# Why the order of operations?

1. ### S_David

601
Hi,

I am wondering why the order of operations where made? Is it the case that mathematics is just a tool of solving physical problems, and hence the order is physically defined? For example, if I say: I took 3 apples from 9 apples and divided the rest to two persons equally, I won't write this mathematically as: 9-3/2 but rather (9-3)/2. Right? In other words, the vague order of operations doesn't appear when solving physical problems using mathematical tools.

Thanks

2. ### micromass

18,440
Staff Emeritus
The "order of operations" isn't really of much mathematical importance. It's just defined to make notation easier (and boy, does it make notation easier!!).

For example, something like ##2+3x^3## is well-defined and requires the order of operations to interpret. If we don't like to define some order of operations, then we should denote it as

$$2+(3*(x^3))$$

So we would need brackets in order to clarify the order we want to do things in.

Obviously, this notation is very much tedious and annoying. And this is exactly where the order of operations comes in.

1 person likes this.

### Staff: Mentor

It's more than just a matter of convenience. It's rooted in the fact that with numbers as most people know them, multiplication distributes over addition: (a+b)*c = (a*c)+(b*c). Addition does not distribute over multiplication (boolean algebra excepted).

4. ### S_David

601
OK, that makes sense. Thanks

5. ### S_David

601
I am sorry, I didn't get this one. Can you explain more, please?

### Staff: Mentor

What D H is saying is that addition doesn't distribute over multiplication. IOW,
(a * b) + c ##\neq## (a + c) * (b + c)

Multiplication does distribute over addition, so it's true that (a + b) * c = (a * c) + (b * c).

7. ### arildno

12,015
Well, there is nothing intrinsically wrong with having in default, non-parenthesis notation addition bind more tightly than multiplication.

But, the law of distribution would then get an UGLY, and not the least, confusing form:

a+b*c=(a*c)+(b*c)

Letting multiplication have priority over addition simplifies the expression for the distributive law, and prevents a lot of mistakes that would be made if we reversed order of operations.

8. ### micromass

18,440
Staff Emeritus
Maybe it's confusing because we're more used to the alternative /DevilsAdvocate

One other motivation is when working with units. You want to write things like ##10m##, and this is some kind of multiplication. So it makes sense to bind multiplication closer together than addition. I mean, you don't want to write

$$10 + 20m$$

$$(10 + 20)m$$

And something like $$(10m) + (20m)$$ is also not very nice.

9. ### arildno

12,015
Of course, if we make default invisible addition sign, we'd get something not half-bad like this:
ab*c=(a*c)(b*c)

10. ### micromass

18,440
Staff Emeritus
I like that! :rofl:

11. ### jing2178

41
The order of operations does make sense in the physical world. In your example there are 9 apples and 3 apples are taken away before the apples are shared between 2 people. Thus the subtraction 9 apples take away 3 apples to give 6 apples is done first then the 6 apples are shared between the two people who get 3 apples each.

Think of the physical space where the apples are kept.

I put 9 apples on a table, 3 are removed from the table (perhaps they have been eaten), the remaining 6 on the table are shared between 2 people.

In the mathematical model how do we represent the physical space, the table. We could use a line for example 9 - 3 / 2 but in practice we use brackets to represent the physical space in which items exist so (9 - 3)/2

Without brackets we can assume that the numbers represent objects in the same physical space so in this case 9 - 3/2 would model 9 apples on the table from which three half apples have been removed, since - is always applied to two quantities or numbers and the two numbers shown are 9 and 3/2.

12. ### S_David

601
Yes, but you are trying to understand the physical world from the mathematical equations. But if I told you the physical problem and write it mathematically as 9-3/2 you know that the answer would be 3. Right?

Wrong.

601
Why?

15. ### lendav_rott

223
9 - 1.5

anyway, a good question here would be: Why exactly does multiplication take priority over addition?

16. ### arildno

12,015
It doesn't. If we use parentheses.

The priority rule is established so that we may escape tedious parenthesis writing in about "half" the cases.

17. ### lendav_rott

223
Yes, but suppose I'm a complete math illiterate. How would you explain without parenthesis, that 10 - 5 / 5 = 9 not 1? There are no brackets, yet multiplication comes before addition, why is that?

18. ### jing2178

41
It could have developed that arithmetic was calculated from left to right so that 2 + 3 * 5 = 25

and to show that the multiplication needed to be done first then you would write 2 + (3 * 5) = 17

So why not this and why multiplication first? Mathematics did arise from the physical world.

Buying some goods from a market stall selling fruit calculations would be done as
(I'm English so cost in pence)

1 apple @ 10p 10p
5 oranges @ 12p 60p
3 bananas @ 15p 45p

Total 115p

In doing this the multiplications have to be done before the adding.

Also

If you have a boat and 3 crew you have a boat and 3 crew. You are counting different things so adding the 1 for the boat and 3 for the crew does not make sense.

So if you have one Two and three Fives you cannot immediately add the 1 and 3 since you are counting different things Twos and Fives.

However unlike boats and crews Twos and Fives have something in common - Ones.
A Two is two Ones
A Five is five Ones

So if I can change three Fives into a count of Ones then I can count all the Ones and work out how many Ones there are in one Two and three Fives

three Fives is five Ones and five Ones and five Ones, since now I am counting the same things (ie Ones) I can add the fives to get fifteen Ones

so one Two and three Fives is the same as two Ones and fifteen Ones which is seventeen Ones or one Seventeen.

Since you can only add quantities together if you are counting the same item you need to change the Two and three Fives to Ones. To do so the multiplication as repeated addition needs to be done first. So giving multiplication priority over addition makes sense.

Last edited: Mar 5, 2014
19. ### arildno

12,015
Because modern math education in arithmetics is WRONG and UNPEDAGOGICAL.

Parenthesis use, for arithmetics between more than 2 numbers SHOULD be taught, and children will take it easily.
I have myself taught one child with a diagnosis of slight mental retardation,in half an hour how parentheses are just "math grammar/syntax", that tells what we should do first.

The child found it fun to see how 5-(3-2)=4, whereas (5-3)-2=0.
That is, understanding how parantheses matter.

The major fallacy in today's school teaching is to introduce parentheses at a LATE stage, i.e, when the kids are to be taught how to REMOVE them in order to get a logically equivalent expression i.e, how to rewrite 5-(3-2)=5-3+2, rather than train kids FIRST in calculating 5-(3-2)=5-1=4, and similarly (5-3)-2=2-2=0

Last edited: Mar 5, 2014