I think writing 1+-1 without any brackets is dumb enough on its own This must either be 1-1 or 1+(-1) but 2 operations next to each other just look and sound and are completely wrong
@@wernerviehhauser94 True. Equal to 1+√(2), not 1+(√2). We have two operations, one after another, no brackets whatsoever, and no one bats an eye. Because we see it all the time, as simple as that. There's nothing inherently wrong with writing two operations without brackets to separate them, it's simply a matter of convention. Pretty much arguing the Oxford comma.
notation should be used to increase comprehension, not decrease it. there is a special place in mathematical hell for people who intentionally use ambiguous or confusing notation
Solve 8/2(2+2) set x = (2+2) 8/2(2+2) = 8/2x Here, two cases arise depending on convention. 1. Implicit multiplication 8/2x = 8/(2x) = 4/x Recall x = (2+2) 4/(2+2) = 4/4 = 1 So 8/2(2+2) = 1 2. PEMDAS (left to right) 8/2x = (8/2)x = 4x Recall x = (2+2) 4(2+2) = 16 So 8/2(2+2) = 16 Ultimately, the question is poorly-phrased and so it is ill-posed. Ambiguity arises in part from the fact that multiplication by juxtaposition has higher precedence than division in a number of textbooks. Algebraically, 1/ab is ambiguous and should not be used without parentheses. The horizontal fraction bar eliminates ambiguity and should be favoured when possible.
About the 1+-1^0 problem, the issue here isn't pemdas/bodmas, but whether the - is a negative sign so -1 is treated as the number 1 below zero (in which case the answer is 2), or a sign indicating we should be subtracting (in which case it is 0)
The issue comes from _why_ there's a double operation, in maths you generally want to understand the intent, in this case the + - is completely unnecessary so it's not unreasonable to think that they wrote it like that because they want it to be read as (-1)^0. Just stop being ambiguous, say what you want to say as clearly as possible (clutter reduces readability too, which is why we use square brackets and spacing to clear up confusion)
In my regards "+-" stands for the plus minus sign "±" (since most keybords don't have this sign). So 1+-1 means 1+1 AND 1-1 at the same time, so 1+-1 equals to 2 and also equals to 0. This is used for measuremts and/or tolerances for a range of values like "The steel bar is 100m +- 1m long" means there is a steel bar that has a lenght of of roughly 100m, but the exact value is unknow and could be any value between 99m to 101m (so it could be 99.1 or 99.232465 or 100.0001 or 100.9999 or...)
Pemdas gets sometimes disregarded even in for example published mathbooks in favour of what common sense would say. Insisting that an educated author would use pemdas can often lead you to a wrong result
They remain ambiguous, because the order of operations isn't "correct" universally. It's a convention, and thus is only "correct" when that convention is being held. A popular convention, but just convention nevertheless. There also exist other conventions, of which "negative signs are a feature of the number not multiplication" and "implicit multiplication/multiplication by juxtaposition has higher priority" are two examples. It is through this that there arises the ambiguity. I (and many others, including some rather notable organizations in fields who may come across equations) personally hold "implicit multiplication/multiplication by juxtaposition has higher priority". So no, the equation doesn't have a single definite value if you follow order of operations, it has a single definite value if you follow specifically the PEMDAS/BODMAS/etc. convention for order of operations. But the problem never said it holds this convention, and there exist other conventions. Thus, ambiguous. Dumb, too, sure. But also ambiguous.
(Some textbooks don't have the spacing for truly stacked fraction bars, and while many require the use of parentheses [a/(bc)], many instead hold the implicit multiplication priority convention [a/bc, with both b and c being on the bottom of the fraction]. Many calculators are also programmed with the implicit multiplication priority rule. I'd find it hard to say either of these have 'done the math wrong', they've just done it with a different convention. Notably, one that is likely spelled out in the style guides for the publisher or the manual of the calculator, unlike a random, ambiguous problem present online with no statement of convention)
indeed, I seem to recall at least programming languages prefix minus binds higher than any binary operator, and so -1^2 would be 1, not -1, because the prefix minus operator binds higher than the binary exponentiation operator. And then there are others where the leading minus (or plus) sign is consumed by the parser for numbers and so it's not even seen as an operator in the language. This latter approach is very much akin to "negative numbers are a feature of the number". And there are also programming languages that are rigidly left to right (Smalltalk) or right to left (APL) without any hierarchy of precedence. PEMDAS is a choice, not a law of the universe.
But really, plus and minus and multiplication and all the signs and symbols our are inventions, so we should be defining how they are structured. Nobody would argue 1+2*3 is ambiguous. Maybe 'adding' or 'subtracting' isn't our invention, but then we would be saying 'The sum of one and the product of two and three' for 1+2*3 and 'The product of the sum of one plus two, and three' for (1+2)*3
@@TimothyRE99 At least for simple expressions, I have always seen the inline division as either "num/(den)[rest of expression]" with denominator clearly delimited by parenthesis, or "[rest of expression]num/den" without parenthesis around denominator; this meaning den is whatever between the slash and the end of the statement. eg "1=4/4" so "1=4/(2+2)" can simplify as "1=4/2+2" although this last step is kind of not ok, still feels obvious with literals, n/a+b feels like "quotient of n divided by sum of a and b, and f*k PEMDAS, you would obviously write b+n/a otherwise" whereas "4=2+2" so "4=2+4/2" or "4=2+4/(2)" and if you really want the +2 at the end "4=4/(2)+2" or "4=(4/2)+2", the last one being the good "obviously correct" one, the other feels OKish, even if the parenthesis are technically useless, they semantically mark the difference with "1=4/2+2" above The good rule being "avoid inline division as much as possible, and if really mandatory, prefer more () rather than too few "
I think even BODMAS/BIDMAS/PEMDAS has been taught differently to different people depending on the era they went to secondary school and the country they went to school, so I'd argue that even 'correctly applying BODMAS/BIDMAS/PEMDAS' has more than one possible answer depending on what you were taught was the correct way to apply it. Basically people are taught what convention their specific exam board wants/wanted them to follow to resolve any ambiguity they might encounter in their exams, so that all students doing the same exam will reach the same answer, making the exam easier to mark than if there are technically multiple correct answers. That seems to be the only place where it even makes halfway sense to have something like BIDMAS/BODMAS/PEMDAS. Any maths, physics or other use of an equation in the real word should never allow any ambiguity to exist in the first place, or even if it does the person in question will know where the equation they are looking at actually came from and what was actually meant by it and therefore won't need an arbitrary rule of thumb to resolve anything.
@@vampire_catgirlA fraction is an expression, division is an operation. Yes, they are not the same, technically speaking. It is possible in different contexts to have fractions without the notion of division and to have division without fractions¯\_(ツ)_/¯
Genuine question: -1 is a value on the number line. It feels weird to say that you have to put parenthesis around it to make it unambiguous. Doesn't making this an order of operations issue imply that you cant actually represent negative numbers without an operation?
Right? At the end of the day it's always just conventions that drive the discussion. And it just shows how we as humans, no matter how hard we try, always have to deal with ambiguity. There is never a fundamental truth, it's just "Let's try to convince a bunch of people to do it like I do" So yes I would say you can represent "negative one" without an operation. And others might say "no you can't". And now again there is no right or wrong answer. It's a decision.
The issue is that the minus sign is both a unary and a binary operation. So context is needed to properly interpret it, and parentheses can provide this context.
After the thousands of hours of algebra I've done, I am firmly in the 8/[2*4] camp, with a formal objection to linear mathematical expressions longer than three arguments. Five for equations.
After also thousands of hours of algebra, my opinion is clear, if you're going to use linear mathematical expressions, they MUST be interpreted as such, that is evaluated with the "PEMDAS" (is there another name, please?) order of operations, FROM LEFT TO RIGHT. 8/2*4 = 16. Hell, even 2/2π = π, BECAUSE YOU CAN ALWAYS USE PARENTHESES, OR EVEN BETTER, NOT WRITE A STRING OF CHARACTERS! You wouldn't write "2/2π" to a computer and expect the result to be 1/π, would you? I thought so.
@DeJay7 your premise, the one you lifted from my comment, doesn't line up with your argument. How do you get from the basis of having a deeply ingrained habit of using expressions like 3(2+5) as a single term to the defense of an impractical, uncommonly used notation that sometimes includes similar expressions?
@@FScott-m1n Who told you I have a deeply ingrained habit of using expressions as such? I've never been forced to use a specific style of mathematical expression, I just tried my best to be accurate and not "lazy", it doesn't take much to write 8/[2(2+2)]. Also, would you say a/b*c is (a/b)*c (1) or a/(b*c) (2)? You don't have to answer, I don't really care about continuing this, but just think about. And if your answer is (1), then why is implicit multiplication different? It's very useful to differentiate the dot product, but why the higher priority? Whatever, you do you, obviously.
@DeJay7 you did when you lifted my premise and used it as your own premise. The fact that you don't know what you did (not to mention the rest of your second reply) pretty much makes my case -- that the premise you used does not fit with the rest of your claim. You should pay closer attention to what is being said, both by me, and yourself. Your carelessness has got you chasing your own tail.
@@FScott-m1n Okay, NOW I understood what you meant by "I am firmly in the 8/[2*4] camp". I first thought that you treat "8/2(4)" as such, for which I replied, and only NOW, after thinking about it for a little while, do I realise you mean that you USE this kind of expression. I guess we agree, then, sorry. Just a misunderstanding from my part, although your statement is ambiguous, just not as much as "8/2(4)" ...
people that argue for implied multiplication bring up examples such as 8 / 4π = 2 / π instead of 2π or 8 / 4x = 2 / x instead of 2x both which are reasonable, but that's why it's important to not be lazy and just add parentheses when typing comments
Why should it be 4 divided by the juxtaposition of 2 and x and not the juxtaposition of 4/2 and x ? Because of this ambiguity I would honestly stop using it.
Can write 2/4 π and 2/x π, use spacing if don't wanna use parantheses. Why link 4π and xπ if not included in the denominator... Many ways to avoid ambiguity, laziness is not one of them 😂
i stand in the 8 ÷ 2(2 + 2) = 1 camp simply because 8 ÷ 2x ≠ 4x , if i had to pick between 16 and 1. the best choice was ofc tell the author to never be an author again
Variables are literally treated different when compared to constants. I see where the comparison lies if you could just simply replace x with 2+2, but the reality of it isnt that simple. The term 8 ÷ 4x is a completely reduced form while 8÷2(2+2) is deductible.
@@tim3line no they are not in this case, we can let x =2+2 you HAVE to then say 2x=2(2+2) so if you’re saying 8/2(2+2) does not equal 8/2x for some x, then you also have to say 8/2(2+2) does not equal 8/2(2+2) which isn’t the case
@@joshii4537 The thing is that 8 ÷ 2x could also be seen as 8 ÷ 2 * x which in that case 4x is a possible answer. This is just a matter of perspective, no right or wrong answer.
@@danielandycandrawira6206 yes ur correct, but I think a majority of perspectives change when you change it to x. The person above me admitted that if it was x then yes it would be 8/(2x)
I can be on the side of the people who want implied multiplication to be its own grouping notation, but that's not necessarily the standard; but it's only in poorly written expressions where the disagreement is really relevant. I like the idea of juxtaposition taking priority.
The Expression 1 + -1 Is not valid, you cannot have two operands after each other. It would either be 1 + (- 1) or 1+ (-1) * (1) And then , when you add the Exponent, it becomes clear, what the actual Result is
I put the first expression into a TI-scientific calculator, and it shows that you are correct. So, the question is not about ambiguity. The problem is wrong, because it cannot be done at all. My other replies in other threads need adjustment relative to this one.
It's NOT a multiplication, though, that is being applied between the negative sign and the 1. Because in the expression " 1 + -1 " the "-" has to (in my opinion), be interpreted as the unary operation that negates a number. And now you have to decide whether this operation is before or after exponentiation in order. And that's just not clear. Personally, the unary minus operation should be done before exponentiation. If that's the case, 1 + -1^0 = 1 + (-1)^0 = 2. And this "ambiguity" is only because someone wrote "1 + -1^0" instead of "1 + (-1)^0" (IF they wanted this interpretation, which is equal to 2), OR "1 - 1^0" (IF they wanted the other interpretation, which is equal to 0), where both expressions are unambiguous.
I would even say "-1" as a whole can be seen as one glyph. It's just the name of a number. So the "-" isn't even an unary operation. To distinguish one could write -(1) when we mean the unary operation . Just as we write -(-1) = 1.
@@Simchen Yeah, you're right. Although we don't know from the expression alone if "-1" is the number -1 or if it is the unary operation minus the number 1. But aside from that, THIS is practically the reason I said that this unary operation should be done before exponentiation.
I disagree with your dogmatic adherence to PEMDAS. As I'm sure you know, it's a mnemonic device intended to summarize mathematical convention. But all useful summaries lose information from the original, and all useful mnemonics rely on an existing understanding of the conventions. They are an aid for information retrieval, they do not serve as the information itself. There are plenty of examples of PEMDAS falling short and failing to explain convention. For an easy example, why do you insist that multiplication and division happen simultaneously, from left to right? Or the same for addition and subtraction? That's certainly not what PEMDAS suggests. PEMDAS suggests that all multiplication should be evaluated before any division. Why is it okay to ignore or selectively interpret some parts of PEMDAS while insisting on others? Here's another example: 3 ^ 3 ^ 3. What does that equal? PEMDAS suggests that the same operator should ALWAYS be evaluated from left to right, right? I think you would struggle to find any calculator that follows PEMDAS in that situation. Why is it okay to ignore or selectively interpret some parts of PEMDAS while insisting on others? Other great examples have been given in other comments, especially those about how π is treated as a pseudo-variable during operations, despite being a known constant. Your interpretation of the first problem of this video relies on a bizarre notion that negative quantities simply don't exist on their own (except negative one, but only when it's convenient for you), but always exist as negative one **multiplied** by a positive quantity (which do get to inherently exist for some reason, unlike negative quantities). That's no different than if I were to suggest that the number four doesn't exist on its own, it is just four ones **added** together. And then I could posit that since addition is the "last" operation that gets evaluated in PEMDAS, then any other operations clearly only apply to the first or last repetition of the addition! Let's see that mistake in action: 1 + -4^0 (the first problem of the video, but with the quantity in question replaced with a four) = 1 + -1 + 1 + 1 + 1^0 (my devil's advocate argument that a four doesn't exist and is simply adding one four times. Obviously bullshit since I didn't include wrapper parentheses during the expansion) = 0 + 1 + 1 + 1 = 3 (only now do I apply the strict adherence to PEMDAS, giving the wrong answer) Now let's see what you did in the first problem: 1 + -1^0 = 1 + -1 * 1^0 (hmmm I think I see some similar bullshit here too. Do you?) = 1 + -1 = 0 (giving the wrong answer, that you claimed was "technically" correct) Hey, wait a second! Why did the first negative one have to be expanded into a multiplication of negative one and positive one, but the resulting negative one gets to just exist without needing its own expansion??? You're right, it is dumb and there is no ambiguity here: PEMDAS should not be brainlessly treated as gospel. Unary operations/NEGATIVE QUANTITIES clearly have higher precedence than exponents. I would guess that is an intuition/understanding of convention that is held by a large majority of mathematicians. When I hear someone trying to argue that PEMDAS is unquestionable and infallible, I hear a child explaining to physicists that there are three states of matter because that's what their teacher said.
Can't read and respond to all of this, but a couple points. 1) You say you disagree with my dogmatic adherence to PEMDAS, and then say how I don't adhere to PEMDAS. 2) 3^3^3 is evaluated without issue by following the order of operations, PEMDAS says nothing about operations always being performed from left to right. It is mentioned one time in this video as something to refer to to settle ambiguous expressions in the case of equal priority operations. 3) Negative quantities don't clearly have higher precedence than exponents, that's why so many people disagree about -1^0, because it is not clear. The convention is to carry out the exponent first, which is why I referred to that as technically correct.
@@WrathofMath 1) You're right, I changed my argument as I was thinking and writing about it, and I should have updated the opening statement 2) Exponents are equal priority with other exponents, no? You're trying to add additional context based on your understanding of convention or based on additional explanations and context that were given to you outside of PEMDAS. That is not actually offered by the mnemonic. In fact, the whole "order" of equal priority operations is never mentioned at all within PEMDAS. It only gives a (slightly misleading) hierarchy of _some_ operations. That is exactly the problem with it. It is both wrong and incomplete without further context. 3) You say it is not clear, but your title says it's not ambiguous. Interestingly, your comment says the convention is to carry out of the exponent first, which I exactly agree with, but your video doesn't seem to. -1^0 = 1. It only becomes a problem if you insist on treating -1 as a multiplication rather than its own quantity. That's the core of my beef. I have never heard any justification for it, and it seems like a thoughtless retrofit to squeeze negation into PEMDAS to quiet students who notice that (-) is also an operator and negative numbers don't get their own special digits separate from positive numbers (but π _is_ special enough to get its own character! Poor negative numbers). "-4" may be mathematically equivalent to "4 * (-1)", but they are different representations and moving between them requires some thought, like adding parentheses if you're going to be introducing a new multiplication operation. Just like moving from a representation of "4" to a representation of "1 + 1 + 1 + 1". Mathematically equivalent, but new operations were introduced. I do actually agree that it is unclear, and that probably wasn't my best opportunity to turn a phrase. The whole point of my comment is that PEMDAS does indeed create ambiguity. When reading math, it is important to know and default to real conventions actually used by mathematicians rather than what your elementary school teacher taught you. When writing math,... just add your dang parentheses!
the problem often isn't the mathematics, it's the interpretation / evaluation of the problem. there is also the problem of around 27 other rules of mathematics that isn't included in the mnemonic like PEMDAS for instance in the 8/2(2+2) that is just a single division, granted it is a complex division, but none the less is is just a division 8 being the numerator and 2(2+2) being the denominator this problem isn't 8/2 * 4 this is 8/2(4) which would be the same as 8 / (2 * 4) this is because the vinculum is a grouping symbol, this is why a/bc = a/(b*c) and not (a/b)*c for the Americans a/b*c it isn't that implied multiplications comes before division, it is that you need to resolve the implied multiplication as part of the denominator before you can do the division.
This reminds me of something I dislike about exponents, which is that without saying the word “parentheses” a bunch, it is ambiguous how the expression is when said verbally. The same goes for fractions. For example, saying “one plus two squared over four” has many different interpretations. (1+2^2)/4, ((1+2)^2)/4, 1+2^(2/4), ect…
For sure, when I speak of these things I try to make my intentions very clear by tone and space between words. For the first expression for example I would say "One.....plus two squared....all - over - 4." It is quite tricky!
An other solution is to read your " (1), (2), (3), etc.." examples as (1) The quotient of the sum of one and the (second power)(square) of two divided by four (2) The quotient of the (second power)(square) of the sum of one and two divided by four (3) The sum of one and the exponentiation of two by the quotient of two divided by four This removes all ambiguity, even if I agree it is kind of annoying to talk this much for a simple expressions like that. (also note the requirement to use the exponentiation operation for non integer "powers". Some people who would have to compute (3) don't even know that operation exist)
I think the issue with these are really whether or not you consider multiplication by juxtaposition, unary positive and unary negative their own operations and if so what order convention you are using.
but it's wrong. Juxtaposition takes precedence over an explicit sign, and it always has. Your third grade math teacher is not a mathematician and PEMDAS is an incorrect oversimplification. You wouldn't say "8÷2x = 4x" -- you know it's 4/x because of implied multiplication. This doesn't change just because you substituted x=4.
@@zacknattackExcept in practice people don’t. I don’t remember the source but I remember someone went through published math papers and did in fact find things like 2/xy meaning 2/(xy), but never (or much more rarely) found 2/xy meaning 2y/x.
You seem very biased against PEMDAS. It looks like you have been taught the other one. For me, I HAVE learnt both, and to avoid confusion, we always use the fraction sign. If not, we use parenthesis, and if either is not applied, we assume it's wrong, and the author gets a lot of attack hate. Furthermore, when you download a calculator app on your phone, it asks, "What do you mean by the division sign and the percentage sign." all of which to avoid confusion.
@ it is intuitive for formulas, the density of dots in a rectangle is D ➗ LW, (dots, length, width) everyone interprets this as LW being a single compound factor term, obviously it’s better to do the fraction bar, but I think in situations with implied multiply it makes sense to think of each implied multiply as a factor of a single object
@@widmo206 in mathematical formulas it's used that way, also we can distribute the product, and also when doing equations you don't separate the x from it's number like this video would imply
Also I think it makes a lot of sense when you think about stuff smooshed together with implied multiplication as factors, because that explains why you can still exponentiate individual terms of it, just like prime factors. A bunch of things next to each other should be treated like the factors of a single term
When I was in school none of the teachers taught us that "-1" is to be _evaluated_ as -1 * 1. Rather, it was treated as simply a negative number. To give an example of how that's handled in manipulating expressions, if you wanted to take 2 - 4² and convert it to addition it would be 2 + -(4²). The parentheses are added because the 4² was given with evaluative priority - the 4² needs to become negative, not the 4 before squaring. Under that consideration, it's not contrary to PEMDAS to apply the exponent to the negative number in -1⁰ because it's not viewed as negative one times one to the zeroth power, but simply negative one to the zeroth power. To the other one, if you are given 8÷2x, where x=2+2, what's the answer? The 2 is a coefficient of x, and that's taken as grouping. I think it a bit silly to not be consistent with that just because you replace the variable with a literal. So it's _not_ unambiguous just because you have an obvious PEMDAS evaluation, which is why you are _correct_ that the thing to do is not write expressions like -1⁰ or 8÷2(2+2).
You are not correct to say that the American order of operations is universally agreed and therefore 16 is the "technically correct" answer. In the UK the correct answer to 8/2(2+2) is 1 not 16. UK calculators evaluate it this way and the UK GCSE exams require evaluation in that way.
It is the universally agreed upon way, though. In programming. People don't write programs with implied multiplication. Also people don't write "8/2(2+2)".
@@DeJay7 Yes, I am coming to realise more and more that the obsession with 8/2(2+2) = 16 comes from people who have done some programming and can't seem to understand that computers and humans don't think the same way.
@@martind2520 Computers and humans do NOT think the same way. So why do you support a style of writing that only makes sense to be used for computers? It's not that people "don't understand the difference", it's that the difference is not accounted for in the first place.
Certainly a big part of this is the fact that our students need to be able to communicate with computers, and so it is important to teach them the mathematical manners to be able to do so.
There is no such thing as US and UK order of operations. If there was then the UK would not be able to build a car, or bridge. This is an agreed upon process that has been accepted for a very long time.
If you treat the expression 6×2÷3 as a sequence of operations and do the division first, then multiply, you will indeed get the same result, 44. Here's the breakdown of your approach: You can first divide 2÷3 which gives 0.666…(or 0.67 if rounded to two decimal places). Then multiply that result by 6: 6×0.67≈4 So, mathematically speaking, doing the division first in this particular case does give the same result. This is because multiplication and division are indeed commutative at the same level in the order of operations - they can be performed in either direction when they're combined in a linear expression. Does this thing always work and if it does can you say that division comes before multiplication?
If you follow the order of operations, it's obvious. When you have 1 + -1⁰, the -1⁰ is treated as (-1)*1⁰, or (0-1⁰). The negative sign applies to the _entire_ of the power, not just the base, simply because "exponentiation" is before "negation", in which "negation" can either be thought of as "multiplied by -1", or "subtracted from 0", either of which, are done _after_ exponentiation. Alternatively, if you remember that x + -y = x - y, you can also see it as 1+-1⁰ = 1-1⁰ Therefore: 1+-1⁰ = 1-1⁰ = 1-1 = 0 1+-1⁰ = 1+(0-1⁰) = 1+(-1) = 1-1 = 0 1+-1⁰ = 1+(-1*1⁰) = 1+(-1*1) = 1+(-1) = 1-1 = 0
I strongly disagree, 1 + - 1^0 is obviously 2 for the following reason: there's two things that could have been meant by that, either 1 + (- 1)^0 or 1 + - (1^0). but it cannot be the second, because *no one* ever writes 4 + - 2. everyone who would wanna write that would instinctively write 4 + (- 2), because otherwise you might confuse it with 4 plus/minus 2. so it has to be the first option also as others have pointed out, for me 8/2(2+2) is obviously 8/(2(2+2)) because (8/2)(2+2) would be written as 8/2*(2+2). although yes, I agree this one is sorta ambiguous and should be written differently if possible math is a language and just as in any other language: if most people aren't conforming to your grammar rule, then the rule is wrong not the people
Well, isn't the notation for the square of the logarithm of the sum of x and 2 "log²(x+2)" ? And "log(x+2)²" the logarithm of the square of the sum of x and two. The same goes for functions. If you want the square of the function f at x, note f²(x). Although here the value of the function at the square of x, f(x²) is not ambiguous, it is with other common functions, such as trigonometric ones: sin²x vs sinx². Because of their heavy repetition, we usually omit the parenthesis, and put the exponent on either the argument or the function name depending on what we square, instead of requiring two layers of parenthesis like (sin(x))² Honestly, if it were not for the power notation x^2 instead of x², I would not see why you were confused by this problem ^^ On another hand, just changing the power notation breaks my habits, and I no longer found the notation easy to understand... Oh habits !!
I think that the division symbol is just a relic of the time when you understand the algebraic operations as 5(exp, +, *, -, /) separate operations instead of the 3(exp, +, *) that are actually there. in defense of the implied multiplication people, in the mind of someone who sees it as only 3 operations, then you should also expect communativity with division within the divisor since it is multiplication. if you're used to fraction bars, it's easy to think that division has implied parentheses around both the divisor and dividend. for example in that 8 / 2(2+2) example, multiplication is communative and therefore you should be able to write 8/ 4 * 2 which should give you 4.
The arguments in the comments about replacing (2+2) with x is a super understandable misunderstanding but is a false equivalency. 8 ÷ 2x is a reduced term 8 ÷ 2(2+2) is a deductible constant. In the same vein we can't equate 8 ÷ 2x to 8 ÷ 2y. Furthermore unknown variables are treated different because they're unknown. But the term within the parentheses is already known.
No it is not a false equivalency at all. The whole point of algebra is that you can replace the letters with any expression in brackets. If x = 2+2 then 8 ÷ 2x = 8 ÷ 2(2+2)
The second interpretation treats negative numbers as second class citizens of the number world. You can never write them without the implied multiplication. Negative numbers should be a separate case in order of operations.
i dont really agree that "-1" inherently means multiplication is involved. to me it's just as much a value as a positive number itself is, using a symbol like - to denote that is just a limitation of how we express that.
When I see that my mind automatically sees it as 1+(-1)^0, it is only ambiguous because it is poorly written. That is like writing "they told them that they were" does the "they were" apply to the they or the them? it could be either because it is a badly written sentence.
Is the first example really ambiguous? The "+" is a binary operator and needs two operands. That leaves (1)+(-1)^0 as the only option with "-" being the unary operator and "-1" being the negative unit.
The reasoning for 1+-1^0 = 0 is just stupid. The same reasoning could be argued for 25+-5^2 = 0, which everyone knows is wrong, as any number squared would return a positive number, otherwise it’s an imaginary number. I don’t agree with the reasoning also because if a negative is explicitly put instead of subtraction, then you usually would include it in the exponentiation (in the parentheses, like (-1), or (-2)).
in the mnemonic, it looks like multiplication has higher precedence than division, and addition has higher precedence than subtraction, which is yet another source of confusion. thats just an artifact of the mnemonic itself though, not the actual standard. where do radicals fit into this? or even fraction bars? they have implied parentheses just like the "implied multiplication" covered in the video, but nobody disagrees about them. i think its only an issue when the entire expression is on one line, like a text comment or a calculator or a computer code. as soon as we're allowed to move numbers around vertically all the ambiguity goes away it seems like
One thing that’s really interesting is the mnemonic in a lot of European countries is “BODMAS” in which division now comes before multiplication, so that’s one of the ways I remember multiplication and division are on the same priority, especially since they are basically the same thing. With radicals, those will have the same priority as exponents. For example, a square root can easily be written as the exponent 1/2. I really like your comment on fractions bars by the way, very insightful!
What makes radicals and fraction bars unambiguous generally is that they have very clear visual grouping communication, with the exception of when written in plaintext on one like. Of course if I'm writing a fraction vertically there will never be confusion, but suddenly 4/x+2 requires me to be careful and use parentheses if I am communicating with a computer or otherwise in plaintext. When the fraction is written in one line, or with the elementary division symbol which forces one line, the expressions are just often unclear visually, and so the disagreements abound. Implied multiplication is not as strong of a visual grouping symbol. Square roots require us to actually put things in a radical, implied multiplication only looks grouped because stuff is close together. So is it grouped? Maybe. Maybe not
The second one remains ambiguous, though keeping in mind that multiplication by juxtaposition takes priority over explicit multiplication and division is useful. But the first one? You're just straight-up wrong. Order of operations doesn't apply here. There is no multiplication, implicit or otherwise. There is only addition of the exponent of a negative number. The number is -1, and it is raised to the zeroth power before the result is added to 1. No ambiguity.
No, your post is *wrong!* Just to mention the second problem, there are no grouping symbols around -1, so it is only 1 raised to 0. 1 + - 1^0 = 1 + - 1 = 1 - 1 = 0
@@robertveith6383 you can't have two separate symbols next to each other. 4 × ÷ 2 is a nonsensical equation. Likewise, 1 + -1^0 makes no sense if you try to interpret the - as a symbol in its own right. Mathematical symbols are almost entirely infix operators taking two numbers as operands. The only way to interpret "+ -1" is as "plus negative one", and thus + -1^0 *must* mean + 1.
This video seems to want to interpret a minus as some unary operation that is equivalent to multiplying by negative 1. You now have one of two problems: 1. you've gotten yourself into an infinite loop -1 = (-1)*1 = ((-1)*1)*1 = ... 2. the minus operation symbol (-) has two different priorities in PEMDAS. In one place it is S and in the other it is M. The problem is that we use the same symbol to represent subtraction and negative numbers. Perhaps negative numbers should be written in red and non-negative in black. Or use a different symbol as done in some programming languages (1+ ⁻1). Since the minus sign is used for both subtraction and negative numbers, I prefer to interpret the minus sign following some other operator as meaning the following quantity is a negative number. Of course you could say that a negation operation (N) should have its own priority. And then a discussion could follow whether N is higher priority that E or N is the same priority as MD (PNEMDAS or PENMDAS) That said, PEMDAS is just a convenient rule to compute expressions with a mix of different operations. Videos on PEMDAS never provide the rationale for this rule. The use of minus to make something negative, the implied multiplication if a quantity precedes or follows something in parentheses (or in algebra between a quantity and an unknown), and the use of a virgule (/) or a horizontal line in place of ÷, make things confusing and leads to the ambiguities discussed in many videos. 1. Exponentiation (to a positive number) is just a way to write repeated multiplication. (Defining anything raised to the zero power to be 1 extends this to zero, it can be extended further to negative numbers and fractions). 2. Division is just a way to write multiplication by the reciprocal (1 divided by the number) 3. Multiplication is just a way to write repeated addition (although this needs a little further definition for negative numbers and fractions) 4. Subtraction just a way to write addition of the inverse (negative). Apply these rules and you get a bunch of additions which can be done in any order and grouped any way you like. PEMDAS is just a way to get the same results without converting to a large number of additions.
First of all i do believe that most people that do a lot of math would say the first one is 2 why else would you just subtract in stead of add for the second one in physics it common to use a / to represent a fraction bar instead of a fraction in exponents to in physics the convention would say the second one is equal to one but we never use a division sign since that is cursed
but a negative number and multiplying by a negative are different things, "-6" isn't "- * 6" or "-1 * 6", it equals the same thing, but it isn't the same thing I feel like bro has no clue what he's talking about
@@WrathofMath It's 0. If you don't have brackets, the negative sign applies to the _entire_ power. Therefore, -xⁿ is the negative of xⁿ, _not_ the nth power of -x. Therefore, -1⁰ is the negative of 1⁰ which means it is -1. The idea of negation (i.e. subtraction, or multiplying by -1), is done after exponentiation. Since there is nothing grouping the -3, you must do 1⁰ first, _then_ do the negative operation. This is how it has always been. Or to put it another way, adding a negative is the same as subtracting. 1+-1⁰ = 1-1⁰ = 1-1 = 0
It's accepted that, e.g. $-e^x$ means $-(e^x)$, not $(-e)^x$. So $1+-1^0=1-(1^0)=1-1=0$ again. No ambiguity. 0:25 $1^0$ is unambiguously not 0 but 1. Where did you get the idea that it's 0?
2:10 i thinkyou should move the parantheses left one unit in that interpretation so (-1^0). Something about + and - side to side with nothing between them just feels wrong...
the trick here is that in the video, the "minus one" is preceded by something. If we write your "totals 0" interpretation with all parenthesis, we get : 1+(-(1⁰)) = 1+(-1) = 0 so you could never reach the statement of the question by simplifying . On an other hand 1+(-1)⁰ = 1+-1⁰ =2 goes through what is asked. Yes, you normally could not remove the parenthesis around (-1)⁰ without confusing it with (0-1⁰) = - 1⁰, but the +- combo removes the (0-1⁰) possibility here, so you can remove the parentesis and reach the equation of the video. Sometimes, when you are not sure about order of operation, start by rewriting the statement with all the useless parenthesis the way you understand, and try to remove them again. If you cannot reach the starting point, it means someone is wrong. If you cannot find any way to get there, this someone probably isn't you
I think writing 1+-1 without any brackets is dumb enough on its own
This must either be 1-1 or 1+(-1) but 2 operations next to each other just look and sound and are completely wrong
I totally agree!
beat me to it
I guess 1+√2 should be written as 1+(√2) then. Or √ is not an operation¯\_(ツ)_/¯
@@vladislavanikin3398 correctly written, the root symbol is equal to brackets
@@wernerviehhauser94 True. Equal to 1+√(2), not 1+(√2). We have two operations, one after another, no brackets whatsoever, and no one bats an eye. Because we see it all the time, as simple as that. There's nothing inherently wrong with writing two operations without brackets to separate them, it's simply a matter of convention. Pretty much arguing the Oxford comma.
notation should be used to increase comprehension, not decrease it. there is a special place in mathematical hell for people who intentionally use ambiguous or confusing notation
@@notacow69 You reminded me of a video made by 3B1B, about the new power notation.
Agreed
Solve 8/2(2+2)
set x = (2+2)
8/2(2+2) = 8/2x
Here, two cases arise depending on convention.
1. Implicit multiplication
8/2x = 8/(2x) = 4/x
Recall x = (2+2)
4/(2+2) = 4/4 = 1
So 8/2(2+2) = 1
2. PEMDAS (left to right)
8/2x = (8/2)x = 4x
Recall x = (2+2)
4(2+2) = 16
So 8/2(2+2) = 16
Ultimately, the question is poorly-phrased and so it is ill-posed. Ambiguity arises in part from the fact that multiplication by juxtaposition has higher precedence than division in a number of textbooks. Algebraically, 1/ab is ambiguous and should not be used without parentheses. The horizontal fraction bar eliminates ambiguity and should be favoured when possible.
thats why its better to use fraction symbols for division.
About the 1+-1^0 problem, the issue here isn't pemdas/bodmas, but whether the - is a negative sign so -1 is treated as the number 1 below zero (in which case the answer is 2), or a sign indicating we should be subtracting (in which case it is 0)
this is why some people write the negative sign indicating a negative number differently than the negative sign indicating subtraction: 1 + ⁻1
The issue comes from _why_ there's a double operation, in maths you generally want to understand the intent, in this case the + - is completely unnecessary so it's not unreasonable to think that they wrote it like that because they want it to be read as (-1)^0.
Just stop being ambiguous, say what you want to say as clearly as possible (clutter reduces readability too, which is why we use square brackets and spacing to clear up confusion)
In my regards "+-" stands for the plus minus sign "±" (since most keybords don't have this sign).
So 1+-1 means 1+1 AND 1-1 at the same time, so 1+-1 equals to 2 and also equals to 0.
This is used for measuremts and/or tolerances for a range of values like "The steel bar is 100m +- 1m long" means there is a steel bar that has a lenght of of roughly 100m, but the exact value is unknow and could be any value between 99m to 101m (so it could be 99.1 or 99.232465 or 100.0001 or 100.9999 or...)
Even when I did pemdas in school we never classified -1 as a multiplication of -
Pemdas gets sometimes disregarded even in for example published mathbooks in favour of what common sense would say. Insisting that an educated author would use pemdas can often lead you to a wrong result
They remain ambiguous, because the order of operations isn't "correct" universally. It's a convention, and thus is only "correct" when that convention is being held. A popular convention, but just convention nevertheless. There also exist other conventions, of which "negative signs are a feature of the number not multiplication" and "implicit multiplication/multiplication by juxtaposition has higher priority" are two examples.
It is through this that there arises the ambiguity. I (and many others, including some rather notable organizations in fields who may come across equations) personally hold "implicit multiplication/multiplication by juxtaposition has higher priority". So no, the equation doesn't have a single definite value if you follow order of operations, it has a single definite value if you follow specifically the PEMDAS/BODMAS/etc. convention for order of operations. But the problem never said it holds this convention, and there exist other conventions.
Thus, ambiguous. Dumb, too, sure. But also ambiguous.
(Some textbooks don't have the spacing for truly stacked fraction bars, and while many require the use of parentheses [a/(bc)], many instead hold the implicit multiplication priority convention [a/bc, with both b and c being on the bottom of the fraction]. Many calculators are also programmed with the implicit multiplication priority rule.
I'd find it hard to say either of these have 'done the math wrong', they've just done it with a different convention.
Notably, one that is likely spelled out in the style guides for the publisher or the manual of the calculator, unlike a random, ambiguous problem present online with no statement of convention)
indeed, I seem to recall at least programming languages prefix minus binds higher than any binary operator, and so -1^2 would be 1, not -1, because the prefix minus operator binds higher than the binary exponentiation operator. And then there are others where the leading minus (or plus) sign is consumed by the parser for numbers and so it's not even seen as an operator in the language. This latter approach is very much akin to "negative numbers are a feature of the number".
And there are also programming languages that are rigidly left to right (Smalltalk) or right to left (APL) without any hierarchy of precedence. PEMDAS is a choice, not a law of the universe.
But really, plus and minus and multiplication and all the signs and symbols our are inventions, so we should be defining how they are structured. Nobody would argue 1+2*3 is ambiguous. Maybe 'adding' or 'subtracting' isn't our invention, but then we would be saying 'The sum of one and the product of two and three' for 1+2*3 and 'The product of the sum of one plus two, and three' for (1+2)*3
@@TimothyRE99 At least for simple expressions, I have always seen the inline division as either
"num/(den)[rest of expression]" with denominator clearly delimited by parenthesis, or
"[rest of expression]num/den" without parenthesis around denominator; this meaning den is whatever between the slash and the end of the statement.
eg
"1=4/4" so "1=4/(2+2)" can simplify as "1=4/2+2" although this last step is kind of not ok, still feels obvious with literals, n/a+b feels like "quotient of n divided by sum of a and b, and f*k PEMDAS, you would obviously write b+n/a otherwise"
whereas "4=2+2" so "4=2+4/2" or "4=2+4/(2)" and if you really want the +2 at the end "4=4/(2)+2" or "4=(4/2)+2", the last one being the good "obviously correct" one, the other feels OKish, even if the parenthesis are technically useless, they semantically mark the difference with "1=4/2+2" above
The good rule being "avoid inline division as much as possible, and if really mandatory, prefer more () rather than too few "
I think even BODMAS/BIDMAS/PEMDAS has been taught differently to different people depending on the era they went to secondary school and the country they went to school, so I'd argue that even 'correctly applying BODMAS/BIDMAS/PEMDAS' has more than one possible answer depending on what you were taught was the correct way to apply it. Basically people are taught what convention their specific exam board wants/wanted them to follow to resolve any ambiguity they might encounter in their exams, so that all students doing the same exam will reach the same answer, making the exam easier to mark than if there are technically multiple correct answers. That seems to be the only place where it even makes halfway sense to have something like BIDMAS/BODMAS/PEMDAS. Any maths, physics or other use of an equation in the real word should never allow any ambiguity to exist in the first place, or even if it does the person in question will know where the equation they are looking at actually came from and what was actually meant by it and therefore won't need an arbitrary rule of thumb to resolve anything.
I take it a step further and disregard any arithmetic that uses the obelus symbol (÷). There's a reason that keyboards have no easy way to input it.
Exactly.
Vinculum > solidus > obelus
@@Grizzly01-vr4pn Please explain? I don't speak latin
It's annoying as hell when people argue that ÷ is division and / is fraction: like hello, do you not understand what a fraction is?
@ the names of various division symbols.
Vinculum: a fraction bar
Solidus: /
Obelus: ÷
@@vampire_catgirlA fraction is an expression, division is an operation. Yes, they are not the same, technically speaking. It is possible in different contexts to have fractions without the notion of division and to have division without fractions¯\_(ツ)_/¯
Genuine question: -1 is a value on the number line. It feels weird to say that you have to put parenthesis around it to make it unambiguous. Doesn't making this an order of operations issue imply that you cant actually represent negative numbers without an operation?
Right?
At the end of the day it's always just conventions that drive the discussion. And it just shows how we as humans, no matter how hard we try, always have to deal with ambiguity. There is never a fundamental truth, it's just "Let's try to convince a bunch of people to do it like I do"
So yes I would say you can represent "negative one" without an operation. And others might say "no you can't". And now again there is no right or wrong answer. It's a decision.
The issue is that the minus sign is both a unary and a binary operation. So context is needed to properly interpret it, and parentheses can provide this context.
After the thousands of hours of algebra I've done, I am firmly in the 8/[2*4] camp, with a formal objection to linear mathematical expressions longer than three arguments. Five for equations.
After also thousands of hours of algebra, my opinion is clear, if you're going to use linear mathematical expressions, they MUST be interpreted as such, that is evaluated with the "PEMDAS" (is there another name, please?) order of operations, FROM LEFT TO RIGHT. 8/2*4 = 16. Hell, even 2/2π = π, BECAUSE YOU CAN ALWAYS USE PARENTHESES, OR EVEN BETTER, NOT WRITE A STRING OF CHARACTERS! You wouldn't write "2/2π" to a computer and expect the result to be 1/π, would you? I thought so.
@DeJay7 your premise, the one you lifted from my comment, doesn't line up with your argument. How do you get from the basis of having a deeply ingrained habit of using expressions like 3(2+5) as a single term to the defense of an impractical, uncommonly used notation that sometimes includes similar expressions?
@@FScott-m1n Who told you I have a deeply ingrained habit of using expressions as such? I've never been forced to use a specific style of mathematical expression, I just tried my best to be accurate and not "lazy", it doesn't take much to write 8/[2(2+2)]. Also, would you say a/b*c is (a/b)*c (1) or a/(b*c) (2)? You don't have to answer, I don't really care about continuing this, but just think about. And if your answer is (1), then why is implicit multiplication different? It's very useful to differentiate the dot product, but why the higher priority? Whatever, you do you, obviously.
@DeJay7 you did when you lifted my premise and used it as your own premise. The fact that you don't know what you did (not to mention the rest of your second reply) pretty much makes my case -- that the premise you used does not fit with the rest of your claim. You should pay closer attention to what is being said, both by me, and yourself. Your carelessness has got you chasing your own tail.
@@FScott-m1n Okay, NOW I understood what you meant by "I am firmly in the 8/[2*4] camp". I first thought that you treat "8/2(4)" as such, for which I replied, and only NOW, after thinking about it for a little while, do I realise you mean that you USE this kind of expression. I guess we agree, then, sorry. Just a misunderstanding from my part, although your statement is ambiguous, just not as much as "8/2(4)" ...
Clearly the only correct way to read that expression is the absolute value of + - to the zeroth power
people that argue for implied multiplication bring up examples such as
8 / 4π = 2 / π instead of 2π or
8 / 4x = 2 / x instead of 2x
both which are reasonable, but that's why it's important to not be lazy and just add parentheses when typing comments
Personally, I consider juxtaposition a higher priority than multiplication.
@@soulsmanipulatedinc.1682 Yes, the order of operations with juxtaposition is usually PEJMDAS
Why should it be 4 divided by the juxtaposition of 2 and x and not the juxtaposition of 4/2 and x ?
Because of this ambiguity I would honestly stop using it.
@TH-cam_username_not_found Not without parentheses around 8/4
Can write 2/4 π and 2/x π, use spacing if don't wanna use parantheses. Why link 4π and xπ if not included in the denominator... Many ways to avoid ambiguity, laziness is not one of them 😂
Embrace chaos: (1+ -1)^0 = no universally agreeable answer
Wdym 0^0 is obviously equal to [insert opinion here]
Engineers would follow the connected rule [2(2+2)=8] and end up with 1. We have no time for pedants.
i stand in the 8 ÷ 2(2 + 2) = 1 camp simply because
8 ÷ 2x ≠ 4x
, if i had to pick between 16 and 1.
the best choice was ofc tell the author to never be an author again
I remember telling this to someone who was super heated at someone who said the answer was 16
Variables are literally treated different when compared to constants.
I see where the comparison lies if you could just simply replace x with 2+2, but the reality of it isnt that simple. The term 8 ÷ 4x is a completely reduced form while 8÷2(2+2) is deductible.
@@tim3line no they are not in this case, we can let x =2+2 you HAVE to then say 2x=2(2+2) so if you’re saying 8/2(2+2) does not equal 8/2x for some x, then you also have to say 8/2(2+2) does not equal 8/2(2+2) which isn’t the case
@@joshii4537 The thing is that 8 ÷ 2x could also be seen as 8 ÷ 2 * x which in that case 4x is a possible answer. This is just a matter of perspective, no right or wrong answer.
@@danielandycandrawira6206 yes ur correct, but I think a majority of perspectives change when you change it to x. The person above me admitted that if it was x then yes it would be 8/(2x)
Being an American who advocates for multiplication by juxtaposition is difficult
I can be on the side of the people who want implied multiplication to be its own grouping notation, but that's not necessarily the standard; but it's only in poorly written expressions where the disagreement is really relevant. I like the idea of juxtaposition taking priority.
The Expression
1 + -1 Is not valid, you cannot have two operands after each other. It would either be
1 + (- 1)
or
1+ (-1) * (1)
And then , when you add the Exponent, it becomes clear, what the actual Result is
I put the first expression into a TI-scientific calculator, and it shows that you are correct. So, the question is not about ambiguity. The problem is wrong, because it cannot be done at all. My other replies in other threads need adjustment relative to this one.
Error: Expected integer, got "-"
@@Petiscorei Some calculators require you to put 0 before the subtraction symbol(It will still give the same answer).
Depends on whether you consider unary minus an operation, as it only takes 1 argument its perfectly fine there.
@@nathansnail That would still technically be 1 + (-1)
When Syntax should have a Sin Tax 🤷🏽♂️
😂😂😂
I'm a strong advocate for using ( ) to disambiguate math statements. (8÷2)x(2+2) or 8÷(2(2+2))
It's NOT a multiplication, though, that is being applied between the negative sign and the 1. Because in the expression " 1 + -1 " the "-" has to (in my opinion), be interpreted as the unary operation that negates a number. And now you have to decide whether this operation is before or after exponentiation in order. And that's just not clear. Personally, the unary minus operation should be done before exponentiation. If that's the case, 1 + -1^0 = 1 + (-1)^0 = 2. And this "ambiguity" is only because someone wrote "1 + -1^0" instead of "1 + (-1)^0" (IF they wanted this interpretation, which is equal to 2), OR "1 - 1^0" (IF they wanted the other interpretation, which is equal to 0), where both expressions are unambiguous.
I would even say "-1" as a whole can be seen as one glyph. It's just the name of a number. So the "-" isn't even an unary operation. To distinguish one could write -(1) when we mean the unary operation . Just as we write -(-1) = 1.
@@Simchen Yeah, you're right. Although we don't know from the expression alone if "-1" is the number -1 or if it is the unary operation minus the number 1. But aside from that, THIS is practically the reason I said that this unary operation should be done before exponentiation.
I disagree with your dogmatic adherence to PEMDAS. As I'm sure you know, it's a mnemonic device intended to summarize mathematical convention. But all useful summaries lose information from the original, and all useful mnemonics rely on an existing understanding of the conventions. They are an aid for information retrieval, they do not serve as the information itself. There are plenty of examples of PEMDAS falling short and failing to explain convention.
For an easy example, why do you insist that multiplication and division happen simultaneously, from left to right? Or the same for addition and subtraction? That's certainly not what PEMDAS suggests. PEMDAS suggests that all multiplication should be evaluated before any division. Why is it okay to ignore or selectively interpret some parts of PEMDAS while insisting on others?
Here's another example: 3 ^ 3 ^ 3. What does that equal? PEMDAS suggests that the same operator should ALWAYS be evaluated from left to right, right? I think you would struggle to find any calculator that follows PEMDAS in that situation. Why is it okay to ignore or selectively interpret some parts of PEMDAS while insisting on others?
Other great examples have been given in other comments, especially those about how π is treated as a pseudo-variable during operations, despite being a known constant.
Your interpretation of the first problem of this video relies on a bizarre notion that negative quantities simply don't exist on their own (except negative one, but only when it's convenient for you), but always exist as negative one **multiplied** by a positive quantity (which do get to inherently exist for some reason, unlike negative quantities). That's no different than if I were to suggest that the number four doesn't exist on its own, it is just four ones **added** together. And then I could posit that since addition is the "last" operation that gets evaluated in PEMDAS, then any other operations clearly only apply to the first or last repetition of the addition!
Let's see that mistake in action:
1 + -4^0 (the first problem of the video, but with the quantity in question replaced with a four)
= 1 + -1 + 1 + 1 + 1^0 (my devil's advocate argument that a four doesn't exist and is simply adding one four times. Obviously bullshit since I didn't include wrapper parentheses during the expansion)
= 0 + 1 + 1 + 1 = 3 (only now do I apply the strict adherence to PEMDAS, giving the wrong answer)
Now let's see what you did in the first problem:
1 + -1^0
= 1 + -1 * 1^0 (hmmm I think I see some similar bullshit here too. Do you?)
= 1 + -1 = 0 (giving the wrong answer, that you claimed was "technically" correct)
Hey, wait a second! Why did the first negative one have to be expanded into a multiplication of negative one and positive one, but the resulting negative one gets to just exist without needing its own expansion??? You're right, it is dumb and there is no ambiguity here: PEMDAS should not be brainlessly treated as gospel. Unary operations/NEGATIVE QUANTITIES clearly have higher precedence than exponents. I would guess that is an intuition/understanding of convention that is held by a large majority of mathematicians. When I hear someone trying to argue that PEMDAS is unquestionable and infallible, I hear a child explaining to physicists that there are three states of matter because that's what their teacher said.
Can't read and respond to all of this, but a couple points.
1) You say you disagree with my dogmatic adherence to PEMDAS, and then say how I don't adhere to PEMDAS.
2) 3^3^3 is evaluated without issue by following the order of operations, PEMDAS says nothing about operations always being performed from left to right. It is mentioned one time in this video as something to refer to to settle ambiguous expressions in the case of equal priority operations.
3) Negative quantities don't clearly have higher precedence than exponents, that's why so many people disagree about -1^0, because it is not clear. The convention is to carry out the exponent first, which is why I referred to that as technically correct.
@@WrathofMath 1) You're right, I changed my argument as I was thinking and writing about it, and I should have updated the opening statement
2) Exponents are equal priority with other exponents, no? You're trying to add additional context based on your understanding of convention or based on additional explanations and context that were given to you outside of PEMDAS. That is not actually offered by the mnemonic. In fact, the whole "order" of equal priority operations is never mentioned at all within PEMDAS. It only gives a (slightly misleading) hierarchy of _some_ operations. That is exactly the problem with it. It is both wrong and incomplete without further context.
3) You say it is not clear, but your title says it's not ambiguous. Interestingly, your comment says the convention is to carry out of the exponent first, which I exactly agree with, but your video doesn't seem to. -1^0 = 1. It only becomes a problem if you insist on treating -1 as a multiplication rather than its own quantity. That's the core of my beef. I have never heard any justification for it, and it seems like a thoughtless retrofit to squeeze negation into PEMDAS to quiet students who notice that (-) is also an operator and negative numbers don't get their own special digits separate from positive numbers (but π _is_ special enough to get its own character! Poor negative numbers). "-4" may be mathematically equivalent to "4 * (-1)", but they are different representations and moving between them requires some thought, like adding parentheses if you're going to be introducing a new multiplication operation. Just like moving from a representation of "4" to a representation of "1 + 1 + 1 + 1". Mathematically equivalent, but new operations were introduced.
I do actually agree that it is unclear, and that probably wasn't my best opportunity to turn a phrase. The whole point of my comment is that PEMDAS does indeed create ambiguity. When reading math, it is important to know and default to real conventions actually used by mathematicians rather than what your elementary school teacher taught you. When writing math,... just add your dang parentheses!
the problem often isn't the mathematics, it's the interpretation / evaluation of the problem. there is also the problem of around 27 other rules of mathematics that isn't included in the mnemonic like PEMDAS
for instance in the 8/2(2+2) that is just a single division, granted it is a complex division, but none the less is is just a division 8 being the numerator and 2(2+2) being the denominator
this problem isn't 8/2 * 4 this is 8/2(4) which would be the same as 8 / (2 * 4)
this is because the vinculum is a grouping symbol, this is why a/bc = a/(b*c) and not (a/b)*c for the Americans a/b*c
it isn't that implied multiplications comes before division, it is that you need to resolve the implied multiplication as part of the denominator before you can do the division.
This reminds me of something I dislike about exponents, which is that without saying the word “parentheses” a bunch, it is ambiguous how the expression is when said verbally. The same goes for fractions.
For example, saying “one plus two squared over four” has many different interpretations.
(1+2^2)/4, ((1+2)^2)/4, 1+2^(2/4), ect…
For sure, when I speak of these things I try to make my intentions very clear by tone and space between words. For the first expression for example I would say "One.....plus two squared....all - over - 4." It is quite tricky!
An other solution is to read your " (1), (2), (3), etc.." examples as
(1) The quotient of the sum of one and the (second power)(square) of two divided by four
(2) The quotient of the (second power)(square) of the sum of one and two divided by four
(3) The sum of one and the exponentiation of two by the quotient of two divided by four
This removes all ambiguity, even if I agree it is kind of annoying to talk this much for a simple expressions like that.
(also note the requirement to use the exponentiation operation for non integer "powers". Some people who would have to compute (3) don't even know that operation exist)
I think the issue with these are really whether or not you consider multiplication by juxtaposition, unary positive and unary negative their own operations and if so what order convention you are using.
but it's wrong. Juxtaposition takes precedence over an explicit sign, and it always has. Your third grade math teacher is not a mathematician and PEMDAS is an incorrect oversimplification. You wouldn't say "8÷2x = 4x" -- you know it's 4/x because of implied multiplication. This doesn't change just because you substituted x=4.
This ^
PEMDAS is just a mnemonic guide to help remember a basic order of operations. It is not a comprehensive guide to order of operations.
actually i would say 8 ÷ 2 x = 4 x. cuz. it is.
if it were meant to be 4 / x it would have been written 8 ÷ (2 x)
@@zacknattackExcept in practice people don’t. I don’t remember the source but I remember someone went through published math papers and did in fact find things like 2/xy meaning 2/(xy), but never (or much more rarely) found 2/xy meaning 2y/x.
You seem very biased against PEMDAS. It looks like you have been taught the other one. For me, I HAVE learnt both, and to avoid confusion, we always use the fraction sign. If not, we use parenthesis, and if either is not applied, we assume it's wrong, and the author gets a lot of attack hate.
Furthermore, when you download a calculator app on your phone, it asks, "What do you mean by the division sign and the percentage sign." all of which to avoid confusion.
@@danielrhouck well 2 / 4 x and 2 ÷ 4 x are different. if i saw the former i'd think "well this is ambiguous" and if i saw the latter i'd think 0.5 x
Implied multiplications should come before division.
Is there any reason for that, or is it just your opinion?
@ it is intuitive for formulas, the density of dots in a rectangle is D ➗ LW, (dots, length, width) everyone interprets this as LW being a single compound factor term, obviously it’s better to do the fraction bar, but I think in situations with implied multiply it makes sense to think of each implied multiply as a factor of a single object
@@widmo206 in mathematical formulas it's used that way, also we can distribute the product, and also when doing equations you don't separate the x from it's number like this video would imply
I cant disagree.
Also I think it makes a lot of sense when you think about stuff smooshed together with implied multiplication as factors, because that explains why you can still exponentiate individual terms of it, just like prime factors. A bunch of things next to each other should be treated like the factors of a single term
When I was in school none of the teachers taught us that "-1" is to be _evaluated_ as -1 * 1. Rather, it was treated as simply a negative number. To give an example of how that's handled in manipulating expressions, if you wanted to take 2 - 4² and convert it to addition it would be 2 + -(4²). The parentheses are added because the 4² was given with evaluative priority - the 4² needs to become negative, not the 4 before squaring.
Under that consideration, it's not contrary to PEMDAS to apply the exponent to the negative number in -1⁰ because it's not viewed as negative one times one to the zeroth power, but simply negative one to the zeroth power.
To the other one, if you are given 8÷2x, where x=2+2, what's the answer? The 2 is a coefficient of x, and that's taken as grouping. I think it a bit silly to not be consistent with that just because you replace the variable with a literal.
So it's _not_ unambiguous just because you have an obvious PEMDAS evaluation, which is why you are _correct_ that the thing to do is not write expressions like -1⁰ or 8÷2(2+2).
Mathematicians shake their head at this, but then fail with:
"Buy a gallon of milk, and if there are eggs, buy a dozen"
You are not correct to say that the American order of operations is universally agreed and therefore 16 is the "technically correct" answer. In the UK the correct answer to 8/2(2+2) is 1 not 16. UK calculators evaluate it this way and the UK GCSE exams require evaluation in that way.
It is the universally agreed upon way, though. In programming. People don't write programs with implied multiplication. Also people don't write "8/2(2+2)".
@@DeJay7 Yes, I am coming to realise more and more that the obsession with 8/2(2+2) = 16 comes from people who have done some programming and can't seem to understand that computers and humans don't think the same way.
@@martind2520 Computers and humans do NOT think the same way.
So why do you support a style of writing that only makes sense to be used for computers?
It's not that people "don't understand the difference", it's that the difference is not accounted for in the first place.
Certainly a big part of this is the fact that our students need to be able to communicate with computers, and so it is important to teach them the mathematical manners to be able to do so.
There is no such thing as US and UK order of operations. If there was then the UK would not be able to build a car, or bridge. This is an agreed upon process that has been accepted for a very long time.
If you treat the expression 6×2÷3 as a sequence of operations and do the division first, then multiply, you will indeed get the same result, 44. Here's the breakdown of your approach:
You can first divide 2÷3 which gives 0.666…(or 0.67 if rounded to two decimal places).
Then multiply that result by 6: 6×0.67≈4
So, mathematically speaking, doing the division first in this particular case does give the same result. This is because multiplication and division are indeed commutative at the same level in the order of operations - they can be performed in either direction when they're combined in a linear expression.
Does this thing always work and if it does can you say that division comes before multiplication?
-x^a = -1x^a, applying rules that must be used for algebra, -(a^x); answer is 0. Now if it were 1+(-1)^0, then the answer would be 2.
If you follow the order of operations, it's obvious.
When you have 1 + -1⁰, the -1⁰ is treated as (-1)*1⁰, or (0-1⁰).
The negative sign applies to the _entire_ of the power, not just the base, simply because "exponentiation" is before "negation", in which "negation" can either be thought of as "multiplied by -1", or "subtracted from 0", either of which, are done _after_ exponentiation.
Alternatively, if you remember that x + -y = x - y, you can also see it as 1+-1⁰ = 1-1⁰
Therefore:
1+-1⁰ = 1-1⁰ = 1-1 = 0
1+-1⁰ = 1+(0-1⁰) = 1+(-1) = 1-1 = 0
1+-1⁰ = 1+(-1*1⁰) = 1+(-1*1) = 1+(-1) = 1-1 = 0
I strongly disagree, 1 + - 1^0 is obviously 2 for the following reason:
there's two things that could have been meant by that, either 1 + (- 1)^0 or 1 + - (1^0). but it cannot be the second, because *no one* ever writes 4 + - 2. everyone who would wanna write that would instinctively write 4 + (- 2), because otherwise you might confuse it with 4 plus/minus 2.
so it has to be the first option
also as others have pointed out, for me 8/2(2+2) is obviously 8/(2(2+2)) because (8/2)(2+2) would be written as 8/2*(2+2). although yes, I agree this one is sorta ambiguous and should be written differently if possible
math is a language and just as in any other language:
if most people aren't conforming to your grammar rule, then the rule is wrong not the people
this is cool, I mean not the problems they are dumb, but you explaining why they are dumb is cool.
It's clearly the absolute value of + -, measured in degrees
I remember seeing a math problem that said something along the lines of log(x+2)^2 which was supposed to be interpreted as log((x+2)^2).
Psychotic
Well, isn't the notation for the square of the logarithm of the sum of x and 2 "log²(x+2)" ? And "log(x+2)²" the logarithm of the square of the sum of x and two.
The same goes for functions. If you want the square of the function f at x, note f²(x). Although here the value of the function at the square of x, f(x²) is not ambiguous, it is with other common functions, such as trigonometric ones: sin²x vs sinx². Because of their heavy repetition, we usually omit the parenthesis, and put the exponent on either the argument or the function name depending on what we square, instead of requiring two layers of parenthesis like (sin(x))²
Honestly, if it were not for the power notation x^2 instead of x², I would not see why you were confused by this problem ^^
On another hand, just changing the power notation breaks my habits, and I no longer found the notation easy to understand... Oh habits !!
I think that the division symbol is just a relic of the time when you understand the algebraic operations as 5(exp, +, *, -, /) separate operations instead of the 3(exp, +, *) that are actually there. in defense of the implied multiplication people, in the mind of someone who sees it as only 3 operations, then you should also expect communativity with division within the divisor since it is multiplication. if you're used to fraction bars, it's easy to think that division has implied parentheses around both the divisor and dividend. for example in that 8 / 2(2+2) example, multiplication is communative and therefore you should be able to write 8/ 4 * 2 which should give you 4.
The arguments in the comments about replacing (2+2) with x is a super understandable misunderstanding but is a false equivalency.
8 ÷ 2x is a reduced term
8 ÷ 2(2+2) is a deductible constant.
In the same vein we can't equate
8 ÷ 2x to 8 ÷ 2y.
Furthermore unknown variables are treated different because they're unknown. But the term within the parentheses is already known.
No it is not a false equivalency at all. The whole point of algebra is that you can replace the letters with any expression in brackets.
If x = 2+2 then 8 ÷ 2x = 8 ÷ 2(2+2)
The second interpretation treats negative numbers as second class citizens of the number world. You can never write them without the implied multiplication. Negative numbers should be a separate case in order of operations.
i dont really agree that "-1" inherently means multiplication is involved. to me it's just as much a value as a positive number itself is, using a symbol like - to denote that is just a limitation of how we express that.
pretty sure these are both wrong because Implicit multiplication is handled before sign multiplication and division
When I see that my mind automatically sees it as 1+(-1)^0, it is only ambiguous because it is poorly written. That is like writing "they told them that they were" does the "they were" apply to the they or the them? it could be either because it is a badly written sentence.
Agreed
maybe it's one person called "they" and another person called "them" and the other "they" is a totally different guy with the same name.
Is the first example really ambiguous? The "+" is a binary operator and needs two operands. That leaves (1)+(-1)^0 as the only option with "-" being the unary operator and "-1" being the negative unit.
The reasoning for 1+-1^0 = 0 is just stupid. The same reasoning could be argued for 25+-5^2 = 0, which everyone knows is wrong, as any number squared would return a positive number, otherwise it’s an imaginary number. I don’t agree with the reasoning also because if a negative is explicitly put instead of subtraction, then you usually would include it in the exponentiation (in the parentheses, like (-1), or (-2)).
I just go with whatever the compiler says it is. And if that throws an error I throw in some parenthesis until it does what it should.
in the mnemonic, it looks like multiplication has higher precedence than division, and addition has higher precedence than subtraction, which is yet another source of confusion. thats just an artifact of the mnemonic itself though, not the actual standard.
where do radicals fit into this? or even fraction bars? they have implied parentheses just like the "implied multiplication" covered in the video, but nobody disagrees about them. i think its only an issue when the entire expression is on one line, like a text comment or a calculator or a computer code. as soon as we're allowed to move numbers around vertically all the ambiguity goes away it seems like
One thing that’s really interesting is the mnemonic in a lot of European countries is “BODMAS” in which division now comes before multiplication, so that’s one of the ways I remember multiplication and division are on the same priority, especially since they are basically the same thing.
With radicals, those will have the same priority as exponents. For example, a square root can easily be written as the exponent 1/2.
I really like your comment on fractions bars by the way, very insightful!
What makes radicals and fraction bars unambiguous generally is that they have very clear visual grouping communication, with the exception of when written in plaintext on one like. Of course if I'm writing a fraction vertically there will never be confusion, but suddenly 4/x+2 requires me to be careful and use parentheses if I am communicating with a computer or otherwise in plaintext. When the fraction is written in one line, or with the elementary division symbol which forces one line, the expressions are just often unclear visually, and so the disagreements abound. Implied multiplication is not as strong of a visual grouping symbol. Square roots require us to actually put things in a radical, implied multiplication only looks grouped because stuff is close together. So is it grouped? Maybe. Maybe not
The second one remains ambiguous, though keeping in mind that multiplication by juxtaposition takes priority over explicit multiplication and division is useful.
But the first one? You're just straight-up wrong. Order of operations doesn't apply here. There is no multiplication, implicit or otherwise. There is only addition of the exponent of a negative number. The number is -1, and it is raised to the zeroth power before the result is added to 1. No ambiguity.
So -x^2=x^2?
Seems like a death of the author take, interesting
No, your post is *wrong!* Just to mention the second problem, there are no grouping symbols around -1, so it is only 1 raised to 0.
1 + - 1^0 =
1 + - 1 =
1 - 1 =
0
@@robertveith6383 you can't have two separate symbols next to each other. 4 × ÷ 2 is a nonsensical equation. Likewise, 1 + -1^0 makes no sense if you try to interpret the - as a symbol in its own right. Mathematical symbols are almost entirely infix operators taking two numbers as operands. The only way to interpret "+ -1" is as "plus negative one", and thus + -1^0 *must* mean + 1.
@@joshii4537 I would say it's ambiguous in any context other than where you have + - next to each other like in the video.
This video seems to want to interpret a minus as some unary operation that is equivalent to multiplying by negative 1. You now have one of two problems:
1. you've gotten yourself into an infinite loop -1 = (-1)*1 = ((-1)*1)*1 = ...
2. the minus operation symbol (-) has two different priorities in PEMDAS. In one place it is S and in the other it is M.
The problem is that we use the same symbol to represent subtraction and negative numbers. Perhaps negative numbers should be written in red and non-negative in black. Or use a different symbol as done in some programming languages (1+ ⁻1). Since the minus sign is used for both subtraction and negative numbers, I prefer to interpret the minus sign following some other operator as meaning the following quantity is a negative number. Of course you could say that a negation operation (N) should have its own priority. And then a discussion could follow whether N is higher priority that E or N is the same priority as MD (PNEMDAS or PENMDAS)
That said, PEMDAS is just a convenient rule to compute expressions with a mix of different operations. Videos on PEMDAS never provide the rationale for this rule. The use of minus to make something negative, the implied multiplication if a quantity precedes or follows something in parentheses (or in algebra between a quantity and an unknown), and the use of a virgule (/) or a horizontal line in place of ÷, make things confusing and leads to the ambiguities discussed in many videos.
1. Exponentiation (to a positive number) is just a way to write repeated multiplication. (Defining anything raised to the zero power to be 1 extends this to zero, it can be extended further to negative numbers and fractions).
2. Division is just a way to write multiplication by the reciprocal (1 divided by the number)
3. Multiplication is just a way to write repeated addition (although this needs a little further definition for negative numbers and fractions)
4. Subtraction just a way to write addition of the inverse (negative).
Apply these rules and you get a bunch of additions which can be done in any order and grouped any way you like. PEMDAS is just a way to get the same results without converting to a large number of additions.
First of all i do believe that most people that do a lot of math would say the first one is 2 why else would you just subtract in stead of add for the second one in physics it common to use a / to represent a fraction bar instead of a fraction in exponents to in physics the convention would say the second one is equal to one but we never use a division sign since that is cursed
but a negative number and multiplying by a negative are different things, "-6" isn't "- * 6" or "-1 * 6", it equals the same thing, but it isn't the same thing I feel like bro has no clue what he's talking about
The answer is either 2 or 0, since there are two ways to solve:
1) 1 + (-1) ^ 0 = 1 + 1 = 2
2) 1 + -(1^0) = 1 + -(1). = 1-1=0
Well interestingly it's DEFINITELY 2, or DEFINITELY 0, it just depends who you ask!
@@WrathofMath Just saying the comment was one of those "Click Comment Close"
@@WrathofMath It's 0.
If you don't have brackets, the negative sign applies to the _entire_ power. Therefore, -xⁿ is the negative of xⁿ, _not_ the nth power of -x.
Therefore, -1⁰ is the negative of 1⁰ which means it is -1.
The idea of negation (i.e. subtraction, or multiplying by -1), is done after exponentiation. Since there is nothing grouping the -3, you must do 1⁰ first, _then_ do the negative operation.
This is how it has always been.
Or to put it another way, adding a negative is the same as subtracting.
1+-1⁰ = 1-1⁰ = 1-1 = 0
@@scmtuk3662 You know I actually have a cool video going over all that
It's accepted that, e.g. $-e^x$ means $-(e^x)$, not $(-e)^x$. So $1+-1^0=1-(1^0)=1-1=0$ again. No ambiguity.
0:25 $1^0$ is unambiguously not 0 but 1. Where did you get the idea that it's 0?
What the fuck is "$"?
Forget about the 0th power someone needs to explain what does the absolute value ot +- even mean?
2:10 i thinkyou should move the parantheses left one unit in that interpretation so (-1^0). Something about + and - side to side with nothing between them just feels wrong...
remember -1^0 is -1 but (-1)^0=1
so 1+-1^0=1-1=0
Exactly, this is how we blend in with the calculators!
the trick here is that in the video, the "minus one" is preceded by something.
If we write your "totals 0" interpretation with all parenthesis, we get :
1+(-(1⁰)) = 1+(-1) = 0 so you could never reach the statement of the question by simplifying . On an other hand
1+(-1)⁰ = 1+-1⁰ =2 goes through what is asked. Yes, you normally could not remove the parenthesis around (-1)⁰ without confusing it with (0-1⁰) = - 1⁰, but the +- combo removes the (0-1⁰) possibility here, so you can remove the parentesis and reach the equation of the video.
Sometimes, when you are not sure about order of operation, start by rewriting the statement with all the useless parenthesis the way you understand, and try to remove them again. If you cannot reach the starting point, it means someone is wrong. If you cannot find any way to get there, this someone probably isn't you
-1^0=1
also, 1^0=1. so either way
1+ -1^0=2
-(1^0) is -1
1:54 have you even seen this part? It explains why it's not
I like javascript's answer
I feel like the problem can be greatly simplified with they put a bracket around the negative one.
1+(-1)^0 = 1+1 = 2
Still 0.
HELLO FELLOW SCHOLARS
Me when doing 8/2(2+2) 8/2=4 and (2+2)=4, 4*4=16 answer
hi
Season's greetings!
third this comment
I feel like the problem can be greatly simplified with they put a bracket around the negative one.
1+(-1)^0 = 1+1 = 2
Yes, that's literally the point of the video.