Eli5: the order of operations

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How did we discover PEMDAS? If math is just representative of real things, why does it matter the order? I know that going out of order yields the wrong answer, but why does going through PEMDAS yield the right one?

In: Mathematics

We didn’t discover it, order of operations is like grammar for math.

When you’re writing a sentence or an equation both the reader and the writer must agree on the correct structure, otherwise the meaning is lost.

The order of operations should be clear and easily followed. If it isn’t, the equation is poorly formatted.

Going out of order gives the wrong answer according to the convention we all follow. There’s nothing special in multiplication or addition etc. The convention was made to set a common set of rules through which it becomes easier to interpret complex expressions uniformly throughout the world.

The order is arbitrary and is no more objectively correct than any other.

However it is crucial that we all agree on the same order to use, otherwise two people doing the same equation will get different results. PEMDAS is just the one that stuck.

It’s arbitrary, an artifact of writing mathemathical expressions as we do in everyday use (“infix notation”). The only thing that matters is that everybody uses the same order or stuff goes tits up.

As an example of a notation that doesn’t *have* the concept of “order of operations”, there is the Reverse Polish Notation / Postfix Notation:

1 2 + 4 *

means the same thing as (1+2)*4 and

1 2 4 * +

is the same as 1+2*4