# Why the order of operations?

by S_David
Tags: operations, order
 P: 599 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
 Mentor P: 18,240 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. Does that answer your question?
 Mentor P: 15,153 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).
P: 599
Why the order of operations?

 Quote by micromass 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. Does that answer your question?
OK, that makes sense. Thanks
P: 599
 Quote by D H 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).
I am sorry, I didn't get this one. Can you explain more, please?
 Mentor P: 21,258 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).
 Sci Advisor HW Helper PF Gold P: 12,016 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.
Mentor
P: 18,240
 Quote by arildno 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.
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.
 Sci Advisor HW Helper PF Gold P: 12,016 Of course, if we make default invisible addition sign, we'd get something not half-bad like this: ab*c=(a*c)(b*c)
Mentor
P: 18,240
 Quote by arildno Of course, if we make default invisible addition sign, we'd get something not half-bad like this: ab*c=(a*c)(b*c)
I like that!
 P: 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.
P: 599
 Quote by jing2178 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.
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?
Mentor
P: 15,153
 Quote by S_David 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.
P: 599
 Quote by D H Wrong.
Why?
 P: 223 9 - 1.5 anyway, a good question here would be: Why exactly does multiplication take priority over addition?
HW Helper
PF Gold
P: 12,016
 Quote by lendav_rott Why exactly does multiplication take priority over addition?
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.
P: 223
 Quote by arildno 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.
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?
P: 41
 Quote by lendav_rott anyway, a good question here would be: Why exactly does multiplication take priority over addition?
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.

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