Incommensurability means “without a common measure” or “not having a common measure.” It may seem a strange thought that there could be, for example, two straight lines, or two geometrical figures (as another example) which could not be measured by a common unit. But it is so, as we shall see.
When this was first realized (or discovered) it caused an intellectual scandal among the Pythagoreans (whose basic principle was that everything is made of numbers.) But that was a long time ago (6th century B.C.), and we are no longer scandalized, partly because we have devised means of working with incommensurables.
The proof that there are incommensurables may be seen most easily in the example of the square and its diagonal. While this is only one example, it shows that, in principle, there are such things.
Let there be a square with side s, and with a diagonal d.
Proposition: To show that s and d have no common measure.
First, a note about the nature of the proof. The proposition cannot be proved directly but must be proved indirectly (often called a reductio ad absurdum). The strategy of the proof is to assume the contrary of what is being proved, and then show that it leads to an absurdity, to a contradiction. This means that the contrary is false—and, therefore, that the contrary of the contrary (that is, the original proposition) must be true.
Assume the contrary of what is being proved. If the side and diagonal of a square are commensurable (the hypothesis), then they may be measured by a common unit, so many units each, and their relative size may be expressed as a numerical ratio s:d. Let the relation be so expressed and in its lowest terms.
2s² = d² (by Euclid 1.47, the so-called theorem of Pythagoras)
Since d² is equal to 2s² it must be even. But if d² is even, then d must also be even. Let it equal 2m.
Then, 2s² = d² = (2m)² = 4m² and 2s² = 4m²
Therefore, s² = 2m² , and it follows that s is even.
Both s and d are even, so that they are not in their lowest terms (both being divisible by 2), which is contrary to the hypothesis. Therefore, they are incommensurable.