58 Chapter 6
Aha! Did you discover the intended proof? The idea, of course, is that the whole unit
square has area 1, and it is successively divided into smaller pieces: a rectangle of area
1
2
, a square of area
1
4
, and so on. Each rectangle is followed by a square of half the size,
and each square is followed by a rectangle of half the size. So the sum of all the pieces is
1
2
+
1
4
+
1
8
+
1
16
+ ..., and since they exhaust the unit square in the limit, the sum is 1, as
desired.
6.2 Binomial square
For another easy case, consider the following diagram, offered as a proof of the binomial
square identity. Can you see how the diagram proves the identity?
a
2
b
2
ab
ab
a
b
a
b
(a + b)
2
= a
2
+ 2ab + b
2
Aha! The whole square, having sides a + b, has total area (a + b)
2
; but this area can also
be realized as the sum of four regions, two squares and two rectangles, whose areas add to
a
2
+ 2ab + b
2
, and so the two quantities are equal. In the exercises, you will be asked to
find a similar proof without words of the identity (a + b)(c + d) = ac + ad + bc + bd.
6.3 Criticism of the “without words” aspect
But let us not be dogmatic about the “without words” aspect of proofs without words.
Proofs of any kind can be improved with insightful explanation, and there is little reason
for a figure to stand entirely on its own. In fact, I know of numerous failed instances
of proofs without words, where a clever diagram is too obscure by itself to constitute
a mathematical argument. I regard these instances instead simply as poorly explained
proofs. These clever diagrams could become part of a successful proof, if supplemented
with proper explanation.