Mathematical proof for order of operations

Question

I was watching this YouTube video and at around 40:40 the speaker himself states that he does not know why we have the order of operations we have today. This got me thinking and I realize that I couldn't prove it or even think of a reason for any of it, except for maybe parenthesis because of how it looks when you read it, but that, I think, is based on syntax and semantics and not mathematics.

I tried finding something on the internet but all I get is how it is supposed to be, but not why.

So, is it possible to provide proof for the order of operations in arithmetic, and if so, what is it?

P.S: I can't find any suitable tags for this question so I'm going with arithmetic and math-history.

Answer 1

There is no such proof. The order of operations to which we are accustomed is really nothing more than a mathematical convention to which most adhere in order to help eliminate the alternative of ambiguity.

But it never hurts to use parentheses to designate operations to perform first (inner to outer), which is virtually universally understood, thus eliminating our reliance on convention in the hopes that others will know the convention!

Answer 2

The order of operation is defined, but not proved.

Answer 3

There is no such proof, but maybe one can find good motivations. The precedence of multiplication with respect to addition is probably a consequence of natural language:

3 cats and 2 dogs

is something like $3\times c + 2\times d$. This has to do with the fact that multiplication is sometimes expressed by adjectives in the natural language, while summation is expressed by a conjunction.

Answer 4

Order of operations is entirely for communication and clarity.

There was an old computer language, APL, that simply evaluated all expressions that were not in parentheses left to right, so that $1+2\times 3$ was evaluated as $(1+2)\times 3$. It's entirely permissible to do this, but the problem is, you end up writing lots of parentheses, which makes your expressions harder to read and your computer code inscrutable.

The purpose of order of operations, then, is for ease of communication and readability. We'd much rather read:

$$1+2x+3x^2$$

than:

$$\left(1+(2\times x)\right) + \left(3\times(x^2)\right)$$

An order of operations can be thought of as a systematic way of determining how to fully parenthesize an expression. The only requirement of an order of operations is that it not allow ambiguous expressions - allowing two different and unequal fully parenthesized expressions. The one we usually use was almost certainly chosen because it simplifies well more of the expressions we write.

Answer 5

There is no proof for that. It just organizes problems so if there is a problem such as 1 x 2 + 4 x 3 people would get the same answer; that's all.

Answer 6

You can do some kind of a weak proof, at least for multiplication and addition.

I will write $|c+$ and $|c\cdot$ for applications of the commutative property, $|d$ for applications of the distributive property, $|lr$ for from left to right and $|+$ for $+$ first.

As a sign of calculating something first I will use parentheses

First if there is no order of operations, so everything is calculated from left to right.

$$ I: a + bc = a + cb \quad | c\cdot \\ II: a + bc = (a + b)c \quad |lr \\ = (a + b)c = (b + a)c \quad |c+ $$

with I:

$$ a + cb = (a + c)b \quad |lr \\ = (a + c) = (c + a)b \quad |c+ $$

So it should be:

$$ (b + a)c = (c + a)b $$

lets use some numbers: $a = 2, b = 3, c = 5$

$$ (3 + 2)5 = (5 + 2)3 \\ 5 \cdot 5 = 7 \cdot 3 \\ 25 \ne 21 $$

So it is at least wrong for some numbers ouf of $\mathbb{R}$

Next: Let $+$ be stronger than $\cdot$:

$$ a + bc = (a + b)c \quad |+ \\ = (a + b)c = ac + bc \quad |d \\ = ac + bc = a(c + b)c \quad |+ \\ = a(c + b)c = (ac + ab)c \quad |d \\ = acc + abc \quad |d $$

So it should be:

$$ (a + b)c = ac(c + a)bc \quad |+ \\ $$

Again lets use some numbers.

$$ (2 + 3)5 = 2 \cdot 5(5 + 2)3 \cdot 5 \\ 5 \cdot 5 = 2 \cdot 5 \cdot 7 \cdot 3 \cdot 5 \\ 25 \ne 1050 $$

So again it is at least wrong for some numbers ouf of $\mathbb{R}$