# How did Order of Operations Come About?

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Or the they can be called "order of precedence"?

I'd like to know why this is necessary -- how was it established.

I'm making it a point to understand the necessity behind these things so I'm not just blindly following rules. You know...so my math skills are more intuitive.

I'm asking for a relative, so Please Excuse My Dear Aunt Sally.

...I couldn't resist throwing a corny joke in there.

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Cause you get the wrong answer if you don't!

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Cause you get the wrong answer if you don't!

JesterArts,

I don't know the answer, myself, but I found this which goes into your question at some length:

http://mathforum.org/library/drmath/view/52582.html

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I'd like to know why this is necessary

This is pretty easy.

Is 2+3*4 = 24 or 20?

(2+3)*4 = 20

2+(3*4) = 24

It is picked so that there is a convention. So that even when someone fails or forgets to put the ()'s in, there is a definitive answer. I don't think that any one method is intrinsically superior to the other ... I don't think that 20 is a better or worse answer than 24 to the above question -- except that my brain goes with the established convention and so in that respect part of me does prefer the 24.

I think that the rules are there to make it as unambiguous as possible.

Maybe someone else will know more about the history or rationale behind them, because I don't...

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sure... but why is it 24 and not 20?

It's not a word problem, so there's nothing inherently better about 24 than 20 as an answer, at least not observationally.

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JesterArts,

I don't know the answer, myself, but I found this which goes into your question at some length:

http://mathforum.org/library/drmath/view/52582.html

Thank you, that was actually very helpful.

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My immediate thought was because 'multiplication distributes over addition', which is exactly what that article said. I think of it this way: distributivity makes an operator greedy. One must use parentheses to reign in a greedy operator.

The article did not touch on the fact that exponentiation evaluates right-to-left: $a^{b^c} = a^{(b^c)}\,\ne\,(a^b)^c$, which is contrary to the (arbitrary) direction in which all other operators are evaluated. This opposite direction for exponentiation is natural because $(a^b)^c = a^{(bc)} = a^{bc}$. The last set of parentheses can be dropped because bc are superscripted.

The article didn't even begin to touch this.

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This is pretty easy.

Is 2+3*4 = 24 or 20?

(2+3)*4 = 20

2+(3*4) = 24

It is picked so that there is a convention. So that even when someone fails or forgets to put the ()'s in, there is a definitive answer. I don't think that any one method is intrinsically superior to the other ... I don't think that 20 is a better or worse answer than 24 to the above question -- except that my brain goes with the established convention and so in that respect part of me does prefer the 24.

I think that the rules are there to make it as unambiguous as possible.

Maybe someone else will know more about the history or rationale behind them, because I don't...

I think you'll find 2 + (3*4) = 14 .

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I think you'll find 2 + (3*4) = 14 .

Ha! that's what happens when my fingers outrace my brain! (and now everyone can laugh derisively at my "Maths Expert" title! I know I am!)

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You're fine; mathematics experts do not need to know how to add and subtract.

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Indeed. My mental arithmetic has declined immensely since completing my degree

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You're fine; mathematics experts do not need to know how to add and subtract.

I detect some humor here, but isn't it true that the abstract (pattern based) ability to do math is not related to the ability to add, subtract, multiply, etc?

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Certainly it's true in my experience. I've found that mental arithmetic is of very little consequence in a mathematics degree, since the emphasis is on proof. So as long as you know how to do addition, you're fine; whether or not you can necessarily do it is not really the point.

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I detect some humor here, but isn't it true that the abstract (pattern based) ability to do math is not related to the ability to add, subtract, multiply, etc?

That's exactly right.

Accountants worry about 1+1=2, plus a whole lot more complex arithmetic manipulations. Mathematicians worry about defining systems in which 1+1=2, plus a whole lot more complex arithmetic manipulations, plus a whole lot of other systems far beyond the everyday set of numbers with which accountants work. Mathematicians don't have to do the manipulations because mathematicians are not accountants.

Take me, for example. I am not a mathematician by training, but I do use a lot of hairy math in my work. Until recently, I also had to fill out my timecard by hand. (Even though I am salaried, I still need to fill out a timecard. Something about billable hours...) Our office administrator would look at my whiteboard, show me my errant timecard, and laugh. I could never get those stupid numbers to add up correctly horizontally and vertically.

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If you don't have order of operation the grammar is ambiguous. In a particular type of computer program which recognizes computer languages (most of which contain arithmetic with implied order of operation) this is known as a shift/reduce conflict. These programs resolve the problem by picking one of the two ambiguous options, but it's generally frowned upon to leave these in your grammar.

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Not that I'd class myself as a mathematician by any stretch, but the only time I use mental arithmetic is when I go shopping.

Incidentally, we had a quiz recently at work, where one of the questions was to provide the sum of the numbers around a dart board. While others were sitting there jotting numbers down, I used Gauss's formula, which was arguably derived when he was 10...

$\sum_{i=1}^{n} i = \frac {1}{2} n (n+1)$

So there are short cuts, which you can do in your head, here's the one I use when I go shopping...

$shopping + me = stress \times 10 ^6$

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• 2 weeks later...

By the way, the Hewlet-Packard calculators used to (I don't know if they still do) use "reverse Polish notation" which uses a completely different "precedence" convention. It was supposed to be easier to enter complicated problem using "reverse Polish" but I always found it difficult to not unconsciously convert to "ordinary" notation (thus getting the wrong answer).

"Reverse Polish notation" requires that you work through a formula from left to right, each time you reach a binary operator, apply it to the last two numbers.

For example, (1+ 4)*6+ 3 would be entered as 1, 4, +, 6, *, 3, +. (Notice that you need a "enter" key to separate numbers. As you entered that, the calculator would save 1 and 4, then seeing the "+" add those, saving 1+4= 5, then save 6, then seeing the "*" do 5*6= 30, saving 30. Then save 3 and, when you enter the last "+", do 30+ 3= 33.

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By the way, the Hewlet-Packard calculators used to (I don't know if they still do) use "reverse Polish notation" which uses a completely different "precedence" convention. It was supposed to be easier to enter complicated problem using "reverse Polish" but I always found it difficult to not unconsciously convert to "ordinary" notation (thus getting the wrong answer).

"Reverse Polish notation" requires that you work through a formula from left to right, each time you reach a binary operator, apply it to the last two numbers.

For example, (1+ 4)*6+ 3 would be entered as 1, 4, +, 6, *, 3, +. (Notice that you need a "enter" key to separate numbers. As you entered that, the calculator would save 1 and 4, then seeing the "+" add those, saving 1+4= 5, then save 6, then seeing the "*" do 5*6= 30, saving 30. Then save 3 and, when you enter the last "+", do 30+ 3= 33.

It's funny, Halls, because once I got used to RPN (Reverse Polish Notation), I almost couldn't work a "regular" calculator anymore. I found the RPN to be very intuitive once I got used to it. And, I found it much easier to put in more complicated expressions than the standard way of entering numbers.

These days, I don't use either. I still have my HP calculator, and I know it still works, but I haven't used it other than to open it up to see if it still works for many years now. I actually just use Excel if it is just numbers, or open MathCAD if I have to do some symbolic math. I've found that I am always close enough to a computer that Excel is never more than a few seconds away.

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• 2 months later...

The nice thing about RPN is that it doesn't need brackets or priority of operations at all; all you need (from a computing point of view) is a stack (a LIFO or last-in-first-out data structure). Essentially, a number means 'push this number onto the stack', + means 'pop the top two numbers off the stack, add them, and push the result', and *, - and / mean something analogous, making sure in the latter two cases to get the operands in the right order.

RPN is terrible as far as human readability is concerned, though.

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• 2 weeks later...

On a more practical note, most programming languages, including the very commonly used C and C++, obey the Algebraic Operating System (AOS) when there are no parentheses, meaning that first parentheses are executed, then ALL multiplication and division in left-to-right order, then addition and subtraction in left to right order. As an example

6 / 2 + 2 * 4 - 12 / 3

=7

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Jasmine

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I'm a programmer and I like to put lots of brackets in so there can be no confusion.

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• 3 weeks later...

To understand the reasoning behind the Order of Operations you have to keep in mind that multiplication is just multiple additions and that powers are just multiple multiplications.

Thus

2+3*4 = 2+3+3+3+3=14 or 2+4+4+4=14

2+3*4 = 5*4=4+4+4+4+4=20 but this would create two more values of 4.

Values that are not in the original problem.

Order of Operations allows us to solve complex problems without having to change everything to addition but still get consistent answers.

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To understand the reasoning behind the Order of Operations you have to keep in mind that multiplication is just multiple additions and that powers are just multiple multiplications.
Only in the very naive sense when all multipliers and powers are natural numbers.

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