What makes non-PEMDAS answers invalid?

It seems to me that even the non-PEMDAS answer to an equation is logical since it fits together either way. If someone could show a non-PEMDAS answer being mathematically invalid then I’d appreciate it.

My teachers never really explained why, they just told us “This is how you do it” and never elaborated.

In: 834

PEMDAS is like grammer for math. It’s not intrisicly right or wrong, but a set of rules for how to comunicate in a language. If everyone used different grammer maths would mean different things

Example

2*2+2

PEMDAS tells us to multiply then do addition 2*2+2 = 4+2 = 6

If you used your own order of operations SADMEP you would get 2*2+2 = 2*4 = 8

So we need to agree on a way to do the math to get the same results

In essence, it’s done for simplicity sake. It’s just something everyone can agree on, it could have easily been DEPSAM or some other abbreviation. However, think about it like this, exponentiation is repeated multiplication, multiplication is repeated addition, and addition is just counting up. It’s placed in the order of highest influence, but is still just done to create uniformity.

PEMDAS isn’t required.

What is required is that everyone agrees to the same order of operation.

Everyone needs to be on the same page in which order a term is processed.

If everyone agrees that we process the terms according to PEMDAS that works. If everyone agrees that we simply go left to right, that works too.

What doesn’t work is if some people want to read a term one way and some other people want to read it another way. That doesn’t work.

It is like finding a word written down and arguing whether reading it as a French word with French pronunciation and meaning or as an English word with English pronunciation and meaning is more correct.

One way of reading a word is not more correct than another, what is important is that everyone agrees on a single way to interpret the word in the context it is in otherwise it has no meaning at all.

For the same reason we require you to treat a + as “addition”.

Yes, the equation would still *look* fine and logical if you decided that a + now means “multiplicaiton” and that `*` means “subtraction”. You could also decide that the symbol “17” now means “two hundred and forty point three”. It would be mathematically valid, it just wouldn’t mean whatever the author *wanted* it to mean.

If I write `2 + 3 * 7`, my intent is for you to read it as “two plus the result of multiplying three by seven”. If we follow the same mathematical rules then you will be able to read it the way I intended it.

These conventions are communication tools. They allow us to write things down and have other people read them *and gain the same understanding*. If you don’t follow the same conventions as everyone else then you won’t understand what they meant by what they wrote, and they won’t understand what you mean with what you write. Then you’re no longer speaking the same language.

Just like any language needs grammar, mathematics is a language that needs rules to be intelligible by everyone. If we resolved operations with any made up order two people would get different results for the same equation, and would write it differently to say the same thing, which is obviously not very practical. As such, everyone agreed to use this one made up order.

You can write words wrong but people will not understand what you are saying, so it is in the best interest of everyone to write words right. Right in this case means “As everyone else”. Same principle

Edit: By the way I had a similar problem the first time I started with technical drawings back in the day. I didn’t understand why one of the drawings was wrong, and it turns out that it was because I didn’t follow certain conventions. Which is vital, but at the time I didn’t understand the concept and the teacher just kept saying “that’s just how it is done”. Looking back it’s just that she was dumb as a rock, a teacher that can’t clearly explain to a kid something so simple yet so vital is a bad teacher