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### Ordering the Operations

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Date: 11/10/98 at 22:51:00
From: Amy Greenburg
Subject: PEMDAS

In my math class we are studying PEMDAS. Why does the order of
operations have to be in that order? Who made up PEMDAS?

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```
Date: 11/11/98 at 12:38:53
From: Doctor Peterson
Subject: Re: PEMDAS

Hi, Amy -

People generally say that the order of operations is nothing more than
an arbitrary convention - that is, there had to be some rule so
everyone would read an expression the same way, so they just chose a
rule. I don't think any one person made the decision, but it just
gradually developed as the modern symbols for algebra and arithmetic
developed. But I think there is a good reason that the traditional
order was agreed upon without any arguments.

That reason is the distributive rule, which we write as:

a * (b + c) = a * b + a * c

If we reversed the order of operations, doing addition before
multiplication, we would write it this way:

a * b + c = (a * b) + (a * c)

Do you see the difference? In our usual form, we can say that the
multiplication distributes over the terms in parentheses. The
parentheses are required because the addition has to be done first.
But in the reversed form, the parentheses aren't needed there, so the
distribution isn't nearly as obvious.

For the same reasons, polynomials would be more awkward to write, since
each term would require parentheses.

To put it more simply, we do multiplication before addition because
multiplication distributes over addition; multiplication is in some
sense "more powerful" by nature.

Similarly, exponentiation distributes over multiplication, so we do
that first:

(a * b)^c = a^c * b^c

would be written as:

a * b ^ c = (a^c) * (b^c)

if we did multiplication before exponents, and that isn't as clear.

Note, by the way, that exponentiation distributes only in one
direction. Because it is not commutative, it is not true that:

a^(b * c) = a^b * a^c

but rather:

a^(b * c) = (a^b)^c

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Elementary Multiplication
Elementary Square Roots
Middle School Exponents

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