Lesson Twelve: In Which We Are Rational

Remember that originally we had the natural numbers, which we obtained by starting with 1 and repeatedly looking at the next whole number, obtained by repeatedly adding 1. Then we saw that it would be more useful to consider the integers, so if we added something that later needed to be undone we could do so by adding a negative number. Now we come to the notion of the Rational Numbers which is all of the fractions that can be written with an integer in the numerator and a non-zero integer in the denominator. Notice that every integer is a rational number because, for any integer n, n = (n/1).

The Rationals and Addition:

Conveniently, the rationals are closed under addition, that is, the sum of two rational numbers is always another rational number. Consider (m/n)+(a/b) = (mb+na)/(nb), and since the product of two integers is always an integer, and the sum of two integers is always an integer, (mb+na) and (nb) must both be integers, so the fraction is a rational number. Furthermore, since addition and multiplication of integers are both commutative and associative, addition of rational numbers is both commutative and associative. There is an additive identity, the rational number 0 = 0/1 = 0/n, and every rational number has an additive inverse since (m/n)+(-m/n) = (m+(-m))/n = 0/n = 0.

The Rationals and Multiplication:

Even more conveniently, the rationals are closed under multiplication. This is even easier to show since (m/n)*(a/b) = (ma)/(nb) and the product of two integers is always an integer. As before, multiplication of rationals is both associative and commutative, by now do you have a firm notion of what associativity and commutativity mean respectively? There is also a multiplicative identity, since 1 = 1/1 is a rational number and (1/1)*(m/n) = (1*m)/(1*n) = m/n. Finally, every rational number EXCEPT all forms of 0 has a multiplicative inverse. Hopefully you can convince yourself that a rational number represents the number 0 exactly when its numerator is 0. Now suppose that m/n is a rational number that is not 0, so n is not zero, since m/n is a rational number so zero cannot be in the denominator, and m is not zero, since that would make m/n = 0. In this case n/m is also a rational number, and (m/n)*(n/m) = (mn)/(nm) = (mn)/(mn) = 1. This is a mathematical way of saying that we can "undo" multiplication by any rational number except for 0.