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were does order of operations come from?

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Well, the basic ones are a derived from mathematics itself:
Parenthesis(sp?),Exponents,Multiplication,Division,Addition,Subtracion. Easily remembered by "Please excuse my dear Aunt Sally." The other ones, I am not so sure about at the moment(it is kinda late).

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In algebra terms, order of operation comes from topology.
Take your favourite algebra book and look up "semigroups", "groups", "rings" and "fields".

In short, you use two basic operations (like addition and multiplication) and define a set of rules for combining them.
The order of operations is a consequence of fullfilling these rules, which define the topology of the space they are defined in.

I hope this gets you started, if not - feel free to ask.

Pat.

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Tradition, I believe. Standard order of operations seems to have come about over time rather than any hard logical reason. It generally makes sense, but there might be a couple more rules than the minimum necessary, like the precedence of multiplicative operators over additive ones. Realise, though, that there are a number of different standards, reverse polish being one, that do not necessarily have this rule (or other rules the standard form has).

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Quote:
Original post by Doc
Tradition, I believe. Standard order of operations seems to have come about over time rather than any hard logical reason. It generally makes sense, but there might be a couple more rules than the minimum necessary, like the precedence of multiplicative operators over additive ones. Realise, though, that there are a number of different standards, reverse polish being one, that do not necessarily have this rule (or other rules the standard form has).

But isn't reverse polish notation just what the name states - a notation (and a very convenient one in some cases) rather than a set of rules? After all it's just a question of definition - as long as the rules are sound you might as well use non-standard order of operation.

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Quote:
Original post by Luke Thompson
Well, the basic ones are a derived from mathematics itself:
Parenthesis(sp?),Exponents,Multiplication,Division,Addition,Subtracion.

Your list is redundant: Substracting a from b is the same as adding the (additive) inverse of b to a. Dividing a by b is equal to multiplying a by the (multiplicative) inverse of b.

a - b ≡ a + -b

a / b ≡ a * b-1

Furthermore, multiplication can be defined as addition as well:

4*a ≡ a + a + a + a

though this only holds for natural numbers [wink].

Additionally, the proper term is powers, not 'exponents'.
Exponents are parts of powers [smile].

Cheers,
Pat.

PS: I admit this is a bit pendatic[smile].

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Yes and no. IIRC, polish (reverse or not, I'm not sure) notation:

2 3 4 + * is equivalent to 2 * (3 + 4). You'll note that I must put the barckets in for the order of operations to be correct, but the order is different in polish notation. Order of operations is pretty much a consequence of the notation used, that's why I thought it could be rooted in tradition (though Group etc. theory is possible).

2 3 4 * + => 2 + 3 * 4

No parentheses necessary. Notice that polish notation does not require parentheses to describe any weird combniations of ops. (I haven't studied much polish notation, so I could be talking uot my arse, but from what I understand of it seen this is the case.)

(8 + 7 * 10) * (2 + 3 + 4 + 5) * (4 + 1)
in standard math notation becomes
8 7 10 * + 2 3 4 5 + + + * 4 1 + *
in polish. Not easy to read unless you've practiced, but very easy for a computer to parse, and parentheses are redundant.

Sure you could rearrange the numbers, but I'm illustrating something here. Besides, you can only do that if the operations commute, which is not he case in general. Anyway, my point is the order of operations very much depends on the notation used.

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Doc: You're right about notation (btw. your examples are reverse polish notation[smile]).
Though it doesn't affect the general of the order of precedence as defined by the rules of addition and multiplication. IMO it's a very convinient way to express explicit statement (e.g. getting rid of parenthesis). It's more of a transformation rule that offers obvious advantages over Peano-Russell-Notation in first order logic for example (e.g. when building syntax trees).

[side note]
Interestingly reverse polish notation was introduced by HP because it was the only simple way to input complex formulas into the their first calculators and is rumored to be still around in some models [grin].
[/side note]

But with regards to the OP's question you are of course 100% correct - order of operations is defined by the notation that is used.

[edit]
Historically, order of operation rooted in notation and was later formalised into modern mathematical concepts like group theory. This did not take happen until the midddle of the 19th century, though.
[/edit]

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Well most math comes from concepts in the physical world, so I'd think the order is based upon the fact that it JUST DOESNT WORK any other way. I don't know, I ain't no math major.

Which brings up the question, could you derive equations for things like velocity, acceleration, etc, where it would always give the correct outcome if you did addition before multiplication? What about if you had to then re-arrange the equation algebraically and solve for something else, would the system still work?

Perhaps the order of ops we have is the only possible way to do it?

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Quote:

Which brings up the question, could you derive equations for things like velocity, acceleration, etc, where it would always give the correct outcome if you did addition before multiplication?


You'd just have to place the parantheses at different places.

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Guest Anonymous Poster
You asked a deep question that most people do not consider.

As with all ideas, if they can't be simply explained after a little bit then the person presenting them isn't doing a good job.

Mathematics is grounded in our sensory experience. That means that mathematics isn't something outside of our experience in everyday life, however much some people may want to think it is.

Mathematics is the application of logic to basic assumptions. The basic assumptions have to be made to avoid circular reasoning. Why are certain assumptions made and not others? Good question! Pretty much because they represent our common intuition about things from living in this world. Sure, the names of things and the exact specification of them make the assumptions not seem to be grounded in our everyday life, but they are. You could make a set of your own assumptions and apply logic to them to discover new things, but what you discover wouldn't be of much interest since it would almost certainly never be applied to anything in the real world.

Mathematics has so many uses in our world because it is based on the intuition of everyday observances. So it is not a miracle that it works out, or that this or that relationship occurs. People that believe otherwise are the mistacle type that want what they study to have some purity or existence outside of our own.

Anyways, now that I got that out of the way. Let us delve into the order of operations.

First, what is an operation? If we just had the concept of number, we could say that there were 1 apple or two oranges. To make things useful, we use the observation that we all have when we are growing up that we can take a group of 1 thing and a group of 2 things to make a group of 3 things. That is the operation of addition. It is just us using our experience of making one collection from two collections in the everyday world become a mathematical rule that is symbolized.

Why is it symbolized? Because our brains find it more difficult to tackle "a added to b minus c minus 2" than "2+b-c-2". The difference is more pronouced when we perform other operations. Also, we can apply typographical rules that require no understanding at all, much like a computer does.

So that is the operation of addition for positive numbers. The numbers that we all understand from daily life.

What is multiplication? It is the grouping of similar collections. 3 x 4 can be viewed as the grouping into a single collection 3 groups of 4 objects, or 4 groups of 3 objects. That is why 3 x 4 = (1+1+1+1)+(1+1+1+1)+(1+1+1+1)= 4 x 3 = (1+1+1)+(1+1+1)+(1+1+1)+(1+1+1). The x operation didn't have to be defined that way, but it is because that is how we reason it and mathematics is based on human experience.

Why do we do multiplication before addition? Lets take an example. 3 x 4 + 1. Multiplication is understood by us as a grouping of similar objects, and grouping is just addition. We can view 3 x 4 + 1 in two main ways. 1) 3 groups of 4 + 1 which would be the same as 3x5 = 15, or 2) the grouping of a 3x4 collection (3 groups of 4 objects) with a group of 1 object, that is 15+1. If we viewed it the 1) way, how would we add another object to 3 x 4 + 1? Lets try, 3 x 4 + 1 + 1 = 3 x 6 = 18. So we intended to add 1 object using our basic notion of addition, but instead added 3. The numbers that we can get will only be numbers that are never prime, if we neglect mutiplication by 1. What if we do it the 2nd way? Our intial value will be 3 x 4 + 1 = 12 + 1 = 13. Adding 1 to that, we expect to get 14 using our notion of addition in the everyday world. We get 3 x 4 + 1 + 1 = 12 + 1 + 2 = 12 + 2 = 14. The number we expect. Here is another way to look at the situation. If we view 3x4 as a single entity representing multiplication, then performing addition before multiplication will ruin that understanding since will we not be adding a single group of 1 object, for instance, but 1 object to every group!

That is why we perform multiplication before addition.

Mathematicians noticed that many things can have two operators, and that if they have those operators behave in similar ways to the arithmetic operators of addition and multiplication many useful properties result. It is not suprising why they do, or why they can later be used to understand properties in the real world. The people that would mention concepts such as groups (not the groups mentioned above), rings, modules, etc, are puting the cart before the horse in explaining what mathematics is really.

So, Mathematics is using the logic of our everyday experience in symbolic form and then abstracting and using logic again, rinse, cycle, and repeat. Concepts such as rings and fields are then created and the origins of the mathematics then expressed in those terms as if the later was the origin of the former.

Mathematics is a lot of fun! Number theory is very interesting! You should study it in university.

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I had always assumed that the precedence of multiplication over addition, at least, came from the fact that mathematics has always been tied to finance. Long before numbers were used for abstract theoretical constructs they were used to calculate payment. And when making up an invoice, you always multiply (unit price times number of units) and then add (each product).

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Nice illustration of how standard order of operations may have been derived. If he weren't an AP he'd get a rate++.

Quote:
Original post by Anonymous Poster
Mathematics is a lot of fun! Number theory is very interesting! You should study it in university.


It is. It is. I do. (Maths, that is.)

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