Squaring Negative Numbers

Date: 02/19/2002 at 10:59:10
From: Thanh Phan
Subject: Squaring negative numbers

Hello,

I would like to know: does -9^2 = 81 or -81?

Date: 02/19/2002 at 12:38:08
From: Doctor Rick
Subject: Re: Squaring negative numbers

Hi, Thanh.

You really should be precise about what you are asking in this case, 
since  (-9)^2 means -9 times -9, but the expression -9^2 could also be 
taken to mean -(9^2), that is, the negative of the square of 9, which 
is -81.

When we're working with variables, if we see -x^2, we interpret it in 
the second way, as -(x^2), because squaring (or any exponentiation) 
takes precedence over negation (or any multiplication; -x is treated 
as -1*x. 

When you have numbers only, as in -9^2, it's not at all clear that we 
should treat it differently from -x^2. However, some will argue that 
it should, because -9 represents a single number, not an operation on 
a number. Thus, some will interpret -9^2 as (-9)^2, while others will 
read it as -(9^2).

Because of the difference of opinion, I highly recommend that you put 
in the parentheses explicitly whenever this situation arises.

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