# eli5: how did PEMDAS came to be? what decided this order and rule?

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like how does it make sense? why is it correct compared to just solving equations left to right? what are some scenarios that prove PEMDAS to be the absolute truth and necessity? @_@

In: Mathematics

PEMDAS is a convention for *writing* mathematical equations. There is nothing better or worse about it. People long ago decided equations looked cleaner that way.

P has to be first because it overrides all the other orders. Putting MD before AS is something you learn in junior high. It makes coefficients and like terms look cleaner. So, where does E go… they just decided to put it there.

The main point of the exercises in math courses is that you have to learn how to *read the equation with the same convention that was used to write it*.

It’s not meant to be confusing. Later on, if something looks ambiguous, then parentheses will be added to clarify. It’s not meant to be a “trick”.

Without PEMDAS, you’d need parentheses everywhere. Equations would look ugly.

PEMDAS is not an absolute truth. It is just a convention that is generally agreed upon so everyone understands each others’ expressions. You could invent another notation that works equally well.

>what are some scenarios that prove PEMDAS to be the absolute truth

It’s not. There is ambiguity in PEMDAS and as such you will notice that parenthesises are heavily utilized in university (higher) level math to avoid any possible ambiguity.

PEMDAS is mostly good enough and works by convention. The goal of order of operations is to make sure that everyone gets the same results.

But without properly using brackets you might find different calculators will give you different results due to this ambiguity.

PEMDAS is a convention that is linked to how we write equations. The order is arbitrary and we could rearrange it, but we would need to write our equations differently to match.

If you don’t follow the PEMDAS order of operations for equations written with that expectation then you will end up with the wrong answer. The equation 5+4×7 should result in the answer of 33 because first you multiply 4 by 7, then add 5. If you just solved it left to right you would add 5 and 4 to get 9, then multiply by 7 to get 63 which is wrong.

You could flip the order of operations to something like PEASMD so you first add then multiply, but you would need to change the equation to something like (4×7)+5 to get the same answer.

It’s doesn’t have to be. It’s just what mathematicians decided.

However it makes sense. Parenthesis groups stuff together so it makes sense to do that stuff first before it interacts with stuff outside. Multiplication is shorthand for repeat addition so it makes sense to do that first before adding another number in (meaning 6•3+2 is just shorthand for 6+6+6+2, so it doesn’t make sense to change it to 6•5).

Its like asking why is reading left to right the absolute truth instead of reading right to left or top to bottom. It isn’t, it was chosen arbitrarily, and it doesn’t matter what convention we chose, just that everyone agrees on it so that we all get the same answer when we solve the same equation. If we had chosen a different convention, we would write the same equations differently, with parenthesis in different places. Just like how if we read right to left, we would write the same words differently.

For example if I have y = 5 * x + 2 under PEMDAS, and I wanted to instead use a convention where + happens before *, I would instead write that as y = (5 * x) + 2, to keep the same meaning.

Parenthesis do have to go first tho because the meaning of the parenthesis character is “do this first” like that’s it’s meaning as a symbol.

As everyone has mentioned, PEMDAS isn’t baked into math but rather it is just how we agreed to write the math. Here is a concrete example.

Say we have the equation 4 x 5 + 10.

Let’s pretend there is no order of operations. This could mean one of two things.

You have four buckets and each bucket has five apples in it. You have ten apples in the side not in a bucket. How many apples do you have? You have 30 apples.

You have four big buckets. Each of these buckets were advertised as being able to hold ten more apples than the previous buckets. How many apples can you now carry? You can now carry 60 apples.

There are some parts of PEMDAS that aren’t arbitrary. MD and AS have to stick together. This is because they are the same things.

5 ÷ 6 is the same as 5 x 1/6. Likewise 3 – 5 is the same as -5 + 3. Since M & D, and A & S are the same they need to be fine together.

E doesn’t make any sense without PEMDAS because it would become arbitrary where it goes. If the order was ASE then 4^2 + 5 would be the same as 4 + 5^2 since in both cases you would add 4 & 5 first.

P likewise makes even less sense as, without being first, it becomes meaningless.

You could do some alternative to PEMDAS like go from left to right but this would make many complex problems impossible or at least extremely difficult.

PEMDAS isn’t an absolute truth of mathematics; it’s just a convention for writing mathematics equations.

If we didn’t use PEMDAS, we’d be using something else.

>why is it correct compared to just solving equations left to right?

Because left-to-right is meaningless.