Which is the worst math debate: 0^0, sqrt(1), 0.999...=1, or 12/3(4)?
2024 ж. 9 Нау.
225 956 Рет қаралды
These are the most debated math topics on the Internet but which one is the worst?
(A) 0 to the 0th power=1 or undefined. No calculus limit here.
(B) sqrt(1) = 1 or both +-1?)
(C) 0.999...=1 or not?
(D) order of operations 12/3(4)=1 or 16
More than 28,000 viewers voted in my recent poll and now let's discuss what each debate is all about.
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Can you solve x^ln(4)+x^ln(10)=x^ln(25)? kzhead.info/sun/q6apiraPenmtZmg/bejne.html
B is hotly debated, true. Is zero odd/even? Is zero a natural number?
@@deltalima6703 Zero is even and non-natural number
@@deltalima6703 What? I don't think that anyone has ever argued that zero is odd.
@sriprasadjoshi3036 some say 0 is even, some say it is neither even nor odd. Recent advances in set theory strongly suggest the ancient mathematicians made an error and 0 is actually a natural number. There is an ongoing debate at the moment.
@@deltalima6703 how come 0 is so bullied all the time? Everyone tells him "you're nothing" and he's always excluded from set parties.
D is just intentionally bad notation. The others at least have some interesting mathematics behind them. So D is indeed the worst debate.
Yeah easily D, the rest are like good debates that are just worn out. D is just faulty written. Barely anyone even uses the division symbol, the just write it fractionally depending on what they want to portray
I really don't think D is that bad of a notation. To me, D is very clear and has a clear answer. In terms of bad notation, tan^-2(x) should take the top spot. : )
Right. I don't like D not so much for mathematical reasons, but rather because its one of the low-effort comment engagement posts that bots post on social media. And I hate that.
@@blackpenredpen probably not to someone who really knows their math, but 12/3(4) would be very different to (12/3)4, give two answers because they are different. And pemdas is not taught well, there’s always confusion on multiplication or division first. The way it’s written is meant to confuse people into thinking it’s 12/(3 * 4) so I would say it’s bad notation, since most people use fraction to mistake division now
I see what you mean, but at that point the debate is not about really about mathematics, it is about syntax.@@blackpenredpen
D is the reason we don't use ÷ after elementary school
The division symbol isn't the problem. It's the juxtaposition of terms that people assume takes priority over the division symbol, that is the problem. We'd have the same issue if it were a simple slash. It would be a much more efficient order of operations, if juxtaposition DID take priority over division, because it would allow you to write your denominators without snaring them, and professional scientists and mathematicians use this order of operations all the time. It's middle school teachers who don't want to deal with this problem, and the curriculum they follow that created this problem, who tell you that multiplication and division have the same priority regardless of notation.
@@carultch here mathematician, I'd tell you to write a parentheses or for me that's undefined.
@@user-os4lj3pi4q scientist here, I see 1/2π all the time, nobody ever writes 1/(2π). Makes sense too, because if 1/2π did mean the same thing as (1/2)π, then you'd just write π/2 to begin with, why write a complex form when it can be simple. By having it this way you have short expressions that are unambiguos
@@Alvin853: You can also write 1/2/π. Actually, you can just write /2/π, although a lot of people won't understand that; but once you get used to it, it's very convenient.
@@carultch Professional mathematicians wouldn't write it like that because it is just intentionally vague. If you want to do it properly without parentheses you use one horisontal dash and put the 12 on top and the 3 on the bottom
Half the comments are saying D is the worst debate, the other half are arguing about how it's really solved. ABC are just about forgotten
I cannot believe the amount of people who don't understand D. Like, if you have 5/4+4*5 do they seriously think that means 5/(4+4*5)? In what world does it make sense to take everything after a division sign and throw it together in parentheses when there are no parentheses... Perhaps PEMDAS needs to be taught a few more times in school.
@@gudadada Actually, both sides of the debate are correct. This expression can be solved either way, as both interpretations of the expression are commonly accepted. When you have a number adjacent to a parenthesis, it's called a juxtaposition, and is solved before other operations. Outside the US, some countries instead learn BEJMA (Brackets, Exponents, _Juxtaposition_ , Multiplication, Addition). It's very handy for factoring. Imagine the expression [(2x² + 4x)], which can be rewritten as [2x(x + 2)] So technically, both forms are correct. Just make sure to use extra brackets when inputting into a calculator (I have two different calculators that solve it differently), or when sharing your problems with others, to make sure everybody is on the same page!
No, you are misinterpreting what juxtaposition is. Juxtaposition refers to a sign being implied, but it does NOT change grouping. For example, 20÷3x,x=5 is NOT the same as 20÷(3x),x=5. The former is 100/3, the latter is 4/3. This is a rule agreed upon by all mathematicians and functionally by all calculators. Of course, this "issue" is usually mitigated by using fractions which have much better visual clarity (everything in the numerator and denominator is contained), but there is really no debate, only one answer is correct. The reason your factored example works is because of left-right order. (2x)(x+2) is the same as (2)(x)(x+2). Really, it's incorrect to view it as (2x)(x+2), because that's one step into expanding. If you instead gave an example with division, say 2/x(x+2),x=5, now you'd have an issue. 2/(x(x+2)) is NOT the same as 2/x(x+2). The lone x does not get attached to the (x+2) without parentheses - that is plain wrong. Try plugging these examples in yourself to an algebra calculator if you don't trust the people who do math for a career.@@brickbot2.038
This is partly because B and C already have solved answers. There’s not anything to argue about there. “A” *also* has a solved answer, but the answer is “both, but it depends on the context”. There are plenty of formulas where 0^0=1 is required because it still outputs correct answers, but there are other cases where the output would be undefined. You could kinda argue there, but you’d still reach the final answer pretty quickly. D gets argued about because it’s an argument about the understanding of syntax. It’s not arguing about a mathematical concept itself outside of parsing syntax correctly. In reality, whoever writes that should use clearer syntax regardless of - even if there is only one way to parse it, it’s still an awkward way of writing an expression.
@gudadada It's not that the first is wrong, it's just that absolutely no one would write it that way outside of inciting a debate. The second you see all the time halfway through a problem. Not everyone is so careful with parens when solving z=20/y; y=3x; x=5.
230 - 220 × (1 ÷ 2) You might not believe me, but the answer is actually 5!
yeah.
I can't believe it's 5!
yeah many people don't get that it's actually 5!
Such a good meme
@@shauryamathbasics You ruined it. You took the funny away by explaining it. I wanted to see people confused.
B is the one I meet most often but D is just stupid
agreed
Especially since the given answer in the video is wrong. The right answer is that its contextual; right-most inner-most is the standard evaluation order since left-descent parsers are problematic. So in most cases, the solution really is 1. But some mathematicians who don't care about formal languages use the order given in this video. The binary division operator is non-standard to begin with.
@@ontoversecorrect. 6÷6 =1 but 6 ÷ 2(3) 6÷3(2) lol. I'm not sure why they ignore juxtaposition or implied multiplication lol
Yeah, so many people always answer that "you forgot the +/-" in the comments of a maths KZhead video, when it's completely inappropriate to the solution development
@@ontoverseWhat? There's no case where it's 1. Only people who don't know how to do arithmetic gets the answer 1.
You missed 1+2+3... = -1/12
Well, but with infinity everything is possible. And -1/12 has real applications. But infinity is also valid answer.
Dat result comes from the foolish assumption of the sum converging in the first place
@@DatBoi_TheGudBIAS No, the sum was established before the current convention of convergence, and shouldn't be interpreted purely analytically. The "identity" depends on an expanded version of algebraic manipulations that are consistent with analytic continuation.
@@caspermadlener4191The issue is that even it is contraintuitive, the result -1/12, has REAL applications. And with other methods we also can end up with the same result. With infinity, rarely there is only one valid answer.
@@radupopescu9977 Yeah, I should have addressed this, since this is the main criticism, and it would definitely be a big deal if it would be possible to get any other real number using manipulations like this. Mathematics of course depends on any two independent parties being able to get the same result. Although I can't proof that there are no such manipulations, I can at least point out some applications of the "identity". First of all, ζ(-1)=-1/12, but this is not the big application everyone talks about. The real application is in algebra, as are the manipulations. You have this formula (Weyl denominator formula) that requires the halfsum of positive roots of a root system (important concept in Lie algebra), and the integers form a root system of a generalised Kac-Moody algebra (kinda like an infinite Lie algebra). Well, the proper constant for that formula is -1/24, which could be interpreted as half of 1+2+3+4+... You can apply the same logic to why the modular form (these are important complex functions) of weight 1/2 looks the way it looks, but this happens to follow from my "main" application.
Both Mathematicians and computer scientists agree that 0!=1
Ah damn I commented a similar thing
Say it the other way around 1=0! Looks more confusing
@@omerdvir1709 != Means not equals in programming so 0 != 1 means 0 does not equal to 1. In maths 0! Means 0 factorial which should be interpreted as (0!)=1 which is also true
@@ghotifish1838you’re right. I’m actually doing programming on school so o should have got that reference but I thought he was referring to the factorial and it would make more sense to put the exclamation at the end
Side note: != as a substitute for ≠ is common (mostly via C) but not universal; Haskell uses /= and is another common inequality operator (e.g. BASIC, SQL, Excel). In C 0==!1 too, but 1=0! fails to parse (! is prefix not in C, postfix factorial in common maths).
B really seems almost more like a communication issue than a math issue. There is no question that 1 and -1 are both square roots of one. it's more that there is confusion over the fact that the square root function is only looking for the principle positive square root instead of all square roots.
Yes, it's a matter of notation: knowing what the radical sign means.
I disagree, because we asume sqr of something to be positive. Because in real life we use only positive that doesn't mean we are right. Think of this: sqr(1-i), it has a principal value 1.098...-i*0.455... (equivalent of sqr(1)=1) and a second value: -1.098...+i*0 455 (equivalent of sqr(1)=-1. What value do you choose? Both are valid. So, we use the positive value in real numbers, but when we deal with complex numbers, we can't ignore the second one. So squareroot means 2 values, third root 3 values and so on. The fact that in real life we 1 only one value, that doesn't mean the other's doesn't exist.
You said it yourself, "the square root function is looking at positive values". It only uses positive values because the principal square root is a function, whereas the +- version is not
Same with D. People are doing the operations correctly, just in the wrong order. Poor communication around the necessary order of operations
@@radupopescu9977um actually complex square root is a different function from the principal square root. just like complex logarithm is multivalued, so is complex square root. But normal square roots and logarithms have one value. No one would say that ln(2) is a bunch of numbers, in fact they would write the answer to the complex logarithm IN TERMS OF normal one! Like Ln(2) = ln(2) + 2n*pi*i or something. Just like the actual answer to square equation is written in terms of principal root with additional symbols, like Sqrt(2) = ± sqrt(2)
".999 repeated is 1 becuase you cant find a number between them" is a really cool observation
And IF .999... is not equal to one, then there must be an infinitude of numbers between .999... and one.
no (for me) because this means its the exact next number. think of 1 quantum of numbers . its 0.00..1 so this is the smallest value .. can you have 1.5 cents? no because 1 cent is the quantum of euros , thats why theres no "cent" between 1 cent and 2 cents
It doesnt make any sense for me because if you look at integer Numbers 2 and 3 you cant find any number between them and they arent equal
@@fab3f2.5
@@namespaced4437 "integer Numbers"
I think D is the worst because writing it in fraction form would clear any debate so I think it's more of a communication/notation problem than a debate really. I can't think of a single situation where I would rather write ÷ instead of just expressing division as a fraction.
t was used in grade 3 then forgotten about until it shows up as a button on a calculator.
I do the division sign when I'm dividing a fraction by another fraction, there just isn't enough space to write 4 "layers"
Not laying out division and multiplication in an intuitive order or using brackets unambiguously should be considered as invalid as not having a closing parenthesis... To paraphrase The Big Lebowski on PEMDAS: "You're not wrong Walter, you're just an asshole."
@@Firefly256id probably multiply by the inverse of the fraction
"I think D is the worst because writing it in fraction form would clear any debate so I think it's more of a communication/notation problem than a debate really." EXACTLY THIS IS WHAT I WAS SAYING. The debate is stupid because it revolves around an ambiguity that should not be their in the first place.
Definitely D. Terrible notation and not ISO compliant (ISO-80000-1 and ISO-80000-2 not followed). It's simply ambiguous notation. A trick. Academically, multiplication by juxtaposition implies grouping but the more programming/literal interpretation does not. Wolfram Alpha's Solidus article mentions the a/bc ambiguity and modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. Even over in America where the programming interpretation is more popular, the American Mathematical Society stated it was ambiguous notation too. Multiple professors and mathematicians have said so also like: Prof. Steven Strogatz, Dr. Trevor Bazett, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter, Eddie Woo etc. Even scientific calculators don't agree on one interpretation or the other. Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation (1) Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation (1). TI later changed to the programming interpretation (16) but when I asked them were unable to find the reason why. A recent example from another commenter: Intermediate Algebra, 4th edition (Roland Larson and Robert Hostetler) c. 2005 that while giving the order of operations, includes a sidebar study tip saying the order of operations applies when multiplication is indicated by × or • When the multiplication is implied by parenthesis it has a higher priority than the Left-to-Right rule. It then gives the example 8 ÷ 4(2) = 8 ÷ 8 = 1 but 8 ÷ 4 • 2 = 2 • 2 = 4 A,B,C I 100% agree with here, but D, no, 16 is not the corrext answer according to the evidence. 16 is 'a' correct answer, along with 1. The expression is wrong. That is the correct answer.
thank you, I'm sure if bprp was given: f(x) = 12 ÷ 3x then he would agree that f(4) = 1
The *internet argument* expression is wrong, because it's intentionally lacking context. The notation makes sense in most articles it is used, but could be clarified, which is what engineering standards like ISO 80000 directs. Casio did state that the reason they made regional models that don't prioritize juxtaposition over division (but not exponents, it isn't parenthesis) is that teachers of lower level maths insisted on it. Meanwhile, in higher level maths it's common to define new notation within an article.
Thank you! I'm honestly disappointed bprp did not say this. His answer is just perpetuating the issue.
OK fine thanks for the thesis 🤓
@@0LoneTech In higher level maths you will NEVER SEE ambiguity. The only time you'd EVER see something like a/bc in real maths is if the journal explicitly states that that is the convention that will be used. And even then, you are very unlikely to see it unless it's involving something like 2pi, where it's a commonly used number, usually a multiple of an irrational number. Otherwise, pretty much no one is going to write a/bc in a real publication. Not if they want to be taken seriously, or if the journal is too broke to format for a fraction bar or to print brackets (which isn't going to happen).
It's (D). The others at least require some mathematical thought. (D) is just dumb and is only an issue because people hate that particular division symbol and assume it means something that it does not.
no I'm pretty sure it's caused by multiplication by juxtaposition being weird on some calculators (i.e. PEMDAS vs PEJMDAS)
@@NOT_A_ROBOT Correct. Multiplication by Juxtaposition is still very common, so that's why we're getting different answers.
So it is 12/3 × (4), right? This sign creates confusion cus it is just next to a paranthesis, right?
@leaDR356 basically. Some people, especially old people learned to multiply the parentheses first so 12 / 4(3). Tbh thats how I learned it and managed to get a math minor... so it matter little when you are calculating on your own cause your not going to write it that way past 6th grade anyway
@@leaDR356 Not necessarily. It is perfectly valid to consider juxtaposed multiplication as higher priority than explicit multiplication or division. In which case the answer absolutely is 1.
D is the least interesting as it's mostly a question on syntax. The people who say the answer is 1, generally do so because they view 3(4) as implied multiplication, which has been taught (by some) to have higher precedence than standard multiplication (using the "x" or "÷" symbols). I wonder how the responses may change if we did some alterations to the question: 12/3(4)=? or evaluate 12/3x where x = 4? or what about: 12 ÷ 3π=? would you evaluate that as (12 ÷ 3) x π or 12 ÷ (3 x π) I'm not arguing for one or the other, it's just that I can see how people would find it ambiguous and I can see an argument for both sides. But all in all, it's just not an interesting problem.
D is not about syntax, it's about associativity of operators of same order. It appears to be "left associative" in USA, and "right associative" where I'm from
Completely agree
@@dfhwze The order is exactly what's in question. Division has been used as lower than multiplication, or specially lower only to the right (meaning the division itself could imply the answer). And implicit multiplication has been higher than explicit. All else was added to fuel the flames. E.g. the parenthesis are there to allow the numerals, which are there to get more people to weigh in without grasping the question.
So people really teach that that would have a higher precedence? That is just weird. That is just a sign people get this wrong because the school system sucks.
@@thomasdewierdo9325 I wouldn’t say that. I would say the fault lies in the person who wrote the question. And of course it was written that way purposely for the controversy.
D is not a math problem it's a notation interpretation issue So Why when it's 12/3X we interpret 3X as a number and not as 3 * X, Prioritising implicit multiplication is more consistent
there is no 'we interpret' such and such, this notation is ambiguous and I would personally ask for clarity if this was given to me, the use of '/' is something that is typically only seen online and therefore the correct interpretation is undetermined however if you were to say 12÷3x this notation is not ambiguous and clearly implies 12÷3*x
Yeah people need to realize that pemdas is just a convention and not a mathematical truth
12/3x=4x right?
Exactly why notation with ''/'' is limited and not typically taught in math classes. Regardless, no it is not more consistent at all. 'Left to right' is more consistent than 'left to right but implicit multiplication before division'...
@@aMyst_1 It can also be equal to x/4. If you use multiplication by juxtaposition. The problem is not that one answer is the correct answer, the problem is that there's no universally agreed upon way to interpret the problem
D is the worst because order of operations is an arbitrary linguistic layer applied over the top of mathematics and not itself pure mathematics, so all arguing about it in a mathematic context is inherently insipid.
Can you please consider making a wordless definition of the limit shirt? Meaning only quantifiers and other math notation exclusively. I would purchase it so fast, by far my favorite calculus topic!
I think D. The ISO 80000-2 standard for mathematical notation recommends only the solidus / or "fraction bar" for division, or the "colon" : for ratios; it says that the ÷ sign "should not be used" for division.
I think the issue with D is that there's disagreement about whether or not implicit multiplication takes priority over explicit division. I remember The How and Why of Mathematics made a couple videos about this debate, that I thought were really good.
What really bugs me is how people pick a position and start making stuff up about all other positions; in fact, this video is guilty of it. It presents it as "do the parenthesis first", but the parenthesis is only there to distinguish 3x where x=4 from 34. In a conversation, this could be a simple misunderstanding and be resolved. But when it moves to lecturing like this video, it's a straw man, misrepresenting the other position(s). I agree that The How and Why of Mathematics presented this well, including actual research.
Because for D multiplication by juxtaposition is often done first. Even in other fields you will catch scholarly papers dividing by stuff and not putting parentheses around their denominators.
PEJMDAS vs PEMDAS moment
@@NOT_A_ROBOTWhen you put it like that 👍🏻😆
It's literally the more natural way without calculators or computers to look at it. It's why it's common in older education
and it's a big mistake to ignore parentheses
@@adamwalker8777 The parentheses are incidental - they're only there to allow for juxtaposition between numerical values (à la “3a”). The contents of the parentheses are trivial: (4) = 4 So that has no bearing on the controversy. What's controversial is whether juxtaposition takes higher precedence than other multiplication and division; i.e. whether “xy” represents “(x ⋅ y)” or just “x ⋅ y”. Its simply a matter of notational convention - either way would work entirely consistently - but people's intuitions seem to differ, so it's best to make your intentions clear using brackets or fraction bars.
"Multiplication denoted by juxtaposition (also known as implied multiplication) creates a visual unit and has higher precedence than most other operations. In academic literature, when inline fractions are combined with implied multiplication without explicit parentheses, the multiplication is conventionally interpreted as having higher precedence than division, so that e.g. 1 / 2n is interpreted to mean 1 / (2 · n) rather than (1 / 2) · n."
What are you quoting? And what is said to be ''conventional'' in certain literature (kind of suspect this is not mathematic literature but rather scientific or engineering literature) does not change the actual rules of mathematics.
@@FrenkieWest32 That's not how that works, you just made that up. There is no rules of mathematics saying what should the mathematical notation look like. 12/3x = 4/x for people that say that implied multiplication is done before division, and 12/3x = 4x for people that say that implied multiplication is done with the same priority as division. There's nothing mathematics tells us about how it should be interpreted, math notation is made by humans and could be completely different
@@F_A_F123 What exactly did I make up? Your comment is dubious, ironic considering the topic. 'Rules' are made by people. I would not refer to the intrinsic nature of reality with 'rules'. Orders of operation in mathematical communication are 'rules' made by humans. Just how one can say it is a rule to use 'x' for your first variable, even though this is completely manmade. With all that said: yes, rules are not set in stone completely. And apparently there is more controverse around this than I thought.
@@FrenkieWest32They're quoting the Wikipedia page for the Order of Operations, but conveniently left out the part right after the quote saying that some academic authors advise against the form a/bn (the form in their comment) and say that you should use the much less ambiguous forms of (a/b)n and a/(bn). In the case of the video, that would give both (12/3)*4 and 12/(3*4).
@@zaleshomeowner3493 You made that shit up, I didn't look at wiki.
For D, i like to change parenthesis into the X. For some reason, no one will tell you that 12 ÷ 4x is 12 ÷ 4 × x
That's because 4x is a monomial. So you treat it the same way you would do 12 ÷ 4
That's why this debate is so stupid, NO ONE in real life would write 12÷4x if they mean 12x / 4
That's what confused me so hard. I thought it was basically 12/4x where you just simplify and get 3/x where x=3 so it's just 3/(3) and therefore 1.
Well 12/4x is exactly 12/4*x. What are you even talking about? I never thaught 1st year arithmetic is so hard
@@sychuan3729 No it is not, for no one writes 12/4x expecting it to be read 12x/4. No one. That doesn't happen.
I think the problem with D is that even with the same operation, it's usually implied that when the operation doesn't appear, it should be done first. e.g. when you write 12÷4a, you kinda want to do "4a" first. Obviously the whole thing with order of operations is just a convention. As a programmer, when I occasionally write math operations in the code, I often add parentheses which are technically redundant, just to make it clearer what is going on. e.g. I write var1+(var2*var3) instead of var1+var2*var3. They're technically the same, but the first is much easier to understand from a quick glance, and unlike what some people think, the point of writing things in math (and code) is to make it *easier* for other people to understand what they're reading. As for the debates, personally I think the worst debate is C. Debates A, B and D are just about conventions. You can define these things however you want, it's just for the sake of convenience, there's no hidden meaning there. Like, you could define square root to be a function that returns pairs of values. It would be less convenient to work with, but nothing would break. C is the only debate that is actually about the *meaning* of something, that actually shows a fundamental misunderstanding of what real numbers are and of how series work.
How is B about convention ?
I really like your point about notation. I always teach that notation in maths is like good punctuation in English. Your main objective is to communicate your intention to your readers not to be technically right but misleading - that's no good for anyone. Things like D only exist for the sake of it. The others are all real things to define or discuss.
@@yann8765 why pick the positive root and not the negative root as the single answer? It's just a convention, like rounding 7.5 up to 8 rather than down to 7. I mean it is sensible but not mathematically forced on us?
@@72kyle None is "picked", the result is ±1 ?
@@72kyle Rounding 7.5 to 8 isn't a convention either ; if it was rounded to 7, rounding would create a bias toward lower values (0,1,2,3,4,5 (so 6 digits) rounded below, but 6,7,8,9 (only 4 digits) rounded above. EDIT : I take that back, actually it seems to me that it is secondary to another convention, which is that when splitting a continuous interval, we tend to do it as [low, up) rather than (low, up]
Thank you - that has helped.
The problem with D is that it when you deal with variables or symbols (𝜋 etc), implied multiplication does take priority. Take this question from a recent GCSE maths paper for example: simplify 12x⁷y³ ÷ 6x³y. The correct answer is 2x⁴y² not 2x¹⁰y⁴. And nobody would see 1/2𝜋 and think it means 𝜋/2 instead 1/(2×𝜋)
A. Undefined unless convenient. B. Principal square root if we're applying the function (and we nearly always do); plus or minus if we're trying to find all solutions. C. Boring debate due to people not understanding how infinite sums work ("but they can't be equal because there's still a difference of .0000000000...001!') or who just outright deny the use of infinity for philosophical reasons (finitism) in the first place. D. Old clickbait tactic used to drive engagement via making people think they're arguing about mathematics when they're really arguing semantics. From my experience D is the least interesting because it's a semantic debate and almost always clickbait but people get really heated about C and, as annoying as it is, people never tire going at it about it because everyone is really convinced of the sense of their argument and the nonsense of the other side. Never really seen people argue about A or B. I think you should have put that approximation of pi meme there instead if you know what I'm talking about.
Tell that to any teacher I've ever had for B. If I didn't write down -1 as well I'd get points off.
Yep, D I'd say, write parentheses or it's your fault.
For B situation: Think of this: sqr(1-i), it has a principal value 1.098...-i*0.455... (equivalent of sqr(1)=1) and a second value: -1.098...+i*0 455 (equivalent of sqr(1)=-1. What value do you choose? Both are valid. So, we use the positive value in real numbers, but when we deal with complex numbers, we can't ignore the second one. So squareroot means 2 values, third root 3 values and so on. The fact that in real life we 1 only one value, that doesn't mean the other's doesn't exist.
"C. Boring debate due to people not understanding how infinite sums work ("but they can't be equal because there's still a difference of .0000000000...001!') or who just outright deny the use of infinity for philosophical reasons (finitism) in the first place." - That's a strawman argument. Those of us who object to this equality claim that you have never proved that 0.999... is a real number. If it is a real number, then, yes, it is most definitely equal to 1. But simply assuming that it's real is circular logic. One could just as easily assume that 0.9 recurring is how we write an infinitesimal (a hyperreal or surreal number that is infinitely close to 1, but less than 1, while at the same time being greater than every real-number-less-than-1).
@brendanward2991 and what level of precision requires an infinite number of 0's in front of it to be accurate? At some point all math is rounded to the number of significant digits.
D is a matter of multiplication by juxtaposition where 3(4) takes precedence over 3x4. It used to be taught that way 100 years ago, and it is coming back. Some calculators are programmed now to do #(#) before doing x / left to right. My calculator can be set to do it either way.
All math guides from professionals state that adjacency notation for multiplication is an implied parenthesis in the order of operations. For example: 12 ÷ 3n, when n = 4. 3n is pre-grouped single operand. 3(4) is the same as 3n, when n = 4. Similarly, fraction notation (which is division) will be performed prior to left-to-right order as an implied parenthesis.
D is the worst because even BPRP is wrong. The answer is that there is no consensus. Different publications disagree on the relative precedence of inline division vs implied multiplication.
Hence why only idiots ever even write that kind of stuff unless the entire reason is the generate clicks.
Exactly. The answer is either "both" or "need more context" (or so forth). The fact that both "12 / 3 * 4" can show one answer while "f(x) = 12/3x" can show another is the whole issue. They're the same equation, but will result in different answers. If someone doesn't recognize that and instead consider it to be a single answer (either exclusively 1 _OR_ 16), that is what is incorrect.
That's why the ÷ symbol shouldn't be taught in school. Use fractions for everything
@@ThomasTheThermonuclearBomb It's still an issue if you use / for division, which some publication style guides recommend if it makes a nested fraction easier to read.
@@codahighland If you must use the / symbol, parentheses are absolutely necessary on both sides like 1/(3+4) or (1/5)*6
0^0 = 1 since 0^0 is the set of all maps from the empty set to the empty set, where there is exactly one such map. It's also the IEEE standard. The often-made flaw is to assume that x^y requires to be continuous, and then argue with lim.
Using limits to prove the value of a function just shows that the person does not understand limits and functions. 0^0 = 1
C is definitely the worst because there’s no debate. one side is objectively wrong
But the only one which is kinda a debate is A The other ones all have objectivly right answers too
ikr everyone is so dumb
In the usual construction of the real numbers, its obviously 1, yes.
Maybe you should do a video about the Monty Hall Problem. Pretty sure that might come in pretty high on your list of "most debated math topics" if you kept getting comments about it :P
Good rule of thumb for square roots, if you introduce the first square root is +/-. If the question gives you the square root, usually it’s just +
I believe your answer for D deserves an "Incomplete" mark. It's correct, inasmuch as PEMDAS is the final word on order of operations. But PEMDAS is *not* the final word on order of operations everywhere. In some contexts "multiplication by juxtaposition" (eg,, signifying multiplication by just putting two objects next to each other) is given a higher precedence than regular multiplication. Don't believe me? Spend some time going through the manuals for many different calculators, preferably of different brands and sold in different countries. You'll find that each manual (usually) has a whole section on "order of operations" and that some do assign higher precedence to "multiplication by juxtaposition" and some do not. Also note that in the submission guidelines for many scientific/mathematical journals, authors are instructed to observe the convention that "multiplication by juxtaposition" takes precedence over regular multiplication. I would argue that D should also get a "no agreement" answer on this basis.
A: maybe also add 0 to the options B*: ³√(-1) (principal root vs real root) B**: arcsin(2) D*: 2x / 2x (options: 1 or x²) to see who picks (2x/2)x instead of (2x)/(2x) by that "order of operations" E: f(x)ⁿ for f that can be written without brackets (e.g. sin(x)² ln(x)³) E*: f⁻ⁿ(x) for f which fⁿ(x) is used as (f(x))ⁿ and f⁻¹(x) being an inverse function F: any notation which limit exists but written without limit (e.g. x³/(5x-x²) at x=0) (whether it equals a value when it gets only one possible value) F*: sum of divergent series (e.g. 1+2+3+...) F**: step function at 0 G: non-integer factorials without using gamma function (e.g. i! , 3.5!) G*: analytic functions / continuations H: Division as inverse (e.g. matrix, modular arithmetic)
Impressive list of ambiguities.
B*: + set of all complex roots
D**: =2^2^2^2 D***: =−2^2
Funny part:: • ²3² • !3! WA somehow assigns a value to the latter.
Follow-up to B*: do irrational equations such as ³√x = −2 or ³√(x−6) = x have solutions over the field of complex numbers, and if so, what definition of the cube root should we use.
0^0 = 1 and I will die on this hill. (1) One definition of n^m is the product of exactly m copies of n. However, I wouldn't consider this a rigorous mathematical definition. Instead, using recursion: n^(m+1) = n*n^m. But recursion relies on a base case. You could start with n^1 = n, but there's no contradiction in starting with n^0 = 1. To leave n^0 undefined is simply avoiding a special case for the sake of unnecessarily leaving it undefined. (2) The set theory definition of exponent is that n^m is the number of functions from a set of m elements to a set of n elements. There is exactly 1 function from the empty set to the empty set, so 0^0 = 1. To leave 0^0 undefined, I would want to know the (rigorous) definition of exponents being used. (3) Just as an empty sum is assigned the value of zero (the additive identity), it makes sense to assign an empty product (such as n^0) the value of the multiplicative identity. (4) The binomial formula, power series, and the general power rule in differentiation rely on 0^0=1. Leaving it undefined makes these theorems (and applications) unnecessarily complicated. To address the main points why 0^0 is left undefined: (1) f(x)=0^x=0 for x>0. This function is discontinuous at 0, and there's no fixing it (except possibly right-continuity, but 0^0=0 would lead to other contradictions). Unlike defining 0/0 to be a real number, defining 0^0=1 does not lead to a contradiction. There are many instances where having a base of 0 leads to an exception to a rule (e.g. rules of exponents, x^-1=1/x). In that regard, 0^0=1 being yet another exception isn't a big surprise. (2) 0^0 is an indeterminate form. However, the indeterminate forms are NOT directly tied to arithmetic calculations. The reasons why 0/0 (arithmetic) is undefined (in reals) are well established--any definition would lead to a contradiction. However, the indeterminate form of 0/0 is not undefined, it's indeterminate in that analysis is necessary to determine the value of the limit, rather than the arithmetic value of 0/0. Having 0^0=1 does not lead to a contradiction here, just another exceptional case.
0^0 = 1 and gcd(0, 0) = 0 are my favorite things in math that look really wrong at first but if you look into it have some justification for it
@@aioia3885 Also, mod(n, 0) = n.
THANK YOU
I don't recall seeing anything like #4 in any of my math classes after elementary school. I remember learning the division symbol in like grade 1. Then probably after grade 6, it is never used again in school or pretty much anywhere. Only time the division symbol is used is maybe some skill texting question in draws/raffles (because of some Canadian law which I won't go into) or in some internet puzzle designed to confuse people.
The order of operations for (D) has nothing to do with giving priority to parenthesis. 3(4) is implied multiplication, not explicit multiplication, and does take priority in this expression so the answer is in fact 1. The confusion with this problem is the result of calculators designed for the United States vs the rest of the world. Calculators designed for the US market will allow you to enter an expression using implied multiplication, but will auto correct and add the multiplication sign making it explicit multiplication when you hit "=". Calculators designed for the rest of the world will not add the multiplication sign and give priority to implied multiplication over division and explicit multiplication (As stated in the order-of-operations section of their respective instruction manuals). If you want proof, enter the expression exactly as it is written into any calculator designed for the global market and you will get the answer of 1. Enter the expression into a calculator designed for the US market, and if the calculator will display what you entered as well as the answer, you will see that it adds the multiplication sign and returns the answer of 16. This is not because we do not give priority to implied multiplication in the US. We in fact do just like the rest of the world. The mindset is that you should not (and usually can not) use implied multiplication when programming, which you are essentially doing on a calculator that allows you to enter the entire expression before hitting =. Matlab for example will not let you use implied multiplication. In any text book written for the US, implied multiplication does take priority and you see this all the time with coefficients. For example; 1 / 2y would never be interpreted as 2 / y. 2 / y would be 1 / 2 * y which uses explicit multiplication and has the same priority as division but the order is left to right.
Hello, A - I agree B- depends if we are in R or C. +1 in R and +-1 in C. If the context is unknown, I would tend to +-1, because C is "better" 🙂 C- I agree. D - this depends on the agreement. 3(4) is implicit multiplication, like 2X. Some systems (calculators) give higher priority to implicit multiplication than the normal one. And I think it is correct. Otherwise you can also say, that sin 2X is sin(2) * X == X * sin(2)
No 2x is a monomial. You treat it like a single number. You do NOT treat 3(4) like a single number. Basic stuff.
For D I recently learned that there are some conventions where multiplication by jusxtaposition (when you have a parenthesis adjacent to a number, with no explicit multiplication sign) comes before both regular multiplication and division. Some calculators even have this as a part of their order of operations. Under this order of operations, PEJMDAS, the correct answer would be 1, but most people don't use this order of operations.
How would you evaluate: 12÷3x It feels very wrong to say this is (12÷3)x now that it's a variable, juxtaposition to me seems like "treat this as one entity" Edit: So the debate basically reduces to whether 3(4) is implied multiplication like 3x or normal multiplication like 3*x. Implied seems like a better convention to me 🤷
They're essentially the same if only one term has a variable.
3X is implied multiplication, 3(X) is not, 3(X) is just normal multiplication, following the syntax of multiplication (left to right) it would be ((12/3)*4). Some people might argue that 12/12=1, 12/(6+6)=1, 12/3(2+2)=1, 12/3(4)=1 but that's not how it works. (6+6) Is after a division sign meaning the value of (6+6) is inverted into 1/12, by pulling (3(4)) out of the parentheses you would have to invert the equation into 12/3/4 and using the division syntax (left to right) you get ((12/3)/4) which is 1, it would be written as 12/12 = 12/(3(4)) = 12/3/4 = 1
@@arno_grnfld455 in my entire career, 3(4) has always been implied multiplication, no different than 3x.
@@arno_grnfld455What is "implied" vs "normal" multiplication? It's juxtaposition in both cases. The reason we add parentheses to get 3(4) instead of just writing 34 is because 34 is a different number.
@@TheUnlocked no, 3(4) is normal explicit multiplication, you can shift the 3 around and multiply it for example, 7(x+y)*6(-2x+y) = 42(x+y)(-2x+y). It is not tied down to a variable like how 3x is (3*x) (implicit multiplication) Implicit or implied multiplication is like 3X where 3 cannot be seperated or shifted around, e.g. 3x/4 ≠ 3/4x, the 3 (or 4) is glued to the variable like: (3*X)/4, 3/(4*X), if 3(4) is implicit multiplication, you'd write it as (3*(4)) not 3(4) which is just normal multiplication. In this case 12/3(4) would follow the normal Syntex of left to right, (12/3)*(4), if 3(4) is implicit multiplication, 12/3X, X=4, it would work like 12/(3*(4)) instead
D is the worst debate because it just a notational thing that boils down to how we want to define it, there are no deep secrets hiding within this. B I have never heard of and is also just notation.....
The fact there’s people on this video that say “it depends on context” or whatever for D is so dumb
It may depend on a tool you are using to evaluate the expression, and conventions may vary across different tools, so, yeah, it sort of depends on context (of conventions).
Some advanced calculators (e.g. CalcES aka Scientific calculator plus 991on Android) simply let you choose how you want them to treat implied multiplication: as *1/2π = 1/2∗π* or as *1/2π = 1/(2∗π).* And letting users choose the way they want their implied multiplication being evaluated is a wise decision - it sort of solves the problem by providing both available options.
In a similar fashion, the value of =2^2^2^2 or =−2^2 also depends on conventions being used, and different tools may give different results, And some tools will simply refuse to return a value for the second expression, stating it is ambiguous and "parentheses must be used to disambiguate operator precedence" - exactly the case with the expression from the video.
It's the correct response though. It's simply ambiguous notation. A trick. Academically, multiplication by juxtaposition implies grouping but the programming/literal interpretation does not. That's the issue. You can't prove either answer since it comes from notation conventions, not any rules of maths. Wolfram Alpha's Solidus article mentions the a/bc ambiguity and modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. Even over in America where the programming interpretation is more popular, the American Mathematical Society stated it was ambiguous notation too. Multiple professors and mathematicians have said so also like: Prof. Steven Strogatz, Dr. Trevor Bazett, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter, Eddie Woo etc. Even scientific calculators don't agree on one interpretation or the other. Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation. Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation. TI later changed to the programming interpretation but when I asked them were unable to find the reason why. A recent example from another commenter: Intermediate Algebra, 4th edition (Roland Larson and Robert Hostetler) c. 2005 that while giving the order of operations, includes a sidebar study tip saying the order of operations applies when multiplication is indicated by × or • When the multiplication is implied by parenthesis it has a higher priority than the Left-to-Right rule. It then gives the example 8 ÷ 4(2) = 8 ÷ 8 = 1 but 8 ÷ 4 • 2 = 2 • 2 = 4
I have a question (of sorts) about D. In math classes, we always replace x by its value in parentheses. So I always interpreted it as inseparable from the number associated with it. Is it just set up badly in math classes? If you were to do it left to right then you wouldn’t get the right answer for if you filled in “12/3x” with 4 in the x place (making it “12/3(4)”
Thank you for this.
With part b, its important to note that sqrt(x^2) is abs(x) BY DEFINITION. Thats where the plus or minus comes from. From "cancelling" the absolute value. Too many people believe it's just voodoo, which you kind of lend credence to by saying "oh, its when we solve this equation." NO! Sqrt(x^2) = abs(x). That is where the plus or minus comes from. Please bprp, I rely on you to note this kind of nuance since you are an authority, so i can point people to your videos when people get real resistant to being told they are wrong.
The definition is actually not a mere abs, the y-th root of x is the z with minimum principal argument that solves z ^ y = x. It just happens to be abs when dealing with non-negative reals. But, for example, cbrt(-8) is not -2, unless you are restricted to real numbers.
Every problem needs to say x ∈ ℍ or x ∈ ℝ or whatever. If it doesnt then whoever wrote the question wrote it for a classroom, not for youtube or for the real world.
Think of this: sqr(1-i), it has a principal value 1.098...-i*0.455... (equivalent of sqr(1)=1) and a second value: -1.098...+i*0 455 (equivalent of sqr(1)=-1. What value do you choose? Both are valid. So, we use the positive value in real numbers, but when we deal with complex numbers, we can't ignore the second one. So squareroot means 2 values, third root 3 values and so on. The fact that in real life we 1 only one value, that doesn't mean the other's doesn't exist.
The identity sqrt(x^2) = abs(x) is not a definition, it's a provable property.
For (B) I think its important to point out that the ± is evaulated separately from the radical. We can see its OUTSIDE the radical, so it is its own thing, done after you get the radical's output. The radical only gives one output because it is a function. That output is defined as the principal root. See: ±√2 means plus and minus the (principal) square root of 2. √2 = ±... would be wrong.
The way I see C is, 0.999... is equal to 0.333... + 0.333... + 0.333..., which is essentially 1/3 + 1/3 + 1/3, which equals 1. (It took me a **very** long time to accept that, by the way.)
D is just the least interesting.
I found out there is no agreement for D. Much of the world, especially academia and Europe, follow the juxtaposition of multiplication which gives 1 because multiplication is implied. North America, especially teachers in the U.S., follow strict PEMDAS, which gives 16. Even calculators don’t agree between the two and some have been known to document the order they use and some companies have been known to change between the two orders over time (and back). In short, don’t write notation like this. It’s confusing and I swear it’s used just to start flame wars. And if order is confusing, add parentheses for clarity.
A is a matter of what branch of math you're working in B is a matter of following a definition of a radical function C is a matter of understanding this in term of limit question D is a matter of confusing notation I'm thorn between B and D as being the worst of those 4
The notation on D is not ambiguous. Some people are confused by it because they either cannot or refuse to understand basic order of operations, or because they incorporate misguided assumptions into the calculation.
@@anewman513 okay, the notation is technically not ambiguous, but it's for sure impractical. there are better ways to write down the same arithmetic operation without causing so much confusion about something that hardly has anything with the math itself
@@anewman513 It is ambiguous because some countries basically teach PEJMDAS by telling students to only remove operators if things are "grouped" So someone who learned that would see 12/3(4) as 12/(3(4)) because omitting the multiplication is an implied grouping. For variables most people do this most of the time. Usually when I see someone write 1/2x they don't mean (1/2)x.
@@anewman513 Multiplication by juxtaposition is still commonly used, so there are two answers.
Think of this: sqr(1-i), it has a principal value 1.098...-i*0.455... (equivalent of sqr(1)=1) and a second value: -1.098...+i*0 455 (equivalent of sqr(1)=-1. What value do you choose? Both are valid. So, we use the positive value in real numbers, but when we deal with complex numbers, we can't ignore the second one. So squareroot means 2 values, third root 3 values and so on. The fact that in real life we 1 only one value, that doesn't mean the other's doesn't exist.
Pretty interesting PROF thanks 👍
I was taught to do PEMDAS left to right one by one, I never knew you combined PEMDAS into PE(MD)(AS).
D) you've been miscommunicated. We don't multiply the parenthesis first because we "misheard" doing the inside first, afaik that's just your strawman. The actual reason is due to juxtaposition which is considered of higher order than a multiplication dot •. Example: 12÷3x vs 12÷3•x where x = 4 The first statement has a juxtaposition of 3 and x, wheras the 2nd statement has a multiplicative dot between 3 and x. Thus the first statement is 12/12 = 1 and the second is 12/3*4=4^2=16 Now do this with a parenthesis instead as you can juxtaposition those as well 12÷3(4) = 1 ≠ 12÷3•4 = 16
Or you can just view the equation as a fraction: 12 - 3(4)
@@jasonnelson9141 EXACTLY just use fraction notation, it clears up all of the confusion.
They are both 16 for the exact same reason. You do it from left to right. 12÷3x=12÷3×x=4×x
@@MrPassigo But 3x is one term. 3(4) isn't, so the notation is ambiguous.
I literally forgot how to write the divide symbol when tested by my friend a few days ago. This just shows how trash it is.
Another reason for defining 0^0 to be 1 is that in general for non-negative integers m and n, m^n is the number of functions from a set with n elements to a set with m elements, and there is 1 function from the empty set to the empty set.
D is less of a math issue and more of an issue with how some people interpret the syntax. It’s not that bad. The first one is also an easy hit or miss because like you said, there’s no agreement. I can’t decide whether I hate B or C more. I’ll say C only because that not equal 1 is a mistake I can’t understand how they’d make (if they are a beginner the +-1 on B is an easy mistake) but 0.99999… not equaling 1 is not justifiable in my eyes (at least not comparatively) so it’s the one I dislike the most. B is a close second however.
No agreement means undefined. So easy. Then in particular cases you can use it however you want (once you DEFINE it).
@@user-os4lj3pi4q exactly, it’s hit or miss depending on how you define it.
The problem isn't how people interpret the syntax. It's how they leap from there to "any other interpretation is wrong, including the author's".
Most of us who talk about math online will type “1/2n”, even though by the order of operations we should write “1/(2n)”. The order of operations is a convention to reduce parentheses and simplify communication. If you write 12 ÷ 3(4) and what you intend is 16, you are communicating poorly. Even people who know the order of operations will rightly wonder what you meant. Making people wonder what you’re talking about is not a sign of good writing. The great thing about parentheses is that there’s no problem throwing in an extra one to clarify your meaning. I would write (12 ÷ 3)(4) because I care about my readers and am not interested in confusing them. But be warned: if you type 1/2n into a graphing calculator, you’ll get the wrong answer. So don’t get me wrong: it’s important to understand the order of operations! But it’s also important to communicate clearly.
That's why JavaScript, when asked how much is -2**2, simply returns "Uncaught SyntaxError: Unary operator used immediately before exponentiation expression. Parenthesis must be used to disambiguate operator precedence"
I'm watching your videos for many days. Very Informative 👍. Love ❤ from India 🇮🇳
÷ and / should be always replaced by division line. We should not use (...) for only one digit or varible - let's use () for expressions.
Option D is not so clear. The KZhead video "PEMDAS is wrong" by "The How and Why of Mathematics" tells some examples why multiplication by juxtaposition should be made before division (at least slashing fractions). It is indicated like that in an article by the AMS, another Physical Review Style and Notation Guide, and is usually used when using x, like 1/2x, which is interpreted as 1/(2x) and not x/2.
The debate surrounding D is pointless when you realize that fraction notation exists. By just using fraction notation it gets rid of all ambiguity.
Saying D) isn't the worst debate out of those is the same as saying that "I saw a man with a telescope" is worth discussing whether it's as seeing a man through a telescope or as seeing a man holding a telescope. It's just ambiguity and the flaw is found on the very phrase
Obviously D is stupid, but in this problem it’s still clear enough where there is a commonly accepted answer. There are some that are like 5/2a, which would be a better debate, but this one is at least somewhat clear. B, in the other hand, is stupid because literally both answers are correct. Like the square root is one is both 1 and -1 because both of them square to 1, but at the same time, it’s commonly accepted that a singular number inside the radical is looking for a single number when solving. Sure, +- 1 is technically correct, but that doesn’t matter. If somebody wants to know the square root of 49, it doesn’t help that it could also be -7 because negatives aren’t as useful in basic math, so we ignore that solution. Either is completely correct, so it’s a complete and utter waste of time arguing. At least arguing over syntax has a reason, because if an accepted solution is found, it will solve arguments in the future. Arguing over solutions to a square root just don’t matter. Like who cares whether or not cereal is a soup, it can be both. Colloquially, we define it separately so it’s not the same thing, but going by definition, it would be. All these stupid arguments just end up creating more confusion than they solve, so I think B is the worst, though D is a close second.
@@Chris.McNicholsyou could of just said the answer was 1..
@@w1111-vs3dd True, but I could’ve also said the answer was +- 1, which is the problem with the debate
Obviously it means a telescope and you visited a man together. :p
If you manage to somehow get to a point where you have 0^0 you honestly don't deserve an answer
I agree with you for all of them. For (A) I'm Team 1. I started to disagree with you about (B) but as you explained your reasoning I came around. As far as which disagreement is the worst, it depends on what is meant by "worst."
Flip the question: which is the best debate? I am prejudiced to computer science; but, my favorite is "Does P = NP?" (Does the set of problems solved in polynomial time equal the set of problems solved nondeterministically in polynomial time?)
You're a computer scientist? Then you'd agree that 0!=1 🤣
@WombatMan64 0! Is defined to be 1.
@@bullinmdYou missed the programming joke. != being used to mean "not equal to" in most programming languages. Therefore of course 0!=1, zero is not equal to 1. And yes, maths people will interpret it as zero factorial, which is equal to 1.
Hi very clear, 2 comments: (1) I think it depends on the type of 0. If we are talking about finite math: combinatorics, graphs etc, and dealing with integers or natural numbers, then 0^0 will always be 1. But if we are talking about reals / complex numbers, then 0^0 usually is defined to be the limit of something, and the result may be undefined. It's not that there is no agreement on the math: instead the generally agreed meanings vary depending upon the context. (4) If I write: 12 - 8 + 8 - 3 - 9, then there is no confusion. We just evaluate from left to right to reach the answer 0. It can be just the same for multiplication/division: 12 / 3 * 4 / 2 / 2 = 4. There's nothing different structurally going on. This also means that people (including YOU! heh) should feel free to write 27 / 3 / 3 = 3. It's certainly very convenient, but for some reason there is a taboo on this that most people aren't even aware of.
I'm pretty sure only the last one is an actual factual problem, the problem is the poor notation. A single symbol within () is no operation, that denotes a group. Very different meaning between 12÷ 3(4) and 12÷ (3*4)
If D was written in algebraic form, a÷b(c) = a/bc, because (c)=c. Since a=12, b=3, c=4, this expression evaluates to 1.
nope, a/bc == (ac)/b using elementary order of operations. Doesn’t matter what form it’s in, a/bc means a* 1/b * c, unless specified otherwise with brackets
@@adw1z Almost all professional mathematicians would interpret a/bc as a/(bc) because if you wanted (a/b)c, you could write it as ac/b instead.
@@adw1z In algebra, juxtaposition of terms implies that they're multiplied together FIRST, so a/bc always means a/(b*c).
@@MuffinsAPlenty firstly, no professional mathematician does that (this video, for example) Secondly, no professional mathematician would ever write such an expression without using brackets anyways Thirdly, I have no idea what this juxtaposition business is, it doesn’t exit and people are making it up like some new order of operation a(b) is equivalent to a*b, no more or no less priority. Order of operations will always mean a/bc == a * 1/b * c
@@adw1z The reason I say what I say is because I have actually looked. And I'll provide evidence for you. If you can find some evidence of textbooks (designed for the college level or higher) or peer-reviewed research articles in mathematics which interpret expressions like "a/bc" in the way you have said, I would love to hear about them. Because, based on my collection, I haven't seen a single one that does that. Anyway, here's a list of references of _textbooks._ I have included every instance I could find from my textbook collection where implicit multiplication (multiplication without an explicit multiplication symbol) immediately follows a division symbol. I have excluded things like "d/dx" from calculus and "M/IM" from abstract algebra, as "(d/d)x" and "(M/I)M" are nonsense expressions, meaning there is only one reasonable way to interpret "d/dx" and "M/IM" in those contexts. In _Commutative Ring Theory_ by Hideyuki Matsumura (Cambridge Studies in Advanced Mathematics 8), on page 21, we see how to multiply fractions together with the equation "(a/s)∙(b/s') = ab/ss' ". It is obvious that ab/ss' is meant to be interpreted as ab/(ss') and not as (ab/s)s'. In _Representation Theory of Finite Groups: An Introductory Approach_ by Benjamin Steinberg (Universitext, Springer), on page 123, we see the set of all tabloids of shape (λ₁, ..., λ_ℓ) [sorry, can't get a subscript cursive lower case ℓ] with entries in {1, ..., n} is described as having cardinality "n!/λ₁!∙∙∙λ_ℓ!". It is obvious based on combinatorics that this expression is meant to be interpreted as n!/(λ₁!∙∙∙λ_ℓ!), and not as (n!/λ₁!)∙∙∙λ_ℓ! In _Mathematical Statistics with Applications_ (7th ed) by Dennis Wackerly, William Mendenhall III, and Richard Scheaffer (Thomson Brooks/Cole), on page 205, they go through a calculation of the moment generating function of a random variable with mean μ and variance σ². Throughout much of the text, they are careful to avoid ambiguity by always writing the exponent in the normal distribution as −(y−μ)²/(2σ²). However, in this derivation where things get messy, we see (1/2σ²) and −(u− σ²t)²/2σ², where the parentheses in the denominator are dropped. In _Commutative Algebra with a View Toward Algebraic Geometry_ by David Eisenbud (Graduate Texts in Mathematics 150, Springer), on page 59, fractions of ring elements are defined with a product of two fractions defined as "(r/u)(r'/u') = rr'/uu' ". Again, it is clear from context how this latter expression is to be interpreted. In _Topology_ (2nd ed) by James Munkres, on page 28, he defines the set "{−1/2n | n ∈ ℤ₊}" and declares that this set has no largest element in (−1,1), but does have a least upper bound of 0 in (−1,1). (Note that on page 26, ℤ₊ is defined to be the set of positive integers.) If we were to interpret −1/2n as meaning (−1/2)n, then the set defined above would have a largest element in (−1,1), which would be −1/2 (in fact, this would be the _only_ element of the set in (−1,1)). However, interpreting the expression as −1/(2n) allows the set to have elements arbitrarily close to but not equal to 0. In _Introduction to Elliptic Curves and Modular Forms_ (2nd ed) by Neal Koblitz, on page 34, we see the following quote when Koblitz is giving a formulaic definition of the group law on an elliptic curve, particularly when adding a point to itself: "In the latter case, we can express m in terms of x₁ and y₁ by implicitly differentiating y² = f(x); we find that m = f'(x₁)/2y₁." From actually performing the computation, it is clear that f'(x₁)/2y₁ is meant to be interpreted as f'(x₁)/(2y₁) and not (f'(x₁)/2)y₁. In _Knowing the Odds: An Introduction to Probability_ by John Walsh (Graduate Studies in Mathematics Volume 139, American Mathematical Society), on page 52, we have the expression "2ℓ/πD" after having written in math mode a few lines earlier using \frac{2 \ell}{\pi D}. In _Real Analysis_ (4th ed) by Royden and Fitzpatrick, on page 117, the function f(x) is defined piecewise with the first piece being "x cos(π/2x) if 0 < x ≤ 1" and the second piece being "0 if x = 0". First, note that if π/2x were supposed to be interpreted as (π/2)x, there would be no reason for a second "piece" here, since x cos(π/2x) would be well-defined and equal to 0 at x = 0 taking the the specified interpretation. So we should, instead, interpret π/2x as π/(2x). In this same example, the partition Pₙ is defined as {0, 1/2n, 1/[2n−1], ..., 1/3, 1/2, 1}. It is clear from context using decreasing denominators that 1/2n here is supposed to be interpreted as 1/(2n), not (1/2)n. In _Introduction to Real Analysis_ (3rd ed) by Robert Bartle and Donald Sherbert, on page 103, the example is given that lim(x→0) sin(1/x) does not exist. The function g(x) is defined as sin(1/x). Then the sequence (xₙ) is defined as xₙ = 1/nπ. It is then claimed that g(xₙ) = sin nπ. The only way that g(xₙ) = sin nπ would be correct is if 1/nπ is interpreted as 1/(nπ). In _Basic Complex Analysis_ (3rd ed) by Jerrold Marsden and Michael Hoffman, on page 84, they start a proof, and say, "By definition, sin z = (eᶦᶻ−e⁻ᶦᶻ)/2i".Just knowing what the definition of the complex sine function is (or using Euler's formula), we can see that the correct interpretation is (eᶦᶻ−e⁻ᶦᶻ)/(2i), not ((eᶦᶻ−e⁻ᶦᶻ)/2)i. In _Elementary Differential Equations and Boundary Value Problems_ (9th ed) by William Boyce and Richard DiPrima, on page 100 (section 2.6 exercise 19), we find the problem asking us to show that the differential equation is not exact but becomes exact after multiplication by the integrating factor. It gives us the differential equation x²y³ + x(1 + y²)y' = 0 and gives us the integrating factor μ(x,y) = 1/xy³. If we interpret 1/xy³ as 1/(xy³), multiplying both sides of the diff eq by this integrating factor gives x + (1 + y²)/y³ * y' = 0. Notice that the partial derivative of x with respect to y is 0 and the partial derivative of (1 + y²)/y³ with respect to x is 0; hence, the equation is exact after multiplying by this integrating factor. On the other hand, if we assume 1/xy³ is to be interpreted as (1/x)y³, then multiplying both sides of the diff eq by this integrating factor leaves xy⁶ + (y³ + y⁵)y' = 0. And now notice that the partial derivative of xy⁶ with respect to y is 6xy⁵ whereas the partial derivative of y³ + y⁵ with respect to x is 0. Since these two functions don't match, the resulting differential equation is not exact. Based on these computations, we must conclude that 1/xy³ is intended to be interpreted as 1/(xy³). In _Introduction to Commutative Algebra_ by Michael Atiyah and Ian MacDonald, on page 36, we get a definition of multiplication of fractions as "(a/s)(b/t) = ab/st." Once again, this latter expression is clearly intended to be interpreted as ab/(st). There is one text I saw which avoided the ambiguity by placing parentheses in all such situations. In _Contemporary Abstract Algebra_ (7th ed) by Joseph Gallian, on page 286, he defines multiplication of fractions as "a/b ∙ c/d = (ac)/(bd)", but also on page 317 in exercise 10d, he writes the polynomial "(5/2)x⁵ + (9/2)x⁴ + 15x³ + (3/7)x² + 6x + 3/14". Both of these expressions, if written without parentheses, would allow him to pick one convention over another, but he avoids doing so. There was not a single textbook I had where I could find an example of inline text where implicit multiplication directly followed division, but where division took precedence over the implicit multiplication was the correct interpretation, based on context. If anyone has an example, I would love to hear it! An honorable mention is _Linear Algebra and its Applications_ (2nd ed, instructors edition) by David Lay on page 356 in exercise 22 of section 5.7. In one of the matrices, −1/(RC) is written instead of −1/RC. I found no instances of implicit multiplication following division in inline math without parentheses (but I could have missed something).
C) I think people have this notion that a real number only has ONE decimal representation. And I find that very understandable. Pretty sure that's the whole crux why people argue about this at all. The idea of 1 = 0.999... would break that notion.
You may be right about that!
-quietly cackling about p-adics -_-actually-_- having only one expansion for every number-
D is interesting because we're talking about the parse tree and which needs to be evaluated first. If I was writing that in C, if you looked at 12 / 3 * 4, you'd have to know how it'll be parsed and evaluated which isn't obvious at first. I can never remember. In the interests of clarity for my future self and my coworkers, I'd write it as (12 / 3) * 4. I wonder if teaching parsers would help people understand PEDMAS. :)
Now if only people would have that interesting discussion instead of screaming "you're wrong" at each other.
Yeah but it's 3(4) Not 3*4... At least im Europe 3(4) means (3*(4))
Relative to D: please evaluate 12÷3x for x=4. The variable x and it's coefficient 3 are so tightly coupled that most people will interpret this expression as equal to 1.
As I understand it, much of the confusion around D comes from the publishers of older math text books using that divide symbol to mean do the operations on the left and right before dividing because typesetting equations and printing them to look nice as a fraction was expensive. I believe they would note somewhere that it was a non-standard usage of the symbol, but of course no one notices those things.
It's more that it has become much easier to publish textbooks that don't cover this topic, so people feel confident pointing at one (of many) rephrased introductory tutorial as authority. When one thorough book was all we could reasonably manage, that book needed to define what it used, even if only by context.
if anyone wrote D on an exam or in a paper, I would consider that reason to flunk them. (They have misunderstood the central requirement in math to explain yourself clearly, not just be technically right.) I'd also like to offer this counterpoint: 12 / 3x in this case it's fairly obvious that the x should go in the denominator, but it's actually the same rule as in (D), an implicit multiplication.
This is why I would guess c as the worst debate... because there is no debate
As far as I can tell, the only actual debate is whether 0.333... = 1/3. Because once the naysayers inevitably are forced to agree, the debate is over.
Suppose you're throwing a dart at the real interval [3,4) , and that the outcome X (= the number where your dart lands) has a uniform probability density across the interval. Then Pr(X ≠ 5) = 1 (as 5 lies outside the interval, hence it's impossible to be the outcome), but Pr(X ≠ π) = 0.99999.... (as it's possible, though highly improbable, that the dart lands on the number π = 3.14159... ; the chance that the first decimal of X and π don't match is 0.9 , the chance that at least one of the first two decimals of X and π don't match is 0.99 , the chance that at least one of the first three decimals of X and π don't match is 0.999 , etcetera.) Since the probability Pr(X=π) must "clearly" be greater than the probability Pr(X=5) (as the event X=π is possible while the event X=5 is not), the probability Pr(X ≠ π) must "clearly" be less than the probability Pr(X ≠ 5) . Therefore, I think it can be argued that 0.99999... does not equal 1 .
@@yurenchu 0.9 repeating is just another form of 3/3, due to base 10 numbers not being able to resolve 1/3. If 1/3 = 0.333... and 1/3 * 3 = 3/3, and 0.333... * 3 = 0.999... then 1 = 3/3 = 0.999... There's no odds or probabilities involved. Equations overcomplicate the whole thing. Think of it like a display glitch in our number system. It's not that it's "close enough" or being rounded or anything like that. It's literally equivalent, just displayed differently.
@@yurenchu Actually the probability of X ≠ π IS 1. Read about the concept of "almost surely". The probability that the dart will land exactly on pi is 0, although the set of points it could land for that result to be true is not empty. It's just weirdness that comes from probabilities with infinite sets, but mathematically, you would say that P(X ≠ π) = 1. Of course, you are always encouraged to note that that 1 is representing an "almost sure" probability and not a certain one. But your argument actually ends up helping the case for 0.999... = 1, because when you know about the concept of almost surely, you know that P(X ≠ π) = 1, and your creative way to calculate the probability by each decimal, also gives the 0.999..., so they have to be equivalent.
What about the debate over the order of operation for powers?
Thank you for giving all correct answers! Personally while I believe that 0^0 is overall undefined, I have no problem with setting the convention of 0^0=1 if your working in an area of math where it always will be 1, which is a whole lot of common areas to work in.
I don't understand the last one, why is 3(4) not prioritary if something like 3x would be calculated first? It seems that just changing the 4 for an x will totally change the order. Maybe I have been tought wrong but I would solve 12÷3x as if 3x was in parentheses because I have been thought, even if that may be wrong, that implied multiplication by parentheses are prioritary
If I have a items that must be divided into b lots of c people, then we are looking for x per c to get how many items each person gets... a ÷ (b × c) = x/c. If I have to figure out how many items I need to give c people so that for every lot of a, each person would get b, then the required items are x... (a ÷ b) × c = x. When given 12÷3(4), there is an assumed "= x." The second solution is therefore correct, since we simply assign the values of a, b, and c to the equation as shown. Since we are looking for x and not x per c, then we get 16 instead of 1.
@@theJACKHAMMER13 this got me even more confused, whynis adding =x changing anything?
@Draconic404 Basically, we have a known quantity in 4, so it acts differently to an unknown quantity in a variable like x. Because we are solving for = x, we can do the order of operations left to right without implied multiplication. The ambiguity of the equation is solved by the word problems since known quantities can be seen as tangible things like items, lots, and people.
I have to disagree on D, 3(4) should be done first like 3x because usually juxtaposition takes higher precedence than multiplication and devision, it should be 1
Or you could just be a normal person and write the whole thing in fraction notation.
Type it into any online calculator and check to see that you're wrong lol.
@@thewhat2 And yet read some academic journals' publication style guidelines and you'll see that in academic papers, it would be interpreted the other way and YOU'd be wrong. But the keyword is "some". It's just a matter of convention, and different places have different conventions. The trick is to realize that it is a dumb argument because you're literally arguing for which definition is better, and the fact there's a disagreement at all shows that neither is good and you should adopt a better way to notate it (which already exists).
It comes down to the semantics of what you mean by "worst." I answered three, as I believe that this gives rise to the most arguments. How can 0.9 reccuring possibly equal 1, as they look totally different. I like your point about naming a number between them.
The same way ...999 = -1, just in a different (more well known) system. 🙃 Two ways of writing the same thing (though if you can minus signs, then ...999 becomes the unique way to write the additive inverse of 1, which is something the Real numbers can't do with their equivalent.
Is it possible to solve this integrate sec(sqrt(3y))dy from 2 to 4
If you are taking the principal square root, then yes, sqrt(1) is 1. I think, though, all square roots should be equally valid. Like how the Lambert W function has 2 answers if you are in a certain set of the domain.
Yeah, the principle square root has one answer. The general square root often has two. Depending on the context, either can be more suitable.
Exactly. Think of this: sqr(1-i), it has a principal value 1.098...-i*0.455... (equivalent of sqr(1)=1) and a second value: -1.098...+i*0 455 (equivalent of sqr(1)=-1. What value do you choose? Both are valid. So, we use the positive value in real numbers, but when we deal with complex numbers, we can't ignore the second one. So squareroot means 2 values, third root 3 values and so on. The fact that in real life we 1 only one value, that doesn't mean the other's doesn't exist.
The reason sqrt(1) = 1 is that sqrt (or √) is a function and in most math (especially in school) it's a rule that functions have at most one output for a given input. I personally have no issue with multi-valued functions, but middle and high school math teachers usually do.
@@TheUnlockedAnd I really don't undersand why. See my example...when you can't ignore the second value. Or the third in case of cube root....
@@radupopescu9977 The sqrt function is defined as giving the principal square root for the input. That's just how it's defined. It doesn't mean the other roots don't exist and aren't useful for a given problem, but they're not what you get when applying the sqrt function. Though I'll admit that when you enter the complex world people do start to get a little looser on notation (or I suppose use notation differently).
Great, now you've added "which is the worst math debate?" to the list of the worst math debates...
This should be fodder for an XKCD if it isn't already!
my understanding was that D was ambiguous notation, the calculator on my phone lets me change it to interpret it either way in it's settings. It makes more sense when you use variables. it seems wrong for 12÷4a = (12/4)a rather than 12/(4a)
That is correct. The problem with D is that it uses the division sign which is infamous for causing confusions. We all instead should use fraction notation.
I originally said B followed by D, but I could see it either way. With C, the question is not whether the limit of that particular series is 1, but whether any sequence (or series) should be considered interchangeable with its limit, assuming it has one.
One more argument that I've seen in the past: 0 is a natural number. Some people say yes, some people say no.
It's not an argument, it depends on what you want the natural numbers for.
0 is natural. If i say 1/x is x for all natural i can say 1/x, x є N, but 1/0 not well defined, then i can say 1/x, x є N*, not including zero
I say no. 0 is a whole number but not natural number. A natural number is a number that you can creat with a string of 1's added together.
@@TheMassacreOfTheBanuQurayzahQu Are you American? In my experience, it seems to be Americans that don't want 0 to be a natural number. Besides, 0 is the sum of a series of 1s. It's just the series with 0 terms. :)
@@omp199 Hmmm. I suppose it would be better to say Positive Integers for 1,2,3,... And use Nonnegative Integers for 0,1,2,... At least that's how Prealgebra, the Art of Problem Solving book gets past the natural vs whole debate.
D is definitely the most annoying. For every one time you see any of the other three, you will see 10 posts with 100x more engangment each about D.
For C, theres another simpler way to explain it, besides infinite geometric series, I learned that while learning about repeating decimals. It went as follows: Let x =0.999... 10x = 9.99999... 10x-x = 9.999... - 0.999... 9x = 9 x = 9÷9 x = 1 This works for transforming most repeating decimals into fractions, as far as I know too.
A deeper "problem" may be that 10x - x may actually not really equal 9 (for a similar reason that we also cannot just say: "infinity minus infinity equals 0"), but rather 10x - x < 9 ==> 9x < 9 . 10x - x = 9.999... - 0.9999... = = (9 - 0.9) + (0.9 - 0.09) + (0.09 - 0.009) + (0.009 - 0.0009) + ... = 8.1 + 0.81 + 0.081 + 0.081 + .... If we accept that (8.1 + 0.81 + 0.081 + 0.081 + ....) equals 9 , then automatically we must also accept that 0.9999... = 1 . However, if we don't accept that (8.1 + 0.81 + 0.081 + 0.081 + ....) equals 9 (but rather (8.1 + 0.81 + 0.081 + 0.081 + ....) < 9 ), then we also don't need to conclude that 0.9999... = 1 . The deeper problem here is: what does a summation of infinitely many terms (or a notation with infinitely many digits) precisely mean?
can we use Feynman's integral technique for ((-1)^x)/x dx from 1 to inf?
not really math "debates" its usually just people who struggled to pass high school maths thinking they suddenly understand anything about the subject and get really loud about it
or people who think math is countable and can somehow be plural
@@zachansen8293or people who don't understand that maths is also a singular noun
@@methatis3013 its not, mathematics, not mathematic
You know maths is not plural for mathematics, right? Right?@@zachansen8293
@@imPyroHD maths _is_ a subject, mathematics _is_ a field
There's a difference between the sqrt(x) function and x raised to a fractional power. Sqrt(x) has a restricted domain. (Unless I'm wrong)
No, they are the exact same thing. Both have only positive values (on positive values) (ok, unless I'm wrong as well). If x^(1/2) would have 2 answers, you would be unable to say (x^(1/2))^2 = x for example.
I still say d is 1 because multiplication by juxtaposition comes before regular multiplication/division
it is hard to distinguish whether you want to let 3*4 in D，the better way to express is 12÷[3(4)] or (12÷3)(4)
D is just poor notation, while B and C are misconceptions, so only A worth a true debate
A isn't even a debate, it's just context-dependent. For limits it's an indeterminate form, for power series it's 1, otherwise just define your terms and run with it.
A is worth nothing. When you are in multivariable calculus and you have 1 limit from the x-axis and one from the y-axis do you say "let's debate"? NO. It's undefined. If sometimes it's convenient to DEFINE it (VERY LOCALLY FOR THIS PROBLEM) to get continuity, fine. Otherwise, UNDEFINED.
Well C is a good debate because the answer is 0.999 does not equal 1 😬😬
I so disagree with C
@@harrisonewer 0.999 does in fact not equal 1 you might have missed the ellipsis, however, as 0.999... does equal 1
Another silly one is whether dy/dx is a fraction or a notation. =P
Or that e^x is actually e being raised to the power of x
It's a quotient if you know enough math (with differential forms, whatever that means). If not, it's just notation.
Physics gang rise up, derivatives are fractions brrrr...
@@vampire_catgirldude what?
@@Termenz1 e is a function, the exp function, and "raising e to a power" is just putting that number through the function. It looks like how normal powers work, which is why the notation is to raise e to a power, but that only makes sense for real numbers; you can also raise e by complex numbers and matrices
There is an argument to be made regarding the final statement. PEMDAS while simple, doesn't express the full order of operations. There is a special type of multiplication called implied multiplication (or multiplication by juxtaposition) where there is no multiplication symbol between the binary elements. Implied multiplication says to group these two as a unit and is done before any other multiplication or division. Because this complicates things, it got overlooked and forgotten. So a/bc, it is explained that bc itself is a unit and should be seen as a/(bc) not (a/b)*c. Don't get me wrong, I'm on PEMDAS train all the way. It just upsets me that it's not taught the correct way.
Good evening sir I am solving a cubic =n I found the roots using the cubic formula and simplified them using calculator but I am unable to simplify then by hand please guide that how can I simplify this. Polynomial is x^3-30x-6=0 Pls help. Regards
I put (D) into a calculatotlr and it corrected the question to 12÷(3(4)) and THEN gave the solution, 1.
This is fun, because some calculators solve from right to left and some from left to right. I wonder if what he says is true only in some countries or anywhere (because that's why we have 2 kinds of answers from calculators).
@@user-os4lj3pi4q It's not solving it right to left, it's using PEJMDAS which prioritizes multiplication by juxtaposition.
No, OP, the calculator did *not* "correct* the question. The calculator *altered* the question into something *different.* So, the alleged solution of 1 is irrelevant here.
Where I'm from we use BODMAS instead and prioritize bracket operations at ALL costs, then throw stuff like logarithms, indices, sqrt, e.t.c. right after.
Some calculators solve this problem different. Casio uses juxpositioning and Texas Instruments do not.
I think you're wrong on D. There's a very significant difference between two objects concatenated and those same objects separated by a multiplication sign. Suppose the (4) were a variable like x. Then you would have 12÷3x, which is clearly 12 / (3x) not (12/3)x. This is the same situation - just replace the x with (4).
That's because a monomial is one term. 3*4 isn't one term.
As the other person said. Monomials are one term and implicit multiplication is two terms. So the are not the same. People keep bring up variables as though "Ha! Found this example that proves you completely wrong!" When really those two things are not comparable.
Please help. At Sacramento State University, this integral was given. The integral of x^4 is the numerator and the denominator is 1+4^x and the bounds are negative 4 (lower bound) to 4 (upper bound). I really do not know how to solve this problem. Can you please make a video on how to solve this problem? Thank You
I threw integral of x^4/(4^x+1) into calculator and what came out was an unreadably long fraction with bunch of logs which for your bounds gave the answer approx. 3.403E-61. I'm not sure you typed your problem correctly.
What about E) integral from -infty to +infty of f(x)= x is zero or divergent? (The correct answer is divergent, but many people argue it is zero because 'the areas cancel each other out')