Former Math teacher here.
It's 9.

But! This is a cause of a lot of confusion, both because ÷ is an ambiguous symbol and because people misunderstand PEMDAS and where it comes from.

Thread. https://twitter.com/TessaDare/status/1327340632894033921

First, let's talk about PEMDAS.
For those of you who don't know, it's an acronym to understand the Order of Operations.

It stands for
(P)arenthesis
(E)xponents
(M)ultiplication
(D)ivision
(A)ddition
(S)ubtraction

The first mistake some students make is that they take PEMDAS *extremely* literally, and do precisely one, then the other.

And that's a problem, see. Because arithmetic operations? They're tricky bastards, and will wear a lot of different hats.

Take for example, addition and subtraction. You can ALWAYS rewrite subtraction as addition. It's not exactly elegant or intuitive, but you can!

5 - 3?
You can think of that as 5 + (-3).

In fact, mathematicians don't even technically think of "subtraction." We just use "additive inverses" and generalize all the properties of subtraction that way.

Okay okay.
Hold on.

This is not a short thread, but I am going to be busy so will be sporadically adding to this. Because kiddos?

I've decided.

We're going DEEP.

So. We're going to Zoom way way out for a sec, but I promise we'll come back to Earth and talk about the pesky symbol ÷, and we'll end with why it actually must be PEMDAS and not, say, PAMSED

You have to understand, in general Mathematicians study algebraic structures; there's very (VERY) deep relationships between what sort of equations you can solve and the underlying structure.

For our purposes we just need the basics.

An Algebraic structure is just 1) a set of objects, and 2) an OPERATION (or more than one) on those objects.

A "nice" structure will be an operation and objects that meet familiar properties.

For example, addition is communitive - that's a fancy way of saying order doesn't matter.

2+3 = 3+2.

For more exotic operations, this isn't always true.

One important property of a space is if an IDENTITY exists for that operation.

Identities have the following property: they're lazy and don't do shit.

The general statement is "there exists an object e such that ea=ae=e for all a."

That looks super scary BUT YOU TOTALLY KNOW AN IDENTITY ALREADY! Zero!

Additing zero never does anything!

5+0 = 0+5=5.

1432+0=0+1432=1432.

For ANY x, x+0=0+x = x.

"But Mike, uhmmm, 1 that's so obvious, 2 wtf does that have to do with the meme?"

Patience! We'll get there!

And as for it being obvious? There are exotic spaces where these obvious things aren't true!

The NEXT important property is INVERSES.

Say you have a object, x.
Does it have an evil twin x'?

You know they have evil twins because they EXPLODE!

If you use the operation on them, they give you the identity.

3 is great.

Under additon, 3 has an evil twin. Namely, -3.

3 + (-3) = (-3) + 3 = 0.

You also have identities and inverses in other places too. Multiplication has an identy: 1.

That's why those times tables were so nice!

Incidentally this is why fractions are DOPE.

6÷2 = 3.
What's 6÷5?

It's... 6/5, the fraction.

(I swear I swear we'll come back to earth shortly)

(The kiddo screams. I will resume this later)

*Kicks down door*
ALRIGHT BEYOOOCCHHESSS the kiddo is passed out asleep, I SLAMMED some soda, I also accidentally made some black tea instead of decalf, so I am WIRED.

Let'S MATH IT THE FUCK UP.

SO. WHERE WERE WE?

That's right, ALGEBRAIC STRUCTURES.

So, algebraic structures have IDENTITIES.
You know and love these: 0 for addition, 1 for multiplication.

There are also INVERSES. And inverses are absolutely brilliant - they're literally how we can solve equations.

You have to understand, as far as general algebraic structures are concerned, the real numbers only have two operations: addition, and multiplication.

I know, I know.
"Mike, where the fuck is subtraction and division?"

They're hiding. In the inverses.

Subtraction is REALLY just "the addition of the additive inverse"; i.e. like I said earlier, 5 - 3 is really 5 + (-3).

Note how much more elegant this makes it! Subtraction on face is NOT commutative; 5 - 3 IS NOT 3 - 5.

But 5 + (-3) does equal (-3) + 5.

Division is similar, though you have to reach back to fractions to see it.
(I know, I saw some of you flinch there. It's okay, I'll guide you through! We'll make it, together!)

You have to remember multiplying fractions.

Good news! It's nice.

To multiply fractions, you simply multiply the tops together, and then you multiply the denominators together (I'll skip the why for now).

As an aside: fractions are NUTS.

Think about it. Six divided by five is.... 6/5.

One side is a verb, a process, an action, a thing you DO, a VERB. The other side is a NOUN, an OBJECT.

This kind of verb -> object is very common in a lot of Math is BEAUTIFUL.

ANYWAY. I digress.

Where do you find division? How are inverses involved here?

Well. The inverse of a fraction is simple: flip it.

You take the RECIPROCAL.

If you have 2/3, you can flip it and take the reciprocal to get BOOM. 3/2.

What was on top is now on bottom, and visa versa.

Who knew "the multiplicative inverse of 2/3" just meant "Yeah you just flipperoo."

We can check ourselves too!

Remember, we know things are inverses if, when they meet, they explode into the original identity.

Well. Let's try:

2/3 * 3/2 = (2*3) / (3*2) = 1/1 = 1.

This is super important because we can USE multiplicative inverses to rewrite all division* as a multiplication by the inverse.

That sounds scary, but here's what I mean:

6 divided by 2 could be rewritten as
6 \\times (1/2).

*Only ALMOST all division. You can't rewrite division by zero because that's ~IMPOSSIBLE~ to divide by zero. And incidentally the reason why it's impossible goes back to these fundamental properties - there is no multiplicative inverse of zero.

I can explain later if you want.

We've been cruising at like 40,000 feet, so here's the summary of what I want you to see from this birds eye view:

Subtraction and Division are really just addition and multiplication wearing fancy hats. Inverses are why you can always rewrite it.

The other important take-away is that if you're going to rewrite division, you take the reciprocal. You flip it.

So, instead of 5 divided by 3, we can rewrite that as 5 x (1/3).

I want to just underscore how absolutely INSANE THIS IS.

Now, let's bring it back down to earth and talk about something very specific that is directly in response to the meme:

÷

The division symbol.

This symbol is actually called the "obelus" and was first used by Johann Rahn in 1659 in a book on elementary algebra. Fun fact if you want to see a picture of the page, well, here it is: http://jeff560.tripod.com/rahn.jpg 

(I can also just paste the image directly)

You have to understand something, and you may have noticed it yourself reading this thread:

Representing fractions in type is HARD. Especially compared to how we write them.

It's even harder if you're working on old school printing presses, and it's still a pain in the ass if you're, say, working on a type-writer.

Hell, mathematicians have an entire system (called LaTeX - pronounced "La-Tech") to type notation like this (and much else).

Now-a-days if I was typing up identities I would go to my software and type \\frac{2}{3}\\times\\frac{3}{2} = \\frac{3\\times 2}{2 \\times 3} = \\frac{6}{6} = 1 and let the computer handle all the layout.

(Now there are fancy visual editors, but they usually use TeX under the hood)

Because fractions are SUCH a pain in the ass to typeset, people used all SORTs of symbols to try and make them clearer while still typing.

Others are a:b
a/b
a\\b

And. Of course, the obelus: ÷

The obelus was a simple symbol. It "reads" visually a lot like the other operators we are familiar with: +, x, -, etc. It reminds people of fractions (SUPPOSEDLY the obelus has the dots to denote "this goes here, that goes there" in the fraction but I can't confirm).

Annnnnnd.

Mathematicians fucking HATE that symbol.

Point of fact, when you look at the international standards for mathematical notations (yes - that literally exists https://www.iso.org/standard/64973.html) they EXPLICITLY SAY "DO NOT USE THIS" for division.

(Alas, the full document is paywalled, but I remember looking up the actual raw documentation back when I was teaching. They were on a prior version then, but even then nobody liked using the obelus for division).

Let me explain why this notation is such shit and why we hate it.

See. When it comes to division, there is the DIVISOR, and the DIVIDEND.

In 5 divided by 3, the DIVISOR is 3, the DIVIDEND is 5.

The dividend is what you're chopping up. The divisor is *how* you're chopping it up.

With an obelus, the division sign, it goes like this:

5 ÷ 3
Dividend ÷ Divisor

Where things get unclear is if you have complicated expressions on the right hand side. Because it really REALLY changes the answer. That right there is, in fact, the crux of entire confusion in this meme.

For example...

2 ÷ 5 + 2.

If we rewrite that using fractions, where is that +2?
Is it (2/5) + 2, or is it 2 / (5+2)?

It's very unclear.

The best way to use the symbol is to use a LOT of parenthesis and explicitly indicate what is both in the numerator (the top of the fraction) and the bottom of the fraction, if you have anything even more remotely complicated than a basic expression.

(2÷5)+2, for ex.

That's really what I hate about this symbol. Oh yes, ÷, you cause such strong feelings. Because it is only used when teaching elementary arithmetic; it is literally never used above that because it is unclear and confuses people.

But people see it... because we show it to them.

When I was teaching students, here is the way I told them to consistently interpret this notation:

Literally replace any instance of (a÷b) by rewriting it using inverses as

a x (1/b).

My prior example, 2 ÷ 5 + 2, I would have students write it as

2 x (1/5) + 2.

Well you get some common denominators and do some fraction Math.

It's okay if you're having flashbacks, the answer is 12/5.

I /have/ heard of other standards. Some say "if there's white space" it means the whole thing is offset. But that's not 100% unambiguous.

Honestly the whole issue is relegated to arithmetic books anyway because people quit using the division simple because it's unclear.

But something I want to stress here:

People deride "basic" arithmetic. Yo, I just showed you some of the deep deeeeeep properties that are underlying even our "basic" arithmetic. Hell, why this symbol is such a problem is because it is unclear in a very subtle situation.

People treat arithmetic as "basic" when in reality there are some realllllllly deep topics lurking just beneath the surface.

Like. Holy shit fraction division is REALLY just a result of ONE PROPERY of multiplication.

OKAY OKAY. So. But hold up.

Let's bring this back around though to another issue I wanted to point out: PEMDAS. Because that is another issue of confusion. Then, lastly, I promised why I told you this IS NOT a simple "we decided it"; there's a reason WHY that order exists.

When students are taught PEMDAS, they are taught "Please Excuse My Dear Aunt Sally"; do all parenthesis, then all exponents. And students who are taught without nuance are then taught
"Then do all multiplication. Then all division. Then all addition. Then all subtraction."

But Mike, why is that wrong? Like. That's what you do. And it's what I did. It's how I get the answer yo.

The problem with this strict PEMDAS order is that - remember, INVERSES.

We can rewrite ALL subtraction as addition.
We can rewrite ALL division as multiplication.

It's not actually PEMDAS.
It's
PE(MD)(AS).

All multiplication / division SAME TIME left to right.
Then + and -.

(Personally I, for one class, taught students just "PEMA" because all division is actually multiplication in a fancy hat, and all subtraction is dressed up addition, buuuut.

That didn't go over well. PEMDAS is what people know, so it's what I worked with).

So for example.

5 - 7 + 2.

If you did it STRICTLY in PEMDAS order, it would be
Parenthesis (none)
Exponents (thank god)
Multiplication (nadda)
Division (nope)
Addition (AH HA! 5 - 7 + 2 = 5 - 9)
Subraction (Dope! 5-9 = -4).

This is FALSE.

The deep theoretical reason is that we can rewrite the subtraction using additive inverses and then just do all the addition.

Practically? Just do addition / subtraction at the same time left to right.

5 - 7 + 2 = -2 + 2 = 0

(More fully: 5 - 7 + 2 = 5 + (-7) + 2 = 0)

Again, repeat after me:

THERE IS NO SUCH THING AS BASIC ARITHMETIC.
Even this innocent example has SOOO MUCH going on from deep deep properties that get summarized a little too compactly into PEMDAS.

Similarly, some people are getting the meme wrong because they're following PEMDAS in a religious too-strict order. They're doing ALL multiplication first, THEN all division; in reality, those should be done with the same priority left to right.

So let's break it down. Here's the meme question. I've added the multiplication sign because it makes it clearer than when you have the two symbols next to each other.
6÷2 x (2+1)

So, start with parenthesis. Okay. No drama.

6÷2 x 3

NOW though, division has the same priority as multiplication.

Again, the deeper theoretical idea is we can rewrite it as multiplication by the inverse. I'll show you that first:

6÷2 x 3
=6 x (1/2) x 3
=(6/1)x(1/2)x3
=(6/2)x3
=3x3
=9

That theory is gross and I made no attempt to explain the internal steps; feel free to ask away if you have questions. I kinda did that at warp speed.

The practical way to do this? Do division / multiplication same time left to write.

6÷2 x 3
= 3x3
=9

(Much cleaner)

Again:
THERE IS NO SUCH THING AS "BASIC" ARITHEMTIC. LOOK AT HOOOOOOW MUCH IS GOING ON UNDER THE HOOD THERE.

Again, that notation is really REALLLY unclear though if you have a lot of moving parts to the expression you're evaluating.

But there are a LOT of people doing PEMDAS too literally and that is factually incorrect.

HOOOKAAAAY. So. Have a rest here for a sec. I told you I wanted to show you the big high level view of inverses; I did that so you could see why doing PEMDAS too literally is a problem.

Now I want to pivot to my third point. Let's talk PEMDAS. Feel free to rest first tho.

(Also, fwiw, I am going to be just writing PEMDAS as a shorthand for "the order of operations"; see above where TECHNICALLY it's PE(MD)(AS) or PEMA or whatever. PEMDAS is what people know. So I use the language people know.

Some folks talk about PEMDAS as if it's just a big agreement or a convention; well, Mathematicians decided to do this first, then this, then so. So we all do it this way so everything is consistent.

That's. Not correct.

It is not a convention. There's a reason WHY.

To understand why. We gotta go back to third grade. So ELEMENTARY SCHOOL UP because YOU GOT THIS.

What does multiplication actually MEAN? What is 5x3?

It's shorthand! It's shorthand for repeated addition!

5x3 is either

5 + 5 + 5

or

3 + 3 + 3 + 3 + 3

(note because of fundamental properties of the real numbers, these two are equivalent)

Cool cool cool cool cool.

What about exponents?

You remember those?

Right. Let's review those real quick.
5^2 (five squared). In paper it would have the two being lil smaller and raised up. Yup. Them suckers. Exponents. That's the one.

THAT is shorthand for repeated multiplication.

5^2 = 5x5 = 25

3^5 = 3 x 3 x 3 x 3 x 3 = 243.

(NOTE: 3^5 does NOT equal 5^3. Try it and see for yourself).

This is crazy.
If you have exponents it's repeated multiplication... which is itself repeated addition.

Literally it's all addition.

(And if you use sigma and pi notations with variable indexes, it is actually possible to translate an expression like 5^2 into the repeated addition it actually is. The result would be....

...something finnicky I'll do later because the notation is a pain in the ass to type)

Okay it bugged the shit out of me to not do it, here is what it looks like. I'll do some more involved examples later because now I'm just curious.

Right? NO SUCH THING AS BASIC ARITHMETIC.
Look at how much shit is going on there when you peer under the hood even a little bit.

Next time you hear someone complain about exponents, think how much *work* it is saving us.

So.
Exponents are actually multiplication....
...which is actually... just addition.

Hmmmm.

Wait. Doesn't that sound familiar?

Order of operations, PEMDAS, starts with parenthesis - and that's by definition. Literally the definition of parenthesis in expressions is to SAY "do this first", so that makes sense.

But then you do exponents.

And then you do multiplication (and division, because division is just multiplication wearing a fancy hat).

And THEN you do addition (and also, the addition in a fancy hat that is subtraction).

Do you see where this is headed?

(NOTE: I did a dumb and broke the thread. It continues below the prior comment)