MagicSquare8-9 t1_iyays63 wrote on November 30, 2022 at 1:26 AM

It's "just a convention", but it's something so ubiquitous that pretty much many mathematician independently came to the same thing.

Historically, it used to be context-dependent, sometimes you add first. For example, look at this picture: https://en.wikipedia.org/wiki/File:JakobBernoulliSummaePotestatum.png . Can you guess which line you add first and which line you multiply first? If you understand the math, you can figure it out, but if not, this can be confusing.

Eventually, various mathematicians made up various different conventions on how to write and interpret formulas. Even though they produce different convention, one thing are shared among them all: multiplication takes priority over addition (unless superseded by other stuff like brackets and lines). And there is a good reason for that. Multiplication distributes over addition, so it's a lot easier to write an expression as a sum of product than a product of sum. In other word, a lot more formulas would end up requiring you to do all the multiplications first to get a bunch of products, and then doing addition to add up those products. So it is a lot simpler to make that the default: by requiring that multiplication takes priority over addition, you don't have to put brackets everywhere.

So even though it's "just a convention", there is a mathematical reason behind the convention.

Graega t1_iyb40dl wrote on November 30, 2022 at 2:06 AM

To put that in even simpler terms, imagine you have 5 boxes of 12 eggs, and an extra 2. That's 2 + 5 x 12. But if you add first, you'd have a lot more complicated way of trying to express that. It's not a law of nature, but there is a practical use case that made selecting multiplication first a clear choice.

Walty_C t1_iybftk5 wrote on November 30, 2022 at 3:38 AM

Why couldn’t you just write it 5 x 12 + 2 and solve it all left to right? Or just stick with parentheses as the operator. (5x12) + 2. Could math not be simplified in this way or would it fall apart with more complex equations?

bovisrex t1_iybhjf5 wrote on November 30, 2022 at 3:53 AM

You can and should. A large function in algebra (including basic mathematics) is in making a long problem easier to understand. The problem you wrote could also be written 12 + 12 + 12 + 12 + 12 + 2, but writing it as 5 x 12 + 2... or (5 x 12) + 2... makes it understandable at first glance. This is all that is meant by the common math problem "Simplify this equation."

Agreeing on a convention to write math problems a certain way just removes ambiguity and makes it possible for any person familiar with the relatively simple Order of Operations rule to understand any other problem written by someone. My biggest problem with the order of operations "riddles" that swarm Facebook from time to time is that they'll intentionally write them in a confusing way in order to trick people. It's the same as posting a scribbled rough draft of a paragraph and then making fun of people who misinterpret it.

Salindurthas t1_iybms9m wrote on November 30, 2022 at 4:39 AM

>Why couldn’t you just write it 5 x 12 + 2 and solve it all left to right?

Well, what if you are writing down the eggs you see, as you see them?

What if you spot 2 eggs, so you write that down, and then spot 2 cartons of a dozen, so you write that down.

Oh, dang, now you've written 2+ 5x12, and under your convention, that isn't what I meant (I've written 7x12, but I mean 2 more than 5x12).

Or, what if you have a formula like: "D = 2*x/y + 5*z^2"

You might be working in a lab, , and your assistant tells you "I've measured x & y, they are 7 and 3.5 respectively."

You're waiting for them to measure z, and figure that you'll simplfiy your equation, and you work out the left term happens to be 4 (2*6/3.5=4).

So you write "D = 4 + 5*z^2" to keep your equations easy to read and lined up.

Oh, dang again! That isn't what you meant, you need to change the order around!

I think it is better to have a convention where the order you write addition in doesn't matter, so that you can add things as you discover them/calculate them, rather than needing to reorder your equation every time.

You *could* do it, but I think if society did, we'd have more errors than using our current convention.