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### Reciprocals

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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

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/
```
Associated Topics:
Elementary Definitions
Elementary Division
Elementary Fractions
Elementary Multiplication
Middle School Definitions
Middle School Division
Middle School Fractions

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