I was a high school math teacher. A former student asked me my opinion on a social media post asking what

8 ÷ 2 (1 + 3)

is, with the common answers being *16* and *1*.

I’m sort of a “math person.” Not Phd math solving a Clay Institute math prize get a million dollars math person, but I can do reasonably well through differential equations and teach physics for part of my living. I also write computer code for a majority of a living. So my answer to what that is: **It’s a context free meaningless computation**, written poorly.

It begs the questions — why are you adding one and three? What do they model? Are there any units on those values?

### What’s Math

If we have the concept of counting, and values of one, three, and four, along with the concept of addition, then one plus three equals four, regardless of whether we use words or 1 + 3 = 4 or uno más tres es igual a cuatro.^{1} That’s math, it’s always going to be true. That’s the way the universe was made.

^{1}. The equation in Spanish, according to google translate.

### What’s not Math

Writing the calculation with the plus sign between the numbers is a convention called *infix*, and it’s what normal people expect.

Infix isn’t the only choice — the *postfix* expression of that equation would be:

8, 2, 1, 3, +, *,

÷

which means put the four numbers in a pile (a “stack”). Then replace the last two with their sum, making the stack 8,2,4. Next multiply the last two, yielding 8, 8, then do the division to get 1. Calculators that work that way are called Reverse Polish notation (RPN) and I still own (and use) the HP-11C I bought in 1979 or 80.

So, back to the original question — because of the ambiguity of infix, there’s a generally accepted convention for what to do first, called order of operations. For a programming language, it can run fifteen levels deep (computer nerd link). In the US, this often taught as “PEMDAS.” ^{2} But this is just a convention, followed sometimes: according Wikipedia, different calculators might calculate the expression differently.

^{2}I hate PEMDAS, but that’s a separate blog entry.

### Back to math

##### Adding some trim

I’m replacing some trim on a cabinet like so:

The wood is $8 / ft, but it’s twice the width I need, so I can cut it in half. Let’s write the equation:

So now I have some units and, because my goal is to communicate clearly, even if just to myself, I’ve written the first division as a vertical fraction rather than the linear form. This makes it obvious we’re going to end with $16 of trim.

##### Friends for pizza

So you and your partner are going to invite 3 couples over for pizza. A pie has 8 slices. How many will each person get?

So now I have some units/labels and, again, the layout of the equation makes the computation clear: each person gets 1 piece.

## Math education and social media

Unfortunately, math is too often “taught” as a set of arbitrary rules to memorize than what it is: a cool and fun recognize of patterns in the universe. Addition: *beauty.* “PEMDAS:” *bleh*.

Then we get social media posts which really have nothing to do with math, and are just math abuse so some people can feel superior to others because they can memorize some rules better than others. What I hope the two examples above show is this: If you’re actually doing *math* — solving problems with a context — then the order of operations should just make sense. If you take the effort to write equations *clearly*, then the reader doesn’t have to rely on some memorized rules to understand what you mean.

Writing:

8 ÷ 2 (1 + 3)

then sneering if someone doesn’t come up 16 doesn’t make you “smart.” It makes you a jerk.

Very clear explanation and I like your conclusion