Reciprocals of Negative Fractions

Date: 03/19/2002 at 02:01:47
From: Melissa
Subject: Reciprocals

What is a reciprocal, and what is the reciprocal of negative 1/2? 
What is the reciprocal of 10?

Date: 03/19/2002 at 09:06:09
From: Doctor Rick
Subject: Re: Reciprocals

Hi, Melissa.

You have learned that the reciprocal of, say, 2/3 is 3/2, right? To 
get the reciprocal of a fraction, you just switch the numerator and 
the denominator. 

So what happens if the fraction is negative? You can consider the 
negative sign to be part of the numerator: think of -1/2 as -1 over 2. 
Then the reciprocal is 2/(-1). But you can make an equivalent fraction 
by multiplying the numerator and denominator by the same quantity. Try 
multiplying both by -1:

   2     2*-1   -2
  --- = ----- = ---
  -1    -1*-1    1

Now, a fraction with a denominator of 1 is just a whole number, so the 
reciprocal of -1/2 is -2.

This last part gives us a clue as to how to find the reciprocal of 
whole numbers like 10. We can write the number with a denominator 
of 1:

  10 = 10/1

Then we take the reciprocal of this fraction: switch the numerator and 
denominator to get 1/10.

Does this help?

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