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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The first. It’s just the order that everyone standardized on, so that we can all write equations and interpret them the same way. The actual order of the operations is not as important as making sure that we’re all doing them the same way.

It’s a purely arbitrary standard. Because we all agree on the order of operations, we can omit parentheses and still unambiguously write complicated equations.

If there was no standard, we could still communicate about math, but there would be substantially more parentheses involved.

Its just an agreed upon standard. Back in the day mathematicians used to just list out whatever the order of operations they used so the math would be consistent. At some point we agreed this is sort of stupid and academia just standardized it.

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I hate PEMDAS as an acronym… It should be PEMA. Multiplication/division are done at the same time, and same with addition/subtraction. It’s four steps. PEMDAS kinda suggests 6 steps, and thats specifically why it’s common to see it get screwed up.

While the standard is just an agreed convention, it’s not really arbitrary. It can be summed up by saying “do the higher-order (more difficult/complicated) operations first”. Addition and subtraction, essentially the opposite of each other, are treated as equal and, within those two, the order of operation doesn’t matter. The same goes for multiplication and division. It’s only when you get to exponents that you need to know to go from right to left, i.e., 2^3^4 is 2^81 and not 8^4.

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).

It’s not entirety a random choice. Parentheses, for example, kinda have to go first. If they appear anywhere else in the order they function as a deprioritizing tool, and this gives you fairly convoluted equations if you need to group certain terms, where you rapidly need to start adding multiple parentheses to ensure the terms are treated as a group. Similarly exponents kinda need to come next (again as they function as a group, or you start to need a lot of parentheses to get them to work.

It’s also not really arbitrary that addition/subtraction and multiplication/division are equal in precedence as division is just multiplication of the inverse of the number you are dividing by (Eg 5/3 is 5 x 1/3. They’re the same functions so it makes sense they are equal in precedence.

Although you could argue the division should be after multiplication as it *could* be taken to mean to include multiple terms before and after- and when writing by hand this is how it is commonly used:

A**x** + B

——————-

C**x** + D

When writing on screens you’d normally have to format it:

(A**x** + B) / (C**x** + D)

But on balance there are a *lot* of cases where you’d have to add in parentheses to treat division like this, so it makes more sense to use parentheses to group terms when needed, albeit this gives rise to the debate over “implicit multiplication” where terms are assumed grouped for multiplication when they’re adjacent to each other without an actual sign.

Eg:

Y = 1/3**x**

Where if x = 2 you’d wind up with 1/6 instead of 2/3 – this is common notation in engineering and science, but not in grade school, which results in those viral math problems you see floating about about the order of operations.

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Standard? I thought it was BODMAS (UK Cambridgeshire 1990s)

>was it the result of higher math only making sense when equations are done in that order

If you think about it, you would realize that isn’t the case. We could just have parentheses around everything a hundred times to communicate whatever we want. Time if we were doing 5 * 2 + 3, we could do the 5 * 2 first by writing (5 * 2) + 3 or we could do the 2 + 3 first by writing 5 * (2 + 3). We just agree on doing PEMDAS in that order because it means that we can use less parentheses in most situations

The “bigger” operations go first. Parenthesis are the exception, they always go first, but, for example, X^Y is a much more dynamic curve than X times Y which is more dynamic than X + Y.

I see a lot of folks saying it’s arbitrary, It’s just math and it doesn’t matter how we do it as long as we all do it the same. That is not correct. Math represents real life. You do pemdas because following this rule guarantees you will get real life results. Here’s the best example I’ve seen in the past (for this example forgive the strange farm that has octopus’ and consider their tentacles legs):
You go to a farm and there are 5 chickens, 3 cows and 2 octopuses. How many legs are there? Well if you count the legs you get 38. If you use math:
5×2+3×4+2×8 = 10+12+16 = 38 legs. This is why you must do multiplication first. It’s not arbitrary. Now the way we write this (symbols of + x etc and use of parenthesis, sure that’s arbitrary) but order of operations is required to have math match real life. There’s still confusion around other elements of pemdas, and memorizing rules is a terrible way to “understand” things. but the bottom line is, it’s not arbitrary. Source: I’m an engineer. Edited for clarity

When did it become PEMDAS instead of BODMAS? Or is this a UK / US thing?

My brother has a maths degree and explained it like this.

Imagine you have a sum, something like 8+8+8+8+2, for the sake of saving some space you could just write 8×4+2. The multiplication symbol here is just shorthand for all that extra additions, so it’s kind of like expanding the sum back to what it “actually” is by using PEMDAS

Now imagine 8+8+8+8+8+8+8+8+7+7+7+2
That can be considered
8×8+7×3+2
or to use another bit of shorthand
8²+7×3+2
You could write these with parentheses if you wanted to be extra doubly certain but PEMDAS means you can trust whoever is reading it won’t need them

TL;DR: expanding shorthand

No it’s not arbitrary.

All operations all just different and handy ways to add. And addition is a way to group things and count them.

So in order to calculate you must reduce everything to additions. And in order to reduce everything to additions you just start doing the most complicated things first.

So 3×2+2(1+3) = 3+3+1+3+1+3 = 16.

That’s it.

3^2 + 2×3+4/2 = 3×3+3+3+4×1/2 = 3+3+3+3+3+1/2+1/2+1/2+1/2=19