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 
number you add in order to subtract your number. The additive inverse 
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/