A summer diversion

It’s summer, everyone is at the beach, let’s have some fun.

Either I was asleep when I was being taught this in school or my senility is progressing much faster than I thought, but the following that I recently read in a book^* came as a complete (and delightful) surprise to me: the sum of any consecutive sequence of odd numbers starting at 1 is a square. So, for instance, 1+3 = 4, which is the square of 2; 1+3+5 = 9, which is the square of 3, and so on.

(^*The book is called “The Joy of X, a Guided Tour of Mathematics from One to Infinity”, by Steven H. Strogatz, Atlantic Books, London. The ideas and demonstrations in this blog all are given in this book. I am not claiming them as my own. Borrow it from your library for a good read.)

My first reaction was, that isn’t right, the equation is not dimensionally correct (the right-hand side has squared units, but the left-hand side just has “units”). Obviously I have been doing physical science for too long. These are natural numbers, with no units. A square number is an integer that is the product of some other integer with itself.

My second reaction was, WTF? How on earth can a sum of odd numbers be related to squares?

My third reaction was, I bet that is difficult to prove.

Maybe so, but there is a wonderful “visual demonstration” of why it is so, which goes like this…

Let’s represent numbers by stones, as follows:

Now add 1 and 3:

The result is 4, which can be arranged in a square, 2 stones wide and 2 stones high. That means 4 is a square number, being, in this case, the product of 2 and 2.

OK, let’s add 1, 3 and 5:

All right! The answer is 9, which again is a square number. Here is the next in the sequence if you haven’t picked up on the pattern yet:

You couldn’t really call this a proof (there is nothing in here that shows that the proposition holds true for every sequence of odd numbers), but it sure is a good visualisation. (It is also a pretty good demonstration of the old aphorism that “a picture paints a thousand words”.)

Geometric proofs, which we all learnt at school (I wasn’t asleep the whole time), are also highly visual, which is maybe why they are so instructive.

Here’s a terrific visual proof of Pythagoras’ theorem, which tells us that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.

Below on the left is a right-angled triangle, with hypotenuse of length c. The lengths of the other two sides are a and b, and our goal is to show that a² + b² = c². Below on the right is what c² looks like; it is the area of the square built along the hypotenuse. (So now we really are dealing with squares.)

We now make three exact copies (coloured green, grey and red) of our original blue triangle and arrange them so that c² is enclosed within a big square. It doesn’t matter, but that big square obviously has sides of length a + b. What is important to note is that the “empty space” within the big square, that is, the area that is within the big square but outside the area of the four triangles, is c². If you see that, you are nearly there…

OK, now rearrange the three copies of the original triangle like this:

The empty space in the top left is a square with sides that are b in length. Therefore, it has an area of b². Similarly, the empty space in the bottom right is a square with area a². The total empty space is therefore a² + b².

But, hang on, we said above that the area of the empty space is c². We haven’t created or destroyed empty space here, so that must mean a² + b² = c². And there is Pythagoras’ theorem.

Have a good summer.

Malcolm Green