.
1^♾️, 0^♾️, ♾️^0 should all be the nullity
It’s 1
I think
🤔
why always 117?
Well only in calculus is this a hard question, because of weird calculus rules
In arithmetic inf^0 is 1
Because of arithmetic rules
1:51 shouldn’t the top limit be equal to e not 1?
No. x must approach zero, not infinity. I noticed the same thing
It is one
That's 1, still- & '2' easy!
If you think that x^0=1 then tell me 1^0+2^0+3^0+...+inf^0=?
0:01
Why can’t we use the logic from your video about why 0^0 is 1? In that video, you talked about how you can use the concept of multiplying by 1, which is why you get 1 * no zeroes, which gives you 1. In this case, you’d get 1 * no infinities, so you don’t even have to deal with the issue of infinity not being a real number, because it doesn’t exist in this equation. Then, yet again, the answer is 1
Did he forget about the nullity element
Infinity is variable
Ex - 1+2+3....=infinity
Also 3+4+5... =Infinity
But both value differ
Sure don't they both reach infinity never.
@@SunShine-xc6dh they’re still being added by infinite amount of values
Both values are the same, there are such a things as different values of infinity and bigger infinities, but not in your example.
2:18 DOES NOT COMPUTE
0 ^ infinity equals error cannot reset
Math library error
......... Zero to the power infinity equals 0
Which software is he using to animate?
But also what is Ωˆ0 ?
I've watched a bunch of these indeterminate form videos, and one question keeps popping up in my head:
Why, in the illustration that there could be two answers for the limit, did you make the exponent 1/n?
I get the illustration, but the question is why have the exponent 'change'?
Surely
Lim (f(n))^0 is 1 always?
Edit: I watched the rest of the video and my question still stands. Why does the exponent have to be part of the "limit variable"?
The issue is the 'infinity' part, not the zero...?
Zero is a well defined number, so why not just use the zero exponent in the limit definition?
Any number to the zeroth power is one
Nullity is just perpendicular, thats it
Hi there i have seen a video i guess in your where you separated both cases in pair or unpair exponent variables used m and n
I haven't watched the video yet but, my opinion is 1 everything to the power 0 is always one.
Looking at the word god as a fraction brought me here
Clearing things up a bit.
Infinity is not a number, yes it is, but it is meant to symbolized as a *number* that is too large.
Therefore it still is 1 in this case.
Even if you put ∞^∞^∞⁰ its still technically 1 since it counts as a number but not a number (because its still a symbol) at the same time (not actually since it favors into the number side).
Why make it harder to deny it tbh.
Infinity is a number
The definition of a number is a value and infinity has a value therefore infinity is definitely a number
But eww your pfp is 🤮🤮🤢🤮
goddamn it dark sonic pfp guy, read my comment before giving your opinion.
Infinity is a symbol symbolizing that it has limitless number.
It was just turned into number after a lot of people not getting what it was supposed to meant.
In shorter terms, misleading or false information.
Exponent towers are evaluated from top to bottom; ∞^∞^∞⁰ is ∞. ∞⁰ works out to be 1, so the resulting expression is ∞^∞^1. Since x^1 = x, we can simplify to ∞^∞. ∞^∞ is most definitely not 1.
@@SsvbxxYT In terms of something like this, getting to 0 is impossible, therefore the result is 1, but, if it's just theoretical and ignore some of the "physics" involved, then just say it doesn't exist, or 0.
What your saying is right if it's by ^1 since itself is = to itself obviously.
But in terms of what I said above, there were arguments about this one whether 0 or 1 should be, but the answer is null, nothing, doesn't exist, and what don't exist? 0.
Mate. This limit calculation at 3:42 with x^0 being 1 algebraic is NOT correct while taking the limit as x approaches infinity. You cannot simplify in such a way when x is the variable which you need to evaluate the limit of. So that is not a correct calculation. Hope you just fooled us with that one.
it's 1.
It’s 2
infinity ⁰=1
julia> Inf^0
1.0
⬡^2 is not nonsense they are ways to define it
What’s ▢/2 I would say it’s half of a square in my opinion
Let me guess. Is it equal to -1/12 ???
hexagon to the 0, noice
Unfined!!!
How you made this video animationss plz share??
I thought for sure I was gonna get to the end of this video and you'd say it's -1/12
Inf^-1
BEHOLD, INFINITY
infinity^-1 = 0.
@@AlbertTheGamer-gk7sn no not really, its actually more like 0.000...001
Bri, I never talk about, what money is, let alone 'God's Existence!
I haven't seen the video yet, but the answer is 1.
φ(0,0)=ω^0=1
this guy loves 117
Bro what's ∞^∞ ?
Infinity is infinitely more than infinity
∞^∞ = ∞
Hexagon to the zero is obviously 1. C'mon. ;)
real
❤nikki uh
1