A Classical Beginning 5

take it as a basic principle that if there is a natural number with a property, then there is a

smallest such number with that property.

1.3 A geometric proof

Let us now give a second proof of the irrationality of

√

2, one with geometric character,

due to Stanley Tennenbaum. Mathematicians have found dozens of different proofs of this

classic result, many of them exhibiting a fundamentally different character from what we

saw above.

A geometric proof of theorem 1. If

√

2 is rational p/q, then as before, we see that p

2

=

q

2

+ q

2

, which means that some integer square has the same area as two copies of another

smaller integer square.

p

2

q

2

q

2

=

+

We may choose these squares to have the smallest

possible integer sides so as to realize this feature.

Let us arrange the two medium squares overlapping

inside the larger square, as shown here at the right.

Since the large blue square had the same area as the

two medium gold squares, it follows that the small

orange central square of overlap must be exactly

balanced by the two smaller uncovered blue squares

in the corners. That is, the area of overlap is exactly

the same as the area of the two uncovered blue corner spaces. Let us pull these smaller

squares out of the figure to illustrate this relation as follows.

=

+

Notice that the squares in this smaller instance also have integer-length sides, since their

lengths arise as differences in the side lengths of previous squares. So we have found a

strictly smaller integer square that is the sum of another integer square with itself, contra-

dicting our assumption that the original square was the smallest such instance.