why in algebra class they teach the order of operations (PEMDAS) in that order. Is this just an arbitrary standard everyone agreed on or was it the result of higher math only making sense when equations are done in that order?

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Answer 1

Contrary to what the others say, it kinda does matter (although not explicitly to the letter).

Consider if you go to buy some drinks from the store but the shelves are nearly bare so all you can get are a couple of six-packs and a few singles.

These are some ways to write how many you got in total:

Using parentheses to group:

* (2×6)+3 = 15
* 3+(2×6) = 15

Using PEMDAS to know to do multiplication first:

* 2×6+3 = 15
* 3+2×6 = 15

Just evaluating left-to-right *without* using precedence:

* 2×6+3 = 15
* 3+2×6 = 30

Note how, in the case without using either the parentheses or precedence, you get different answers depending on the order, one of which is clearly wrong. This is because multiplication and division are higher-order operations than addition and subtraction.

More specifically, multiplication is just repeated addition and division is repeated subtraction. Now see if you did the repetition first:

* 6+6+3 = 15
* 3+6+6 = 15

They’re both the same, regardless of the order! (Since you’ve reduced it to a single level of precedence, the order doesn’t matter.)

(Also, exponentiation goes in front of the others because it’s a yet higher-order (repeated multiplication).)

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So, we can either agree to do higher-order operators first, which applies the repetition to reduce the levels of precedence, or we can use a lot of parentheses to specify the order.

But when I said at the start they don’t have to be explicitly to the letter, that’s because within a given level of precedence it doesn’t matter.

Suppose after you got those drinks you drank two, now how many are left?

* 6+6-2+3 = 13
* 3+6+6+(-2) = 13

Because subtraction and addition are just inverses of each other, which is to say subtraction is just addition of a negative number, they’re at the same level of precedence – it doesn’t matter what order you do them in within that single level of precedence.

And while multiplication and division are a higher precedence, they’re just repeated addition and repeated subtraction, so they are also inverses of each other, and on the same level as each other, so it doesn’t matter which order you do them in within that level of precedence.

It’s easier to just say always do them in PEMDAS order. But kinda useful to see how the orders of precedence arise. It might be a little more clarified by noting it with precedence levels: (P)(E)(MD)(AS).