system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: Why does the mathematical Order of Operations exist?

We multiply and divide before we add and subtract. Why? Please don't guess. Back it up. Everybody answer I've read is a guess.
gpt
The people who are saying "it's arbitrary" are wrong. Recall that multiplication is repeated addition, and exponentiation is repeated multiplication. Order of operations ensures that the answer you get when you solve "3x2+3" is the same as the answer you get when you solve the *mathematically identical* problem "2+2+2+3."

If order of ops were any different, then our definitions of addition, multiplication, etc would also necessarily be completely different.

Let's look at an example. Keep it simple: only multiplication and addition. Make it concrete: let's look at 3x2+3 again. Multiplying first gives us 9. Adding first gives us 15.

Why multiply first?

We could have easily written out the problem as 2+2+2+3.

Multiplying first gives us 9. The written-out version gives us 9. So *multiplying first gives us the right answer.*

If we added first, *we wouldn't get the right answer.*

Same thing is true for higher-level operations like exponentiation and whatnot.