Definition of Order of Operations

[V0ODO0CH1LD]

I realize now that I took something for granted when I first learned it god knows when.. So I though of starting a discussion as to why were the order of operations defined the way they were? I mean, is there some kind of natural explanation as to why we should compute exponents first and additions and subtractions last? Or is it one of those "right-hand rule" dilemmas where we are just trying to avoid one question having multiple answers?

 

[Diffy]

The order of operations was defined by mathematicians. From what I know, most likely to make writing polynomials easier.

Certainly you can define the order of operations in a variety of ways. But it makes things look unnatural.

For example

Take 5x^2 + 3x + 10 under the normal order of operations.

Now say that addition should come before muliplication.

Let's try to re-write that same polynomial.

(5x^2) + (3x) + 10

I have to add parenthesis here to get the same result.

I guess my point is that in the math we use, we made the order of operations the way it is for convenience. I think that it kind of makes sense the way it is. Without going into too much details think about how you first learned multiplication.

You were probably told that 5*4 = 5 + 5 + 5 + 5.

When you learned about exponents, you were probably told that 5^3 = 5 * 5 * 5 = (5 + 5 + 5 + 5 + 5) * 5 = 5 + 5 + 5 ... + 5

There is a nice hierarchy here that is well preserved in the way we think about them.

 

[Tac-Tics]

Order of operations is a notational shorthand that was created for convenience of working with polynomials.

In some computer languages, notably Lisp, there is not such concept. Instead, you have to parenthesize EVERY operation.

Instead of x^2 + 2x + 1, you have to write (+ (+ (pow x 2) (* 2 x)) 1). It's rather obnoxious! But, from a computer science standpoint, it is absolutely trivial to parse.

Note that the order of operations is completely arbitrary, with the exception of parentheses which have a special syntactic interpretation. If instead of PEMDAS we chose PSADME, we would instead write the above expression as

(x^2) + (2x) + 1

The PEMDAS order is such that the normal form of any polynomial (the form: a_n x^n + a_n-1 x^(n-1) + ... + a_0) is always parentheses-free.

 

[SteveL27]

I realize now that I took something for granted when I first learned it god knows when.. So I though of starting a discussion as to why were the order of operations defined the way they were? I mean, is there some kind of natural explanation as to why we should compute exponents first and additions and subtractions last? Or is it one of those "right-hand rule" dilemmas where we are just trying to avoid one question having multiple answers?

I found a couple of interesting links.

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

and

http://jeff560.tripod.com/grouping.html