Reciprocals

Date: 12/19/96 at 18:48:09
From: David Gloeckner
Subject: reciprocals

Help!  I'm home-schooling my 12-year-old and we're reviewing
multiplication and division with fractions.  He asks, "Of what use is 
the idea of a reciprocal?"  Thanks for any help you can offer.

Ken's befuddled mother, Carolyn Gloeckner

Date: 12/21/96 at 04:48:03
From: Doctor Pete
Subject: Re: reciprocals

Hi,

The reciprocal is really little more than division in disguise.  As 
you may know, the reciprocal of a number is 1 divided by that number. 
So the reciprocal of 5 is 1/5, for example.  (Does 0 have a 
reciprocal?  1?)

When students are introduced to division, they are not immediately 
told that division is really the same thing as multiplication; that 
would be somewhat confusing, since the methods are different (e.g., 
finding 375/5 is quite different from finding 15 x 5, although they 
both equal 75).  But they are indeed the same operation; and the 
concept that explains this fact is the reciprocal.  

Division by a number is the same as multiplication by its reciprocal.  
To emphasize this, mathematicians sometimes refer to the reciprocal of 
a number as the *multiplicative inverse*.  This is quite important, 
because this phrase describes one of the fundamental propeties of 
reciprocals, which is that the product of a number and its reciprocal 
is always 1.  1 is the *multiplicative identity*, the number which 
when multiplied to any other number, does not affect it.  Hence the 
name "multiplicative inverse."

The notion of reciprocals, then, explains the oft-quoted rule "invert 
the divisor and multiply."  For example,

     12/5
    ------ = 12/5 x 10/3 = 4/1 x 2/1 = 8.
     3/10

When first learning this rule, many students learn the rule but not 
its basis.  The reason why it works is simple - it is simply the 
application of the idea of the multiplicative inverse.  To see what's 
going on, there is a simple intermediate step I can insert in the 
above:

     12/5     12/5     10/3     12/5 x 10/3
    ------ = ------ x ------ = ------------- = 12/5 x 10/3 
     3/10     3/10     10/3          1

Clearly, any number divided by itself is 1.  But what value do we 
choose for the numerator and denominator?  We choose the unique value 
that when multiplied by the denominator, 3/10, gives 1.  This is 
precisely the reciprocal, 10/3.

What is interesting about all of this is that it is not unfamiliar, 
since addition and subtraction have similar properties.  While 1 is 
the multiplicative identity, 0 is the additive identity (0+n = n for 
any number n).  Then the *negative* of a number is to addition as the 
reciprocal is to multiplication.  So the negative of 5 is, of course, 
-5, and their sum is always the additive identity 0.  Naturally, 
"negative"'s synonym is "additive inverse."  Hence the relation to 
multiplicative inverse.

However, this analogy is not perfect, and the only place where we 
need to be careful is when the reciprocal is undefined.  In 
particular, 0 has no reciprocal, whereas any number has a negative.  
(What is the negative of 0?)  Suppose 1/0 was indeed some number.  
Call it n.  Then 1 = 0 x n, but 0 x n = 0 for every n, so 1 = 0, a 
contradiction.  

Similarly, if we suppose 0/0 = n, then 0 = 0 x n = 0, but this is true 
for *any* n we choose, so n is not unique, and it makes no sense to 
assign it any one particular value.  Hence division by 0 is undefined.  
There is no such exception for addition.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/