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08 August 2003

By the Numbers: Going Irrational but Not Crazy

by Peter Torrione

Last time we showed a chart which "showed" all the rational numbers. Recall that a rational number is simply the result of dividing an integer by another integer; this is called a ratio; hence rational number. Nothing particularly sane about it.

Are all the (positive) numbers that exist to be found on the chart? Can we find numbers that do not appear on the chart? In other words can we find numbers which cannot be expressed by dividing one integer by another?

Let's take a look at square roots. Square roots are the reverse process of multiplying a number by itself. 3 times 3 equals 9; so 3 is the square root of 9. (The term "square" is used since it relates to the area of a square.) Obviously both 3 and 9 are on the chart of rational numbers, so no problem yet.

However, how about the number that multiplies by itself to give 2? What is the square root of 2? If it is on the chart, then it is of the form A/B where A is a number from the top row and B is a number from the left column. Also A/B is reduced to its lowest form so there are no common factors between A and B (e.g. 3/6 would be reduced to 1/2).

The ancient Greeks created a proof that both A and B had to be even! Hence no such fraction could exist. If both numbers must always be even then A/B can never be in its lowest form!

We start with A/B = sqrt (2); where sqrt means square root.

The proof is to first square both sides, resulting in (A x A)/(B x B) = 2. (x means multiplied by.)

Multiplying both sides by (B x B) gives

This means that A x A is even since it is twice a number and A must be even since the square root of an even number is always even. Hence A can be written as A = 2 x C. That means that A x A = 4 x (C x C). Using (1) above:

(A x A) = 4 x (C x C) = 2 x (B x B).

Dividing by 2 gives 2 x (C x C) = (B x B). But, this means that B x B must be even and hence, as before B must be even. Both A and B must be even!

The conclusion is that there cannot be such a ratio as A/B that equals the square root of 2!

The square root of 2 is not to be found on our chart! Where is it? Remember Closure! Any number we produce by doing a mathematical operation on one or two numbers must result in a number in our mathematics.

The number line comes to the rescue again. Clearly 107/76 and 109/76 are on the chart as well as on the number line at 1.4079+ and 1.43422+ (the + means that more numbers follow). The square of each is 1.982+ and 2.057+. Our elusive square root of 2 is somewhere between 1.4079+ and 1.43422+ on the number line. Of course we can produce ratios that get us even closer, as close as we please, but we can't get there with rational numbers!

We can construct it however, with a little geometry. Just built a right triangle with sides of one unit and the hypotenuse will be the square root of 2. Just lay that length on the number line.

Numbers such as square roots, cube (third) roots (the cube root of 27 is 3) etc. often result in numbers which cannot be expressed as ratios. How irrational! It really bothered the ancient Greeks!

We noted in the previous article that the infinity of rational numbers is the same as the infinity of natural numbers (1, 2, 3, 4….). What about our newly found irrationals?

They are part of a larger infinity!