Proofs without Words

The sum of the first ${n}$ odd numbers is equal to the square of ${n}$ :

$\displaystyle 1 + 3 + 5 + \cdots + (2n-1) = n^2 \,.$

We can check this for the first few: ${1 = 1^2,\ \ 1+3=2^2,\ \ 1+3+5 = 3^2}$ . But how do we prove it in general. Actually, there is a straightforward proof by induction:

$\displaystyle \sum_{k=1}^{n+1} (2k-1) = \sum_{k=1}^{n} (2k-1) + (2n+1) = n^2+2n+1 = (n+1)^2 \,.$

But, in many cases, the algebraic proof may not be so easy.

Left: Sum of odd numbers. Right Sum of even numbers.

Sometimes there is a simple geometric illustration of the result. To get the sum of the first ${n}$ odd numbers, we can arrange them in a square pattern (see Figure, left panel).

The proof for the sum of the first ${n}$ even numbers is almost as easy: just add a column of height ${n}$ to the left side; the right panel in the figure should make it clear that

$\displaystyle \sum_{k=1}^{n} 2k = n(n+1) \,.$

Proofs Without Words

Proofs without words are diagrams or illustrations that make the truth of mathematical theorems obvious. They are most commonly used for geometric results, but are also of great value in elementary algebra as well as for inequalities, and for summing infinite series. They can be very useful for helping students of mathematics to think visually.

A proof without words is not equivalent to a mathematical proof, because the details of the systematic and logical argument are lacking. Moreover, an illustration can sometimes be misleading, and can fail to catch subtleties of special cases.

However, proofs without words can provide valuable intuition to the viewer that can aid in understanding and show the way to a more systematic proof.

Archimedes and Circles

You would search in vain in Euclid’s Elements for a proof that all circles are similar, in the sense that the ratio of circumference ${C}$ to diameter ${D}$ has the same value for all? It required the genius of Archimedes, who lived a few generations after Euclid, to prove that ${C / D}$ is constant, and he needed to introduce axioms beyond those of Euclid to achieve this. There is an excellent account in an article by Richeson (2015), including details of the additional axioms required.

The area of a circle can be estimated by cutting it up into a collection of slender segments (see Figure). In the right panel, the segments are rearranged in a shape that is approximately rectanglular, with height ${r}$ and width ${\pi r}$ , and so with area ${\pi r^2}$ . This provides an estimate of the area. Of course, reducing the angles of the segments we arrive at the limiting case, where there is equality between the circular and rectangular areas.

The Pythagorean Theorem

We are all familiar with the theorem of Pythagoras, and some may remember a quite complicated proof of the result. It involves constructions that are far from obvious, and is difficult to remember. But there are a very large number of proofs — in the hundreds — and some of these are very simple.

Here is an example. We make four copies of the triangle, and arrange them with the four right angles at the same point, in the pattern of a windmill or St Brigid’s Cross (see Figure to left).

Now move the blue and green triangles down a distance ${a-b}$ , as shown in the Figure to right. Finally move the red and blue triangles left a distance ${a-b}$ , to form the pattern in the panel below.

The result is a large square, of side length ${c}$ and area ${c^2}$ . This is equal to the area area of the central white square ${(a-b)^2}$ , together with the areas of the four triangles, ${4\times\frac{1}{2}ab}$ . So, we get

$\displaystyle c^2 = (a-b)^2 + 2ab = a^2-2ab+b^2-2ab = a^2+b^2 \,.$

which is the familiar Pythagorean result

$\displaystyle a^2 + b^2 = c^2 \,.$

There are numerous videos online that illustrate proofs without words and many of these are excellent.

Sources

${\bullet}$ Richeson, David , 2015: Circular reasoning: who first proved that C/d is a constant? College Math. Journal, 46, 162–171. Preprint (2013) online .

${\bullet}$ Wikipedia article Proofs without Words.

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