20150912 THE SECOND INVERSION:

Which I prefer to call reciprocation.

Multiplication is repeated addition. Division is repeated subtraction. We can always multiply two integer numbers. Occasionally, we can also divide one number by another one and get yet another integer number.

To make division always possible, we need rational numbers. Rational numbers are combinations of integer number and fractions. Once we admitted them to the scene, we understood that every integer number can produce a pair of rational numbers: We only need to divide it by one and divide one by it. Where we had 9, we now have 9/1 and 1/9.

Remember the meaning of the words numerator and denominator: 9/1 means nine of ones. Like any number, 9 is just a bunch of 1s.

This brings about another unary operation of inversion, which I like to call reciprocation. It turns 1/N to N/1 and back.

Forget fractions, what do you do to divide by a whole number? You reciprocate it, so that 9/1 becomes 1/9, and multiply. Dealing with any other rational number, you just do the same.

The first inversion - the one turning negative to positive and positive to negative - could be understood as subtraction from 0 or multiplication by -1. Reciprocation of a number can be understood as division of 1 by this number or as power -1 of this number.

School shows mathematical machine wiggling its limbs. Make the cover transparent. Let kids see how it works.