Negative Numbers Combined with Exponentials

Date: 03/09/2001 at 01:07:20
From: Dee Ryno
Subject: Negative numbers combined with exponentials. 

I know that -3 ^2 {to the second power} is negative 3 because the 
order of operations tells us exponentials are done first and then the 
answer {9} is multiplied by (-1), but why isn't the negative attached 
to the -3 to say -3 * -3, which would make it positive, since 3 ^2 
{to the second power} is 3 * 3?  Help.

This has confused a lot of teachers too!

Dee

Date: 03/09/2001 at 15:29:24
From: Doctor Peterson
Subject: Re: Negative numbers combined with exponentials. 

Hi, Dee.

You are aware that the order of operations is the key. Because 
negation is taken as a multiplication, and exponentiation is done 
before multiplication, we read

    -3^2
as
    -(3^2)

rather than as

    (-3)^2

The exponent holds on to the 3 tighter than the minus sign does.

I think you're asking why we have this rule, when the minus looks so 
close to the 3, and it seems so much more natural to think of it as -3 
squared. I could just say this is a choice that has been made, and we 
just follow the convention. But there's an additional reason besides 
the logic of seeing negation as multiplication by -1. When we get to 
algebra and want to write polynomials, we find ourselves working with 
-x^n, which has to follow the same rules as for -3^n:

    x^3 - 3x^2 - 3x + 5 = 0

There's little question here; we know that 3x^2 is taken as 3(x^2), 
and then we subtract that. But what about

    -x^3 - 3x^2 - 3x + 5 = 0 ?

Do we take this as (-x)^3 or as -(x^3)? Because this is a polynomial, 
we know that x is meant to be the base of all the exponents; we don't 
want to have to write -(x^3) to make it mean what we intend; so we are 
perfectly happy to follow the logic where it takes us and treat the 
negation as a multiplication done after the exponentiation, rather 
than as a part of the base. The rule helps here, rather than seeming 
odd: In a polynomial we want exponents to come before everything else, 
because they are the stars of the show.

I suspect that polynomials drove much of the development of the order 
of operations; many of the early examples of algebraic notation in 
which those rules can be discerned are polynomials.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/