Which is the worst math debate: 0^0, sqrt(1), 0.999...=1, or 12/3(4)? 

Can you solve x^ln(4)+x^ln(10)=x^ln(25)? ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-xBpZRWCGw30.html

D is just intentionally bad notation. The others at least have some interesting mathematics behind them. So D is indeed the worst debate.

D is the reason we don't use ÷ after elementary school

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

B is the one I meet most often but D is just stupid

230 - 220 × (1 ÷ 2) You might not believe me, but the answer is actually 5!

Both Mathematicians and computer scientists agree that 0!=1

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.

You missed 1+2+3... = -1/12

D is the worst cause it’s never ending debate where you will not convince anyone ever.

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.

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.

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.

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

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.

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.

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.

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

"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."

Great, now you've added "which is the worst math debate?" to the list of the worst math debates...

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.

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 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.

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)

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.

".999 repeated is 1 becuase you cant find a number between them" is a really cool observation

This video 'I hate internet math' by doorwaydude is what I consider to be a 'sequel' to this vid. It attempts to explain why these debates exist in the first place (and why some of them are pointless). I really think people in this comment section should have a look if they are still unsure about things like 0.999...=1

Thank you - that has helped.

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

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.....

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 🤷

1/3 = 0.333333... multiply both sides by 3 1 = 0.9999999...

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.

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

D is just the least interesting.

Thank you for this.

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 +

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)

The problem with the root, is that people don't understand that it's not the root itself that gives us two options x²=1 √(x²)=√1 |x|=1 x=±1 you either use absolute value x²=1 x²-1²=0 (x-1)(x+1)=0 x=±1 or the difference of squares

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.

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.

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.

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.

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)”

I like thinking that 0.999… just approaches 1 in such way that there isn’t any number between it and 1

-1/12 is mising from your list.

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.

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.

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.

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.

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.

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

C is definitely the worst because there’s no debate. one side is objectively wrong

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.

D is one of the reasons I left Facebook, seeing the stupidity of people fighting over it

I was taught to do PEMDAS left to right one by one, I never knew you combined PEMDAS into PE(MD)(AS).

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?)

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.

The problem with D is that its not taught everywhere that in the order of operations in cases like this, you always go from left to right, thus you get 1

Pretty interesting PROF thanks 👍

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.

D is just poor notation, while B and C are misconceptions, so only A worth a true debate

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.

What about the debate over the order of operation for powers?

This is why I would guess c as the worst debate... because there is no debate

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.

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)

I'm watching your videos for many days. Very Informative 👍. Love ❤ from India 🇮🇳

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.

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

Math or Maths. Thats the most contested maths debate. I used to math debate quite a lot.

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.

The fact there’s people on this video that say “it depends on context” or whatever for D is so dumb

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.

C is also a case of bad notation. It seems that it is easy to get confused whether 0.999.. is the sequence of partial sums of the geometric series or whether it is the LIMIT of those partial sums. I think the latter is correct. But you apparently have the same confusion when you pose the challenge "show me a number between 0.999... and 1" as if it's a sequence of partial sums. If it's a limit then that's like asking show me a number between 1 and 1. If 0.999... is meant to represent the limit then you should make that clear by writing it as a limit or at least clarifying in words.

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

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.

D is basically PEJDMSA vs PEDMSA

÷ and / should be always replaced by division line. We should not use (...) for only one digit or varible - let's use () for expressions.

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.

surprisingly, his solution for D isn't correct. admittedly, it's not really recommended to write it down like this, but it looks like that the international physics literature overwhelmingly agrees upon that, say, "2x" refers to an "implied multiplication" which cannot spilt up any further and thus has predominance over anything else of seemingly equal (!) order. or in order words, "2 × x" is treated differently from the other notation. no one has to believe me, but that's also how most scientific calculators work (which isn't by mistake but design.) but what I actually tried to say is that in those rare instances where it actually says, "x / 2y," the expected operation to be performed is, "x / (2 × y)." at the end of the day, it's just a convention and doesn't break actual math.

Option D is not so clear. The RU-vid 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.

D : 12/3(4) is the worst math debate, because it's actually not math, but rather about (intentionally) ambiguous math notation and ill-defined notation interpretation. It should/would always be clear from the context which interpretation the author of a mathematical expression had in mind. (Moreover, it should be noted that programmers, software designers, and calculator manufacturers are _not_ math authorities, and hence don't get to define how math is to be notated and how math expressions should be interpretated.) We won't really gain any new mathematical insights by resolving the "D-bait". In my view, C is the most interesting math debate (of those four), because it's intrinsically about math. (B is just a matter of definition; A is simply clear: there's no reason to not define 0^0 = 1 .)

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')

One more argument that I've seen in the past: 0 is a natural number. Some people say yes, some people say no.

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) I: Symbols that can act as both prefix and suffix operators (e.g. 3³3, 3!3) I*: Symbols that can also used multiple times at a row (e.g. 3!!3) J: Iterated binary operations without ending mark, unlike integration (e.g. Σx+1, Π2x) J*: Operators which are not associative, but the operation orders or associativity directions are not well known (e.g. P → Q → R)

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.

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)

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.

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)

For question C, you could just do: 1/3 = 0.3... 3 x 1/3 = 3 x 0.3.... = 0.9... 3 x 1/3 = 3/3 = 1 therefore 1= 0.9...

If you manage to somehow get to a point where you have 0^0 you honestly don't deserve an answer

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. :)

I blame D on teachers who are themselves mistaken.

I still say d is 1 because multiplication by juxtaposition comes before regular multiplication/division

I hate maths debating, one of my friends does it all the time and almost got sent out of college for maths debating in class

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.

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.

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."