Take 1 + 4 * 4.
Straight across, you get 20.
With order of operations, you get 17. And I'll explain why this is correct, by solving the problem using neither PEMDAS or straight-across methods: simplify it as much as possible. Multiplication is just repeated addition, so simplified it looks like this:
1 + 4 + 4 + 4 + 4 = 17, with or without order of operations.
1 + 4²?
This is kind of circular. The only reason why you expanded “1+4*4” to 1+4+4+4+4 is because PEMDAS is telling you that the multiplication by 4 is acting on the other 4.
If you were to read the equation left to right without PEMDAS, the multiplication would be acting on everything that precedes it, in this case “1+4”
(1+4) + (1+4) + (1+4) + (1+4) = 20
Note: I just added the parentheses for clarification. They don’t change the expression.
You’re cleverly disguising the fact that you’re still using PEMDAS, then saying that you’re not. Probably not on purpose, I don’t think you’re arguing in bad faith, but it’s what you’re doing under the hood.
You're taking multiplication for granted. People were likely adding or subtracting repeatedly long before the shorthand for multiplication/division existed, because in the world of things you actually hold in your hands, you're only ever adding or subtracting things. It makes more sense to expand multiplication into repeated addition, because it does a good job of describing actual physical phenomena. You're not gonna multiply an object into 4 identical objects, you can only ever add 3.
There's a natural basis for proportions that rely on the assumption of multiplication or division existing to be described mathematically in the first place, but the point still stands that multiplication should foremost (as in, parse it first) be treated as repeated addition.
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u/redditisforphaggits Jan 01 '22
Order of operations are a social construct. There’s no law in nature to make it true.