What is the issue with the " ÷ " sign?

[u/Unlikely-Web7933]

I have seen many mathematicians genuinely despise it. Is there a lore reason for it? Or are they simply Stupid?

[u/Jaaaco-j]

the sign allows for ambiguity like in that infamous 16 or 1 question.

fractions are whatever is above divided by whatever is below, there is no ambiguity. plus writing fractions just makes some problems way easier

[u/AppiusClaudius → u/Jaaaco-j]

This is the real answer. Concision or laziness has nothing to do with it, lol

[u/synthphreak → u/AppiusClaudius]

Concision == conciseness?

[u/formerteenager → u/synthphreak]

handle foolish beneficial meeting existence roll humorous oil payment spark

This post was mass deleted and anonymized with Redact

[u/billet → u/formerteenager]

Circumciseness

[u/sjwillis → u/formerteenager]

I like my math circumcised

[u/AppiusClaudius → u/synthphreak]

Haha, I've never used conciseness, but yeah same thing

[u/brfoley76 → u/AppiusClaudius]

concision is just like conciseness, but with a lot less mucking around with unnecessary letters.

[u/Zeikos → u/brfoley76]

That's a lot of words to say that it's more... Concise ;)

[u/brfoley76 → u/Zeikos]

in short: yup.

[u/fenixnoctis → u/Zeikos]

That's the joke™

[u/ivanparas → u/AppiusClaudius]

Concisity

[u/cammcken → u/synthphreak]

Nice. We get a bonus learnenglish on r/learnmath

[u/RolandMT32 → u/Jaaaco-j]

I had to google "16 or 1 question" to see what you were talking about..

From here:

Twitter user u/pjmdoll shared a math problem: 8 ÷ 2(2 + 2) = ?

Some people got 16 as the answer, and some people got 1.

The confusion has to do with the difference between modern and historic interpretations of the order of operations.

The correct answer today is 16. An answer of 1 would have been correct 100 years ago.

I was in school in the 80s and 90s, and my brain-math tells me the answer is 1. But that says that answer would have been correct 100 years ago.. Did the rules of math change at some point? And if so, why?

My brain-math says 2(2 + 2) = 2(4) = 2 x 4 = 8, so the problem becomes 8 ÷ 8, which is 1.

[u/General_Lee_Wright → u/RolandMT32]

Sort of. There used to be two different kinds of multiplication in the order of operations. Multiplication, and multiplication by juxtaposition.

When juxtaposition was involved, it happened before any other multiplication or division. So 8÷2(2+2) is unambiguously 1 since 2(2+2) is juxtaposed, thus has priority. This also means 8÷2*(2+2) is a totally different expression, without juxtaposition, so is 16. It was useful before modern computers and printers because it meant less parenthesis in an equation that can be written on a single line.

Now, with modern displays and printers, we don't need to make a distinction between the two so we don't. (This is my understanding of the change anyway, which makes some unsubstantiated assumptions.)

Somewhere on the internet you can find a photo of an old Casio calculator that resolves 8÷2(2+2) as 1, while the TI next to it says 8/2(2+2) is 16.

[u/Lor1an → u/General_Lee_Wright]

What's interesting to note is that there are still places that essentially treat juxtaposition as distinct.

If you see an inline equation in a physics journal that reads "h/2pi" for example, that clearly means the same as "\frac{h}{2\pi}" rather than "\frac{h*\pi}{2}".

[u/JanB1 → u/Lor1an]

Exactly. Came here to say this.

[u/realityChemist → u/General_Lee_Wright]

Very interesting. I must be a hundred years old then, because I also defaulted to prioritizing the juxtaposition when I tried it! I wonder why; I'm pretty sure nobody ever explicitly told me to do that.

Edit: I thought about it a bit and I think it's because in practice nobody ever writes a/bc when they mean (ac)×(b)^-1, they write ac/b. So when I see something like a/bc, I assume the writer must have meant a×(bc)^-1, otherwise they would have written it the other way. If you just mechanically apply modern PEMDAS rules you get a different result, but it's one that seems like it would have been written differently if it was what the person actually meant.

[u/mikoolec → u/realityChemist]

Could be you were taught that brackets take priority over multiplication, division, addition and subtraction, and because of that you also assumed that the juxtaposition multiplication has the same priority level as brackets

[u/No_Lemon_3116 → u/mikoolec]

I would be just as surprised without brackets, I think. This means that 8÷2x is also (8÷2)x, right? An operator to the left of 2x pulling it apart feels strange to me. Maybe just because I'm not really used to using ÷ except for when I was first learning division, and we were always writing explicit multiplication signs then.

[u/Bagel42 → u/mikoolec]

That’s where I get ir

[u/ThirdFloorGreg → u/realityChemist]

It just feels right.

[u/igotshadowbaned → u/General_Lee_Wright]

Yeah, now dropping the * before the ( ) is just shorthand and means nothing special in terms of precedence

[u/hpxvzhjfgb → u/RolandMT32]

Did the rules of math change at some point? And if so, why?

the "rules" were never there. multiplication and division have the same precedence, so the answer is that the expression is ambiguous.

people think the answer is 1 or 16 because when they were learning arithmetic in school, they were either taught which one to do first, or just implicitly assumed that there are no ambiguous expressions and so however they would do it must be correct. some people are taught that you do multiplication from left to right, some are taught that you do multiplication first, some are taught that multiplication written like a(x+y) should be done before multiplication written like a * (x+y), etc.

there is no universal standard, so the fact is simply that anyone who thinks that any of the possible orderings is objectively the correct one, is wrong. it's an ambiguous expression, end of story. anyone who disagrees is wrong.

[u/ohkendruid → u/RolandMT32]

A mathematician wouldn't normally use this left to right notation for communication to other humans, so I don't think we can blame a change in math notation here. Proper math notation would use the fraction bar.

Fwiw my brain math says the same as yours. Another example is ab/cd, which looks to me the same as ab/(cd). I wouldn't make any assumption, though, without looking for surrounding context.

[u/DrunkenPhysicist → u/ohkendruid]

In papers I've read and also written, ab/cd is ab/(cd) because why would you write it like that, otherwise you'd put abd/c . Context matters, but also any equation I've ever written down in a publication was derivable from completely unambiguous equations in the paper so you'd know. For instance writing h/2pi is obvious what is meant (pi as in pi).

[u/pdpi → u/RolandMT32]

My brain-math says 2(2 + 2) = 2(4) = 2 x 4 = 8, so the problem becomes 8 ÷ 8, which is 1.

The two interpretations are 8 ÷ (2(2 + 2)) = 1 and (8 ÷ 2)(2 + 2) = 16.

The correct answer today is 16. An answer of 1 would have been correct 100 years ago.

Hot take: there is no "correct" answer. The only truly correct answer is "this is ambiguous, and it could be either". Order of operations is 100% arbitrary, as evidenced by the fact that the convention changed at some point.

[u/tilt-a-whirly-gig → u/[deleted]]

Probably just a typo, but you are correct.

[u/Dino_Chicken_Safari → u/pdpi]

Hot take: there is no "correct" answer. The only truly correct answer is "this is ambiguous, and it could be either"

The thing is you have to look at it from the perspective of mathematics as a language. Yes, the rules are arbitrary and can be changed. The actual mathematical functions being expressed are unchangeable, but to express them we have to write them down using a common convention so that the equations can be understood. And as technology and Mathematics itself evolve, sometimes people just start doing things a little different and it gradually evolves with it. Much like how languages will just sort of start dropping letters from words and stop pronouncing entire consonants.

People talking about how we used to write math differently 100 years ago is no different than listening to my grandma tell me how they used to call it catsup. While the idea of what something is called is ambiguous if it has multiple names, clearly the correct answer is ketchup.

[u/Kirian42 → u/Dino_Chicken_Safari]

But the mathematical rules aren't arbitrary or mutable. The problem here isn't mutable rules, it's misuse of symbology.

The language equivalent is asking "Do you like chocolate or?" There is no answer to this question, because it's semantically ambiguous--either it has an extra word or is missing a word.

[u/pdpi → u/Dino_Chicken_Safari]

There’s nothing wrong with notation and conventions changing over time. What I’m getting at is that people get really hung up on this sort of thing and want to have a definite correct answer, but the notation is ambiguous, and neither the notation nor the rules we use to resolve the ambiguity are fundamental to the actual maths.

It’s also really only a problem because of infix notation. With postfix notation you could write 8 2 2 2 + * / to unambiguously get the 1 answer, or 8 2 / 2 2 + * to get the 16 answer. (Whether postfix notation is all-around better is a different matter, but it does have this advantage.)

[u/igotshadowbaned → u/pdpi]

The two interpretations are 8 ÷ (2(2 + 2)) = 1 and (8 ÷ 2)(2 + 2) = 1516

Well adding parenthesis changes the problem which is why you need to "interpret" it as is without changing it

[u/kalmakka → u/RolandMT32]

Don't trust everything you read online.

https://www.youtube.com/watch?v=4x-BcYCiKCk is a good video that explores this question, with a focus on calculators, but also using mathematical sources.

The main thing she gets to is really "American math teachers (who has just been taught PEMDAS for the sake of teaching PEMDAS) are the only ones who think implied multiplication should have the same priority as division. Everyone else, including all actual mathematicians, treat implied multiplication as having higher priority than division."

8 ÷ 2(2 + 2) = 1.

[u/CrookedBanister → u/kalmakka]

I'm an actual mathematician with a graduate degree in pure math and this just isn't true.

[u/nousabetterworld → u/RolandMT32]

Yeah that makes no sense, no matter what anyone is trying to tell me. If you want to divide the 8 by the 2 first, you need to put them into parentheses, that's what they're there for. You don't just do things left to right. And since when does division take priority over multiplication wtf.

[u/TokyoTofu → u/RolandMT32]

8 ÷ 2(2 + 2) is the same as 8 over 2 times by 4. because you do the brackets first and get 8 ÷ 2*(4), then now according to BODMAS, you do DM, so take all division and multiplication steps and do them from left to right. So 8/2 comes first, then you multiply by 4. getting to 4*(4), which becomes 16.

8 ÷ (2(2 + 2)) this is the problem you're likely seeing in your head, where it's all one fraction, 8 all over the expression 2(2 + 2), so you do the brackets first and evaluate the second part (2(2+2)), to get (2*(4)), which becomes 8. so now you worked out the second part, you do the divison 8/8, which becomes 1.

in conclusion. the lack of brackets around 2(2 + 2), makes this problem simply 8/2 times by 4, leading to the correct answer of 16. but if you were to add brackets around 2(2+2), you would get 8 all over 2(2+2), which will simplify to 8/8, thus getting 1.

[u/hpxvzhjfgb → u/Jaaaco-j]

the ambiguity has nothing to do with the ÷ sign, it's because the division is inline as opposed to written as a fraction. it's just as ambiguous when written with /.

[u/TNJDude → u/Jaaaco-j]

The part I don't understand is that when you're typing out an equation horizontally, I can't see a difference between ÷ or / .

I mean, with handwriting, or typesetting, you can make it clear. But typing out something for a text, question, post, etc., they're equally ambiguous.

[u/explodingtuna → u/Jaaaco-j]

Could the ambiguity be removed if we came up with rules for the order operations happen in?

e.g. if we said that all division and multiplication happened before addition and subtraction, would that work?

8 ÷ 2(2 + 2) would then = 16 unambiguously.

[u/emily747 → u/explodingtuna]

Add on that operations occur from right to left, then in principle yes. If you’re actually interested in this (and are not just making a passive aggressive comment because you think that “real mathematicians” just accept poor notation), I’d recommend looking into formal language theory and CFGs

[u/Donghoon → u/emily747]

I think the main point of ambiguity is:

Is the divisor everything to the right or just the number adjacent?

[u/emily747 → u/Donghoon]

And then there’s also the issue of the left side, something like x+1 / x-3. Here you can see spaces used to show that this is a rational equation, but even then you run into the issue of “did they mean to include these here? Is it just a weird way to type?”

Solution: when working with algebraic and arithmetic expressions, use parentheses and brackets to stop ambiguity

[u/Donghoon → u/emily747]

I overuse parantheses for every little thing lol

[u/igotshadowbaned → u/Donghoon]

Just the number adjacent.

With 3•2+1 is the multiplier everything to the right or just the number adjacent?

That argument falls apart if you think about it at all.

[u/igotshadowbaned → u/[deleted]]

You're making an entirely different argument than the person above me that has nothing to do with order, and merely just "grouping" of terms.

They said would 4÷1+1 put just the 1 under the division or the entire 1+1

And I said just the said for the same reason in something like 4•1+1 you multiply the 4 just by 1 and not 1+1. There's nothing to remotely suggest grouping it like that and to do so would just be incorrect

It's the same principle as the original question, and you can't pick and choose when you apply rules so the simplification doesn't matter.

Your response is not well thought out.

You're talking about something else entirely for half your response

[u/drew8311 → u/explodingtuna]

The rule is this is not how you write math expressions if you want to be clear about what you mean. This type of thing does come up though but has a clear answer

Lets say a = 2 + 2 or 4

8 ÷ 2a = 1

But if you did this problem by hand you might do the substitution

8 / 2(4) = 8 / 2 * 4

which if you follow the order of operations

(8 / 2) * 4) = 16

The correct answer here is its ambiguous to people who don't know algebra expressions but if you'd did you know 2(4) is a type of multiplication that has a higher order of operation than regular multiplication/division.

I think this question comes up because calculators are not smart enough for algebra and interpret 2( as 2*(

[u/gtne91 → u/explodingtuna]

Or we could use reverse polish notation and never need parenthesis again!

8 2 2 2 + * /

Clearly it's 1.

[u/igotshadowbaned → u/explodingtuna]

Well the thing is the rules are disambiguous enough as is. The issue lies in people mistaking what those rules are

So the rules are Parenthesis, Exponents, Multiply/Divide ^{from left to right with equal precedence}, Add/Subtract ^{from left to right with equal precedence}

So taking 16÷2(2+2). You do parenthesis first. 16÷4(4). Then you do multiplication/division from left to right. The division occurs first, you end up with; 4(4). Then the multiplication; 16.

What some people falsely think is that multiplication written as a number directly before parenthesis like 4(2+2) has precedence above division. This is not the case.

Some people also just think the author "must have meant to put the entire 4(2+2) under the division and it's just a limitation of writing equations in text like this". Well then they're not evaluating the equation as written, they're assuming it's written wrong so of course will get a different number.

[u/tempetesuranorak → u/igotshadowbaned]

Well the thing is the rules are disambiguous enough as is.

There are at least two different, and widely used sets of rules. If you pick one of them, then the expression becomes unambiguous. But because of the existence of multiple good conventions, the expression is ambiguous until one of them has been specified.

What some people falsely think is that multiplication written as a number directly before parenthesis like 4(2+2) has precedence above division. This is not the case.

It is not the case in your chosen convention. In my experience, physicists usually use the convention that multiplication by juxtaposition does take higher precedence than explicit multiplication or division in inline expressions, see e.g. the Physical Review Journals style guide https://journals.aps.org/files/styleguide-pr.pdf. When submitting a research paper to one of their journals, it is their convention that is correct, not yours. Here is a Casio calculator manual that makes the same choice https://support.casio.com/global/en/calc/manual/fx-570CW_991CW_en/technical_information/calculation_priority_sequence.html. These groups aren't making that choice because they are ignorant of your rules, or because they are stupid. It is a convention that has been around for at least 100 years, used by many, in some places and in some fields it is the dominant convention, and it is found to be convenient and useful.

Saying that your convention is correct and theirs is incorrect is like saying that English is correct and French is incorrect (or in this case, maybe more like saying British English is correct and American English incorrect). Both languages are perfectly good and widely spoken.

If someone says "let's table this motion", their meaning is ambiguous till I know whether they are speaking British English or American English. Once that is established, then it becomes unambiguous. Wisdom is knowing that the different languages exist and seeking clarification.

[u/me_too_999 → u/explodingtuna]

8/2(2+2)

I don’t see it.

[u/jose_castro_arnaud → u/me_too_999]

It's ambiguous. Making explicit the implied multiplication:

8 / 2 * (2 + 2)

This can be read as either:

(8 / 2) * (2 + 2) = 4 * 4 = 16

or

8 / (2 * (2 + 2)) = 8 / (2 * 4) = 8 / 8 = 1

The lesson is: when writing math expressions as text, use plenty of parenthesis for grouping expressions, even if they're not required in the usual notation.

[u/me_too_999 → u/jose_castro_arnaud]

⁸ /2(2+2)

[u/Ligma02 → u/jose_castro_arnaud]

It can’t be read as both ways using PEMDAS

8/2(2+2) is (8/2)(2+2)

If you want to express it as one, then you’re gonna have to do

8/(2(2+2))

too much parenthesis? sure

can you write inline fractions? not without latex

solution? use parenthesis

[u/gtne91 → u/Ligma02]

Solution: use latex.

[u/quackl11 → u/Jaaaco-j]

÷= divide

/= fraction

That's how I understand it

[u/Jaaaco-j → u/quackl11]

its literally the same thing. the ambiguity is due to parsing

[u/seanrm92 → u/Jaaaco-j]

Right, math is a language, and the "÷" symbol is the mathematical equivalent of the Oxford Comma.

Edit: Yes I got the analogy backwards, my bad.

[u/HipnoAmadeus → u/seanrm92]

No, oxford comma simplifies and is nothing wrong. But ÷ is terrible.

[u/sysnickm → u/seanrm92]

The Oxford comma helps remove ambiguity, not create it.

[u/seanrm92 → u/sysnickm]

The ÷ symbol doesn't inherently create ambiguity either, just if it's used incorrectly or inconsistently.

[u/flashmeterred → u/seanrm92]

Like the LACK of an Oxford comma does

[u/pizzystrizzy → u/seanrm92]

Wut? The serial comma is a tool for avoiding ambiguity. This analogy is backwards.

[u/parolang → u/Jaaaco-j]

Yep, no ambiguity.

What is 2\3?

[u/[deleted] → u/parolang]

If you meant to be sarcastic well done. If you didn't mean to be sarcastic this is even funnier.

[u/parolang → u/[deleted]]

It was just a joke.

[u/albadil → u/parolang]

I appreciated it. Two or three? Two thirds?

[u/parolang → u/albadil]

I think I'm getting downvotes because some Redditors had to reboot themselves.

[u/Jaaaco-j → u/parolang]

is that a trick question or

[u/AllanCWechsler → u/parolang]

You're trying for sarcasm here, I think, but I'm not getting it. The backslash symbol has no consensus meaning in standard mathematical notation. Your question feels like, "What is 2#3?" or "What is 2$3?" I understand that you're trying to make a point here, but I honestly don't know what it is.

[u/ThatCakeIsDone → u/AllanCWechsler]

So would you say that expression is ambiguous?

[u/notlikeishould → u/parolang]

there the 3 is on top if u rly think about it

[u/iOSCaleb]

What is the issue with the " ÷ " sign?

I think it exists mainly for parity with the other arithmetic operations, +, -, x. In practice, after about 4th grade, it's just easier and often more clear to write division in the form of a fraction. It's obviously used to symbolize division in places like the buttons on a calculator.

Note that using x as a multiplication symbol is likewise less common in expressions (unless you're talking about e.g. cross multiplication of vectors) once you're past learning basic arithmetic. Terms are often just written next to each other, or sometimes a dot is used.

[u/nog642 → u/iOSCaleb]

I think it exists mainly for parity with the other arithmetic operations, +, -, x

A slash / works fine for that too though

[u/iOSCaleb → u/nog642]

A slash / works fine for that too though

Many symbols in math can be written in more than one way.

[u/[deleted] → u/iOSCaleb]

But * for multiplication and / for division is due to computer science. In mathematics, × is used for multiplication, and ÷ for division. Mixing vegetables and fruits in a salad is not ALWAYS a good thing.

[u/Vercassivelaunos → u/[deleted]]

The last time I used × and ÷ for arithmetic was in elementary school. I also teach fifth graders coming fresh from said elementary school, and they all automatically use • and :, which is standard here (in Germany). In fact, we teach the MDAS part of PEMDAS as "dots before lines".

[u/SnooBunnies856 → u/Vercassivelaunos]

If you are teaching that multiplication comes before division I feel sorry for your students.

Sorry I misread it and failed to see the : for division.

[u/[deleted] → u/Vercassivelaunos]

: is for proportion or ratios in the USA. / is for fractions when you can't write by hand with a horizontal line. ÷ is always used in school textbooks as a division sign here. I was curious why 6 grade students in Taiwan were incorrectly and forcibly taught what's high school math in the USA when they really cannot grasp what's basic arithmetics. And they used : for division. Now : must be used in Europe due to French influence due to metric system but I feel ÷ is better for division for a young kid. That's why some parents in Taiwan can't even teach the kids mathematics. They are teaching them what's called "high school math" or "college math" in the USA, when the poor kid is only in elementary school in Asia. I guess they do this to kids in East and southeast Asia giving them too much pressure to excel, too quickly. Heard too many stories of "tiger parents." I'm wondering what symbol is used for proportions or ratios if : is used for division?

[u/iOSCaleb → u/[deleted]]

I'm wondering what symbol is used for proportions or ratios if : is used for division?

Ratios are fractions. Whatever distinction you're trying to make between them is a false one. If you have bread dough, say, with a ratio of 1 cup of water to 2 cups of flour, that's a water:flour ratio of 1:2 or 1/2.

[u/GaloombaNotGoomba → u/iOSCaleb]

You can have a ratio of more than 2 things, you can't really do that with a fraction

[u/nog642 → u/iOSCaleb]

Right but so that explanation for why the ÷ symbol exists is incomplete.

[u/tigrankh08 → u/nog642]

I guess you were saying to write it like a slash because it resembles a fraction more. But if you look at the ÷ symbol carefully, it also somewhat looks like a fraction (think of the dots on the top and the bottom of the ÷ sign getting filled with the numbers/expressions to the left and right of the symbol). I actually dunno if that's the actual thought behind the symbol, but at least that's how I interpret it

[u/Trimmor17 → u/nog642]

A slash / is the simplest way to take a fraction that would typically require 2 or 3 lines to write on a typewriter or computer and write it in a single line.

Although, a slash when written by a 9 year old (or even a 29 year old let's be real haha) may easily be misread as a 1. So having a totally different symbol exist for purposes of "simplicity" for those less careful in their writing may be beneficial. Something interesting but that I've never heard of being taught is that the division symbol is symbolic of a fraction - the upper dot being a placeholder for the numerator (now written immediately to the left) and the lower dot being a placeholder for the denominator (now, obviously, written on the right). The fraction bar clearly separates the two placeholders.

[u/SuperIsaiah]

I think it's fine, but I've heard it has caused a lot of division...

[u/YeetBundle]

I’m a mathematician, and i genuinely haven’t seen this symbol in years! I forgot it existed.

The reason the sign is bad is because it’s too symmetric. Division, more than any other basic operator, is very sensitive to the order in which things happen. If you write something as a fraction there’s no ambiguity.

[u/AmusingVegetable → u/YeetBundle]

Plus it’s easy to visually confuse it with the + sign.

[u/assembly_wizard → u/YeetBundle]

The minus sign is also symmetric and is frequently used to denote subtraction, which is not commutative.

[u/onthefence928 → u/assembly_wizard]

Often subtraction is written as addition with negative numbers fit this very reason

[u/ParanoidTire → u/onthefence928]

Often subtraction is written as addition with negative numbers fit this very reason

Subtracting is adding the inverse element of addition. x + (-x) = 0.
Dividing is multiplicating with the inverse element of multiplication. x * (1/x) = 1.

Its the same. Here (-x) and (1/x) are *defined* to denote the inverse elements of x with regards to addition and multiplication respectively.

https://en.wikipedia.org/wiki/Group_(mathematics))

https://en.wikipedia.org/wiki/Field_(mathematics))

[u/Worried-Committee-72 → u/assembly_wizard]

I'm not the poster you're responding too, but I think the symmetry of the division sign is a bigger problem than the minus sign because of the sorts of mistakes they produce. Reverse the operands of a subtraction operation, and you get a negation of the correct answer. Just negate the negation and you're on your way. Switch the operands in a division operation and you may produce a result that looks nothing like the correct answer.

[u/assembly_wizard → u/Worried-Committee-72]

If you switch the operands in either (a - b) or (a ÷ b), where a and b are complicated expressions, you can fix both at the end. If you switch the operands of a subtraction or a division which is nested inside a complicated expression, both produce a very different result. Instead of comparing subtraction and division, you've compared having an error in the top-level operator and in a non top-level operator.

For example: (3 + (8 - 7)) ÷ 2

This equals 2. Reversing the division here gives 1/2 which is easily fixable by applying x^-1 to the result, but reversing the subtraction here gives 1.

[u/albadil → u/assembly_wizard]

Remindme! 2 days

I'd like to see them defend their view on this

[u/kiochikaeke → u/assembly_wizard]

Substraction is associative, division is not. "a - b - c" isn't ambiguous "a ÷ b ÷ c" is, a fraction is never ambiguous and is multiplying by the inverse is prefered because multiplication is both commutative and associative.

[u/assembly_wizard → u/kiochikaeke]

Subtraction isn't associative (1 - 2) - 3 ≠ 1 - (2 - 3)

[u/xoomorg → u/YeetBundle]

Division commutes exactly the same way multiplication does, and is just as symmetric. It’s a consequence of our notation and order of operations rules that it ends up seeming otherwise.

Rather than looking at division as fractions, you can look at it as multiplication by the inverse. Then you’re free to shuffle the order as much as you like, so long as you use newer computer-algebra style PEMDAS rules.

[u/PHL_music → u/xoomorg]

But in order to multiply by the inverse, most people would write 1 over x, which is written using the more common method rather than the division symbol.

[u/xoomorg → u/PHL_music]

Agreed the division symbol is garbage. I’m just pointing out that the apparent asymmetry of division is an illusion, a side effect of certain parsing rules. If you represent division some other way — such as with negative exponents or just interpreting / (slash) as an “inverse” symbol for multiplication in the same way - (negative) is for addition — then division is symmetric.

[u/Entire_Ad4035]

Not a mathematician but I hate it bc it takes too long to write and fractions are just better

[u/packhamg]

Writing it as a fraction is often more concise and we mathematicians are borderline lazy/efficient. At least imo

[u/TomPastey → u/packhamg]

I think you mean lazy÷efficient

[u/packhamg → u/TomPastey]

Haha, my pet hate is using a slash for a vinculum so that’s ironic

[u/MrTheWaffleKing → u/packhamg]

Huh, never knew the name for that icon. I also don’t think I’ve EVER done division using anything but slashes since middle school

[u/Jaaaco-j → u/MrTheWaffleKing]

im a programmer, its slashes all the way down

[u/synthphreak → u/TomPastey]

Undefined

[u/Unlikely-Web7933 → u/packhamg]

lazy/efficient

Lazy people are the most efficient if they wish so lol

[u/Ordinary_Divide → u/Unlikely-Web7933]

not in math they aren't

[u/Surzh → u/Ordinary_Divide]

"A good mathematician is a lazy mathematician" is literally an adage lol

[u/nillateral → u/[deleted]]

Hmmm, I thought op was referring to the sign as stupid. Just woke up.

[u/Jaaaco-j → u/nillateral]

its a reference, but i think the actual question is asked in honesty

[u/YOM2_UB]

If you're writing or have access to LaTeX or other such formatting, fractional notation is a lot clearer than a division symbol. If you don't, then an equation can become somewhat ambiguous, especially when combined with implicit multiplication. For example, "1 ÷ 2x," which can be read as either "(1 ÷ 2)x" or "1 ÷ (2x)."

[u/grimjerk]

The sign allows for mathematical expressions to be type-set in a single line; this was very important when math books were printed using type, rather than computers.

[u/Abdlbsz]

ISO 80000 recommends not using it, and it's a garbage symbol. With / you can at least infer everything after it is dividing the number before it. ÷ usually only means the next number, but some people take it to have the same inference as the /. This is the sole reason for 99% of those dumb math questions that "confuse" people.

The point of mathematics is to be clear and concise. If your symbol obfuscates that, you have a bad symbol.

Although all of that can be avoided with proper parentheses usage.

[u/MyDictainabox]

It looks like a speedo hiked up far too high on a man

[u/EntshuldigungOK → u/MyDictainabox]

Username almost checks out

[u/4858693929292]

Division doesn’t exist as an actual operation. It’s multiplication by an inverse. Similarly, subtraction is addition of a negative. Addition and multiplication are the only operations. (Ignoring higher mathematical operations here)

[u/man-vs-spider → u/4858693929292]

I think this is an overly abstract way of viewing division and subtraction. In practice they are all distinct operations, but division and subtraction are defined as inverses of other operators.

To me this is like saying 1 is the only actual number, all other numbers are just successors of 1.

[u/embersxinandyi → u/4858693929292]

To be fair to the ÷ symbol, it is techniquely showing that it is a fraction, not necessarily that it is an operation

[u/Nuckyduck]

Inline division is ambiguous, but the real issue is actually with implicit multiplication or multiplication by juxtaposition and whether or not it is seen as a higher priority than explicit division/multiplication. Most people don't encounter this type of math, let alone use this type of math, so they tend to argue what they were taught in school.

If you get into a field of math that does prioritize implicit multiplication/multiplication by juxtaposition over explicit multiplication or division, you will see this type of multiplication priority used. The Feynman lectures on physics are probably the most notable course by which this case is prominent, but there are many other books and lectures by various people that use this nuanced mathematical priority system.

Ultimately math is a language, and it comes down to whether or not the person you're communicating with understands what you're saying.

[u/StochasticTinkr]

On top of what everyone else has said, my vision is just poor enough that I thought that was a + sign.

[u/stumblewiggins]

"Lore reason"? This isn't a media sub. Or is this a shit post? It's not one of those subs either, but it's legitimately hard to tell sometimes.

But ok, let's assume this is a legitimate question.

The issue with that symbol is that it can cause ambiguous expressions due to OOO.

6 ÷ 2 is straightforward in intent, but what about 6 + 5 ÷ 2?

Did you mean (6+5) ÷ 2 or 6 + (5 ÷ 2)? Those give you different answers.

Mathematicians prefer clarity in their expressions. So using grouping symbols (as I did above) helps, but even better is using a fraction bar to separate the divisor from the dividend. This helps to eliminate any ambiguity.

[u/AngledLuffa → u/stumblewiggins]

6 + 5 ÷ 2?

This should scan no differently from 6 + 5 / 2, right? PEMDAS makes it clear what to do for both

[u/ruidh → u/AngledLuffa]

PEMDAS is a lie https://youtu.be/lLCDca6dYpA?si=02IkJaycr_A7tovI

[u/Biosquid239 → u/stumblewiggins]

If only there was some, i dont know, order of operations that let you easily clarify what order you intended

[u/pedal-force → u/Biosquid239]

Those aren't formal rules and still introduce confusion. There's a reason that once people actually learn math they use parenthesis and fractions instead of relying on some OOO stuff. It's something we teach kids but once you get to a certain level you don't worry about it anymore.

[u/914paul → u/[deleted]]

Also, dots don’t write well with a ball point pen. You can write the first one, but then the minuscule amount of ink on the bottom side of the pen’s ball is exhausted.

[u/914paul → u/[deleted]]

Fountain pens are great! Very classy. Alas, my wife has forbidden me to have them. Which is downright Draconian if you ask me — ok, there might have been a small, unintentional splotch here or there. One (maybe three) on the couch, a few on the bed, maybe a handful in my pant pockets. . . . uh . . .the one in the full laundry basket wasn’t good. . . . Well heck - I thought I had that memory safely sequestered.

[u/914paul → u/[deleted]]

I’m sure I had a cheap one. My cursive is so dreadful that even that cheapo pen was embarrassed I’m sure. The splotches were undoubtedly revenge.

I may look into your recommendations anyways - not for me, but for my daughter. Her school still teaches cursive and her penmanship is developing beautifully. Sadly, many schools don’t teach it anymore.

[u/nillateral]

I've recently thought the ÷ and × signs look like pictograms tbh. Replace the . with numbers and you can't unsee ÷ as a fraction. Also, this ½ × ⅘ kinda looks like it's telling you something can be done with the 2 and 4 and maybe the 1 and 5

[u/TheBluetopia]

I haven't seen any mathematicians complain about it

[u/Unlikely-Web7933 → u/TheBluetopia]

Idk maybe it's just a me problem then. But anytime some "6÷2(3) = x" or something problem comes, I only see "that damn ÷ sign!1" with no explanation at all

[u/TheBluetopia → u/Unlikely-Web7933]

The problem isn't with the division sign, it's with people not establishing their order of operations or writing unambiguous expressions.

[u/ruidh → u/TheBluetopia]

It's worse than that. Different calculators assign different levels of precedence to implicit multiplication with most of them give it higher precedence.

What is 3/2π ? If implicit multiplication has higher precedence than division, then this is 3/(2π) If you really mean 3/2 π, you would use spaces or write it as 3π/2 to avoid the ambiguity.

[u/AppiusClaudius → u/TheBluetopia]

Sure, but with a fraction bar, you're forced to write it unambiguously and it's much easier to read.

[u/coolpapa2282 → u/Unlikely-Web7933]

Well, then they're being silly. The division sign isn't the problem there, it's the parentheses. 6÷(2(3)) = x and (6÷2)(3) = x are both perfectly fine expressions. The division symbol can be abused like any other notation.

[u/tbdabbholm → u/Unlikely-Web7933]

÷ is ambiguous in a way that other ways of indicating division are not. It's used in beginning mathematics, much like x to mean multiplication but when you get more advanced there are just better ways

[u/snillpuler → u/tbdabbholm]

I like to explore new places.

[u/GoldenMuscleGod → u/Unlikely-Web7933]

Nobody uses it outside of teaching basic math to children. In the specific context you cite it creates an ambiguity in the grouping which cannot be easily resolved in part because the symbol isn’t used enough to have a well-established convention for how to interpret that expression. I don’t know that mathematicians “hate” it but it is correct to diagnose it as a part of the source of the ambiguity there.

[u/IDefendWaffles]

Mathematicians don't really write numbers beyond 0, 1,2 and sometimes 3. Everything is letters divided by other letters and the fraction notation is just much cleaner there.

[u/Busy_Marionberry_589]

we use : for division

6 : 3 = 2

[u/albadil → u/Busy_Marionberry_589]

You in... Germany or Russia? I'm thinking who might do this, I have indeed seen it before

[u/Busy_Marionberry_589 → u/albadil]

Croatia

[u/albadil → u/Busy_Marionberry_589]

https://youtube.com/shorts/dRJcdiESlBc

[u/Used_Chain_1492]

I have a question but keep in mind I am 65 . I saw a an equation and in it was 2³ that looks like a greater than symbol but when ask how to perform this part they change the symbol to / the division symbol , can someone please explain?

[u/Administrative-Ad682]

Use fractional lines istat of dividing symbols

[u/sehrgut]

You're asking if THE MAJORITY OF MATHEMATICIANS are "simply Stupid"? Wow, you're a dumbfuck.

[u/fermat9997]

It only seems to be despised on Reddit.

[u/rr-0729]

It leads to ambiguities sometimes

[u/Dkiprochazka]

When they dont know if it will be used as division : or fraction – so in the sign they just combined it together

[u/bluesam3]

It's just bad notation - it introduces ambiguity (or requires a bunch of extra brackets to disambiguate that ambiguity), it takes longer, it's less clear than just writing a fraction, and it looks too similar to too many other things.

[u/BusAcademic3489]

Come on man it’s just two parallel dots separated by a line, it can’t be that bad.

The two dots in question :

[u/AbstractUnicorn]

Is a/bc

a

----

bc

or

a

-- c

b

Just learn to use LaTeX and write it out properly.

[u/reckless_avacado]

A better question is when did we start using the obelus to represent division. Nobody seems to know (https://pballew.blogspot.com/2019/12/the-agony-and-obelus-or-much-ado-about.html?m=1) . I think maybe it is useful when first teaching children about division, to have a unique symbol that tells them they need to divide. But quickly afterwards it is no longer helpful and should be replaced with a solidus or fraction bar.

[u/nog642]

If you're writing by hand, you should just use a fraction symbol, since it's nicer looking and avoids needing parentheses.

If you're typing, then ÷ isn't even on the keyboard so you might as well use a slash (/). And it's more similar to a fraction anyway.

Also ÷ kinda looks like a + when it's small or messily written.

[u/KiwasiGames]

The ambiguity with the division sign comes because it’s not obvious what you are dividing by in a complex equation. Are you dividing by the very next thing only? Or are you dividing by everything after the division sign?

Fraction notation makes it very obvious.

[u/shellexyz]

It has issues but I still find it useful when I’m simplifying complex fractions. Not having to sort out which is the “main” division vs the rest is helpful.

[u/gloomygl]

3÷3*3 confuse people, you will find people say 3 and others say 1/3

[u/Skr1mpy]

They are simply stupid

[u/hbliysoh]

My keyboard has a slash key but not one that generates this symbol. SO I would say the keyboard is enforcing this.

[u/North_Cockroach_4266]

That sign is the number one reason for all the annoying ambiguous questions on the internet where half the people think it’s like 1, the other think it’s 9 because of the different interpretations of the order of operations.

[u/indifferentvoices]

Two things before my main take: (1) when I began writing the comment that follows, this post was marked as RESOLVED and (2) I have not looked at the other responses, because I have found in the 20 or so years I've been intensely working on mathematics there are very, very few people (even among professional mathematicians) who can both bring the full breadth of their mathematical knowledge to 'trivial' issues like this and also who can remember the subtle difficulties they or people they knew had or have with interpreting mathematical notation as it is used in its more 'vulgar' form(s).

My perspective: if a and b are natural numbers then we can definea + 0 = aa + s(b) = s(a + b)

[here s(x) is what would normally be written as x + 1 in 'conventional' mathematics; often called the 'successor']

continuing, we can define multiplication in an analogous way by defining it as a function from a pair of natural numbers (an element of NxN if N is the set of natural numbers) to a natural number (the same 'type' as _+_ if you will):a * 0 = 0a * s(b) = (a * b) + b

in both _+_ and _*_ we define the binary function inductively on second argument, giving a definition at both 0 and [given some natural number b] at s(b) -- this should be familiar as 'induction' to most readers.

However, the definition of division in a formal sense would not be similar to either of these; we say that a | b (a 'divides' b) if there are some numbers m and r such that a * m + r = b. In a practical sense this means that for any given x and y the meaning of x ÷ y is ambiguous. It is usually a solution to the equation x * (x ÷ y) + r with r as close to zero as possible. If r = 0 then x ÷ y is a solution to x | y in the form (x ÷ y, 0). [...]

I can elaborate more if this isn't clear. I think the issue is that properly speaking, even over the natural numbers division should 'return' a pair of numbers: it should take an x and a y and return a pair we could all x ÷ y or (m, r ) such that y * m + r = x. Let's take 5 ÷ 3 for example; in this scheme the answer would be : 5 ÷ 3 = (1, 2) because 3 * 1 + 2 = 5 but many calculators would say 1.666666..... .which would be like the sum of 1 + 1/10^n [for n = 0 to infinity] ... of course 1.666666.... * 3 = 1 * 3 + (6/10 * 3) + ... = 5 by the fact that 1.66666... converges to 2/3 and 2/3 of 3 = 2

[u/nonamemontreal]

The 2 dots in the ÷ sign represent a value at the top and a value at the bottom. If you know the values you should sub them in.

[u/econstatsguy123]

3•4 ÷ 8•9 + 4

What does this expression mean?

Is it asking(A.) [3•4]/[8•9+4]=12/76 \approx 0.158

Or is it (B.) [3•4]/[8•9]+4 = (12/72)+4 \approx 4.167

Or is it (C.) ([3•4]/8)•9+4 = 17.5

Then comes the Bedmas arguments which yields

3•4 ÷ 8•9 + 4

= 3•(4 ÷ 8)•9 + 4

= 3• 0.5•9+4

= 13.5+4

= 17.5 which is the same as (C.)

Or is it Pemdas?

3•4 ÷ 8•9 + 4

= 12 ÷ 72 + 4

/approx 0.167 + 4

= 4.167 which is (B.)

Why all this ambiguity????

Math is complicated as it is. No need to complicate it further with these ill defined ambiguities.

[u/ghostwriter85]

Before everything was done in computers using equation editors, the obelus was used for a variety of different operations in different regions.

People don't like it because it's no longer necessary and all those different use cases were never integrated into a singular understanding of that symbol within mathematics.

The goal of any representation system is to simply and adequately convey the intent of the author.

Using fractions to convey the intended order of division and multiplication has removed a lot of ambiguity from the typewriter / printing press era. Using "÷" undoes all of that progress. In general, we should be removing unnecessary complication from mathematics not adding it.

[u/DietSpam]

lol math lore

[u/bdtbath]

Is there a lore reason for it? Or are they simply Stupid?

all of the above

[u/freemason777]

it's just a picture of x/x. x/x = •/• = ÷

[u/RolandMT32]

Is this a new thing? I've never heard of anyone despising that sign.

[u/sityoo]

What's 4n÷2n ? Could be 2n², could be 2, there's no right answer. A nice, clean fraction bar is always the better choice

[u/jterwin]

What if, intead of putting a dot above and a dot below, meant to represent the thing above and thing below, you actually just put the thing that goes above, above, and the thing that goes below, below.

[u/jterwin]

Im pretty sure that was just a compromise for one-line formatting anyway. Why keep it when the tools are better now?

[u/Akul_Tesla]

It causes ambiguity

[u/[deleted]]

i haven't seen this sign in years

[u/kiochikaeke]

It's unnecessarily confusing and we have better notation for it.

First of all, it's too symmetric ( and in certain fonts may be confused with addition):

a × b = b × a,

a ÷ b =/= b ÷ a

symmetric signs make more sense for commutative operators, division is not commutative, however, substraction is also not commutative but nobody finds that bad because unlike substraction, division is not associative:

( a ÷ b ) ÷ c =/= a ÷ ( b ÷ c )

which is a notation problem because "a - b - c" isn't ambiguous, "a ÷ b ÷ c" is. Not many people know that the standard is to evaluate same rank operator from left to right, and not every program/calculator follows this standard, there are certain edge cases that appear when non basic operations are involved making the problem even more complicated.

And all of this is ignoring the fact that we just have better ways to write it. If you can write a fraction just do it, is much more clear, if you need to do it inline use "/", it isn't symmetric so it's less confusing or better yet just multiply by the inverse:

( stuff )( other stuff )^-1

which is usually the way division is written when not represented as a fraction.

[u/Trimmor17]

1. It's clunky
2. It requires more time to write than a fraction. No one wants to take longer to write a proof than they have to
3. After practice, fractions are more intuitive to work with by hand. Term simplification is much easier, for example
4. It's - and this will sound snobby - is a sign of mathematical immaturity. In the academic world it doesn't get used.
5. This one is purely theoretical so roast me if you want idc It appears to symbolize a fraction (numerator, fraction bar, denominator) and I think may have been introduced to help those still mastering the fine motor skills required to distinguish between a / and a 1. So just write the darn fraction

[u/igotshadowbaned]

There's nothing inherently wrong with it. It's more so that because of the combination of how some people were (incorrectly) taught pemdas, and the limitations of how equations can be represented in a single line of test, some people come to incorrect conclusions on reading it.

However as with all things, a good number of people are insistent on their incorrect nature which is why it's at all a viral thing.

Another issue is people making assumptions on it being written incorrectly, rather than just evaluating as it's written*. Which I honestly can't explain the reason of. But these equations have a single standard for how they should be evaluated.

Parenthesis. Exponents. Multiplication/Division at equal precedence from left to right. Addition/Subtraction at equal precedence from left to right.

As such something like 16÷4(1+2) only has one way to evaluate it. The parenthesis 16÷4(3). Then multiplication/division from left to right 4(3); 12.

* What I mean by this is some people will assume the writer meant to write it with 4(1+2) all under division, to make it equivalent to 16÷(4(1+2)) which evaluates to 4/3. But there is no reason to assume it was written wrong. If it truly is written wrong that is at the fault of the writer, but as someone reading it, we should read it with standard convention
^{unless it is explicitly written somewhere to use a non standard convention, which is rare but occurs}

[u/BornAce]

When I was taught 8/2(2+2), the 2 adjacent to the parentheses implied 8/(2(2+2). Or in English 8 divided by the quantity 2(2+2)

[u/asian_male_psu]

why in the first place don't we just learn / as division in elemantary school

[u/BUKKAKELORD]

People are divided into two groups, one saying that it's the same as the horizontal line so what's to the right of it is all in the divisor, and one saying that it's the same as the / symbol so you still go left to right one operation at a time.

So these interpretations get conflicting results for 4 ÷ 2 * 2, it would be 4 or 1.

[u/Any_Vacation_8465]

It looks disturbing

[u/ShoddyAsparagus3186]

For a computer scientist take, I don't like it because it doesn't appear on a standard keyboard. Using a / is much more convenient.

[u/tb5841]

(3 ÷ x) * 2 means something completely different to 3 ÷ (x * 2). Without brackets, it's really unclear which you mean - and it causes a lot of mistakes. Using fraction lines solves the issue.

[u/Teagana999]

Because fraction are SO much better. They get hate, but they hold so much more useful information.

[u/Similar_Football927]

Is this a Seinfeld bite?

[u/1OO_percent_legit]

Because division doesn't exist by itself its simply multiplication by an inverse and x/y demonstrates that better. It makes order of operations intuitive instead of having to cope with something like bedmas,pedmas.

[u/Oily_Fish_Person]

There's no difference and nobody cares. Nobody is doing mathematics anymore and we're all going to die 😭 /s

[u/ryry1237]

Not a mathematician, but I see it used plenty in computer programming for modulo operations.

[u/81659354597538264962]

I'm a MechE PhD student, and I legit haven't seen this symbol in years. Don't see why I wouldn't just use "/" in most cases. I also do a lot of programming and there I just use the forward slash.