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### What is a Reciprocal?

```Date: 10/10/2002 at 11:44:59
From: George Best
Subject: Nature of the Reciprocal

Dear Dr Math,

I was thinking about reciprocals because they seem to pop up in a
lot of areas of everyday mathematics, physics, and electronics. I was
wondering about their nature. I am trying to formulate a rule in my
head as to what they do. Dividing something into 1. What exactly am
I achieving by this in plain English?

Here are my thoughts so far:

1        The Whole
-  =  ---------------  = How many parts you would need to
x        The Part        make a whole.

Is it some kind of ratio or the inverse of its fraction? Can you
clarify what the reciprocal is?

George
```

```
Date: 10/10/2002 at 13:06:40
From: Doctor Peterson
Subject: Re: Nature of the Reciprocal

Hi, George.

The reciprocal is also called the multiplicative inverse, which means
it is what you multiply by to undo a multiplication (that is, to
divide). So 1/2 is the number you multiply by in order to divide by 2;
that makes it the multiplicative inverse of 2. And of course, you can
find the reciprocal of a number either by just dividing 1 by the
number, or by writing the number as a fraction (2/1) and flipping the
fraction over (1/2).

Similarly, the negative of a number is the additive inverse: the
of 4 is -4, since if you add -4 to something, it is the same as
subtracting 4.

Now these two concepts have a lot in common. The negative can be
thought of as a reflection in a mirror placed at the origin:

|
|
<-----------2---1---0---1---2---------->
|
|
mirror

The negative is on the opposite side of the mirror and equally far
back, just as your image in a mirror is opposite to you.

The multiplicative inverse can also be seen as a sort of reflection,
but in a curved mirror centered at 0, passing through 1 and -1:

|
|
|
....+....
...    |    ...
.       |       .
.        |        .
.         |         .
<----------+---------+----o----+----------o-->
-1         |   1/2   1          2
.        |        .
.       |       .
...    |    ...
....+....
|
|
|

This reflection makes very distant objects look as if they are near
the center of the mirror: the reciprocal of 100 is 1/100, and the
reciprocal of 1,000,000 is 1/1,000,000. You can imagine that the image
of "infinity" is right at the center, at 0. Numbers at the mirror's
surface, at 1 and -1, are left unmoved: the reciprocal of 1 is 1.
Remember also that the reciprocal of the reciprocal takes you back
where you started; the reciprocal of 2 is 1/2, and the reciprocal of
1/2 is 2.

So the reciprocal turns the number line inside out: everything farther
than 1 unit from the origin moves inside, and everything inside goes
out. This is called "inversion" in geometry, and is a very useful
concept. (I should mention that a real circular (or cylindrical)
mirror would not do exactly this; the mirror idea is just a useful way
to picture what is happening.)

If you have any further questions, feel free to write back.

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