Exponents and negative numbers

["W Lake" <wlake@xxxxxxxxx>]

I have checked Mathematica, MathCAD and Derive. All three say
that -2^2 = -4, that -2^4 = -16 and that all of the data points from any
series, say out to -2^64 are on a negative power curve.

There is no mention of using brackets or any special "input" techniques to
handle negative numbers.

Attached is an explanation of why -3^2 = -9 from a web site. If the
explanation is correct, then Excel (-2^2 = 4) is wrong and I need to rethink
my usage.

Can anyone add to or refute the logic given in the emails below.

Walter

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Exponents and Negative numbers

Date: 03/02/97 at 11:15:14
From: Anonymous
Subject: Exponents and negative numbers

Dear Dr. Math,

In different texts about this same question, I can find two different
answers. The solution to: (-3)squared = 9. But when -3 is squared
(without the brackets), one source may say 9 while another source
says -9.

In context, the -3 squared used in sequence will always be -9; why
would the exponent apply to the negative sign unless it is enclosed
by a sign of grouping? In short, why wouldn't the answer to -3
squared, standing alone and without parenthesis or brackets, be -9?

Thanks very much.

Sincerely,
Marvin E. Crim


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Date: 03/09/97 at 14:53:39
From: Doctor Ken
Subject: Re: exponents and negative numbers

Hi Marvin -

After a lengthy discussion among the Drs. Math to make sure we had our
facts straight, I think we have an answer for you.

If you ever see the expression -3^2 evaluated as 9, that's incorrect.
The exponentiation is always done before the negation unless there are
parentheses there to indicate otherwise.

However, there are some contexts in which it _looks_ like texts are
saying that -3^2 = 9, but a closer inspection will either reveal a
subtle interpretation or a misunderstanding. For instance, what is the
difference between the following statements:

   "If I take negative three and square it, I get nine."
   "If I square negative three, I get nine."
   "If I evaluate negative three squared, I get negative nine."
   "If I take the opposite of three squared, I get negative nine."

All of the above statements are correct. The reason some of them
end up with 9 as the answer and some end up with -9 is that some of
the statements have groupings implied in their phrasing.  The first
two statements translate into algebraic notation as (-3)^2 = 9, the
third statement translates to -3^2 = -9, and the fourth statement
translates to -(3^2) = -9.

So the confusion here is not really about mathematical notation, it's
about how to translate English into mathematical notation.  Either
that, or your textbook is incorrect!

I hope we've cleared up some confusion.  The bottom line is that
-a^b is always evaluated as -(a^b).

-Doctor Ken,  The Math Forum
 Check out our web site!  http://forum.swarthmore.edu/dr.math/