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Why the order of operations? 
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#1
Mar114, 06:02 PM

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: 93/2 but rather (93)/2. Right? In other words, the vague order of operations doesn't appear when solving physical problems using mathematical tools. Thanks 


#2
Mar114, 07:49 PM

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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 welldefined 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 [tex]2+(3*(x^3))[/tex] 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? 


#3
Mar114, 08:44 PM

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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
Mar114, 10:15 PM

P: 599

Why the order of operations?



#5
Mar114, 10:15 PM

P: 599




#6
Mar214, 11:28 AM

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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
Mar214, 02:22 PM

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Well, there is nothing intrinsically wrong with having in default, nonparenthesis 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
Mar214, 02:31 PM

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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 [tex]10 + 20m[/tex] instead of [tex](10 + 20)m[/tex] And something like [tex](10m) + (20m)[/tex] is also not very nice. 


#9
Mar214, 04:50 PM

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


#11
Mar414, 03:41 AM

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. 


#12
Mar414, 09:25 AM

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#13
Mar414, 02:42 PM

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#14
Mar414, 02:44 PM

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#15
Mar414, 03:09 PM

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9  1.5
anyway, a good question here would be: Why exactly does multiplication take priority over addition? 


#16
Mar414, 03:40 PM

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


#17
Mar414, 11:08 PM

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#18
Mar514, 03:56 AM

P: 41

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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