What is the reason behind the current Order of Operations? (PEMDAS)

Question

After reading through a few other questions I was just asking myself: How was the Order of Operations defined, and why is it this specific order and not a different one?

Most of us know things like multiplication/division before addition/subtraction, parentheses first, etc - but what's the actual reason behind it? I'm probably biased by following those rules since childhood, so I can't really think of any other way.

2 + 2 x 2 = 6 and not 8

But if the order would be changed, let's say to "addition/subtraction before multiplication/division" would that order still work if we assume that mathematics would build up on it? Or is there some strange mathematical problem if we would be using a different order?

Of course, parentheses have a grouping function so they should always come first - I'm mainly talking about the exponents, multiplication/division, addition/subtraction (and maybe other operations left out).

Answer 1

As Zach Stone pointed out, the order of operations is just a convention, and if you choose to change the order, all that would happen is you would need to use parentheses in different places. Everything would work out fine as long as you made the correct adjustments. That being said, there is a reason for the convention. In some sense multiplication is just repeated addition. Furthermore exponentiation is just repeated multiplication(as long as we restrict ourselves to integers) therefore it makes sense to first turn all exponents into multiplication, then turn all multiplication into addition, and then compute the addition problem. Thus, at least as far as the integers are concerned, there is a natural ordering of the operations based on their definition. It gets more complicated when you start dealing with all real numbers, but the order is inherited from integer arithmetic.

Answer 2

Polynomials are important in their own right, regardless of notation. We have settled on a notation that makes them easy to write. We 'could' just use parenthesis between every operation, but that would be terrible. It's just a convention to simplify reading and writing. Changing the convention wouldn't break anything, we'd just need a lot of parenthesis to express what we want.

One place where polynomials occur organically is in field extensions. One could actually do quite a bit of Galois theory without ever explicitly writing down a polynomial. Similarly, polynomials occur organically in linear algebra (one can use tensor products to abstract abstract away things like characteristic polynomials). It would be harder, but equally powerful. But this helps to justify why we care about polynomials and why we might want to write them down.

Answer 3

In the minds of men and demons alike, multiplication is a more important operation than addition. Also, humans developed algebraic notation from their main language, which was Latin when I was born but is now English.

I still remember when Lincoln gave the Gettysburg address. "Fourscore and seven years ago..." That's $4 \times 20 + 7$ years. In German he might have said "Sieben und achtzig Jahre ­ſind verfloſſen, ſeit unſere Väter auf dieſem Continent einer neue Nation..." That's $7 + 8 \times 10$ years. Dozens, scores, hundreds, grosses, that's how you address a lot of integers, and then you adjust with little additions or subtractions.

A lot of this is just commonsense. But it was the development of computers that forced humans to codify a lot of this commonsense. Whereas a human understands what is meant by $$\prod_{n = 1}^\infty 1 - \frac{(-1)^n}{\phi^{2n}}$$ a computer needs to be given this as $$\prod_{n = 1}^\infty \left(1 - \frac{(-1)^n}{\phi^{2n}}\right).$$

Look into almost any book about a computer programming language, or even a scripting language like Javascript, and one of the very first few things you will see will invariably be a table of operator precedence. A computer, in its glorious stupidity, needs to be told in what order to do operations.

Answer 4

Let's pretend multiplication came after addition. Try writing this without parentheses: $$(a\times b)+c$$ You'll find this very hard. With the current order of operations, we can almost always get rid of parentheses, by using the distributivity laws.

In fact, distributivity is what determines the order of operations. Exponents distribute over multiplication (i.e. $(a\times b)^c=a^c\times b^c$), so exponents come before multiplication. Multiplication distributes over addition (i.e. $(a+b)\times c=a\times c+b\times c$), so multiplication comes first. With PEMDAS, we can get rid of parentheses using distributivity. With a different order ("PEASMD"?), we can't.

Answer 5

This is really a linguistic question, so the answer is a typical linguistic answer: the order of operations is as it is because it made communication more efficient.

We change the format of our notation to suit our needs. In the case of operator orders, it was generally found that formulae were more readable with the order of operations (likely due to the reduction in number of grouping symbols).

Consider the equation for motion with a constant acceleration $x = 1/2at^2 + vt + x_0$ If we did not have some order of operations similar to today's rules we'd have to write $x = (1/2a(t^2)) + (vt) + x_0$ Could we write it that way? Sure, but it's harder.

Over the years, mathematicians found the current order of operations to be extremely convenient, so they stick to it.

This fuzzy process is also the explanation for the famous brain teaser $6/2(3) = ?$. Some believe this should equal 9 because it's the same as $6 / 2 \cdot 3$. Others believe it should equal 1 because the multiplication by parentheses binds "tighter" than normal division: $\frac{6}{2\cdot3}$. They have some leg to stand on because most of us agree that $6/xy == \frac{6}{xy}$, so its not clear which way this should go. The real answer is that this is a linguistic ambiguity which exists because it hasn't been important enough for the greater body of mathematicians to agree upon it. If it ever actually became important, we'd decide one way or another.