Proofs by picture

Adding up odd numbers

What happens when you add up a bunch of odd numbers in a row, like 1 + 3 + 5 + 7?

What a nifty way to make my favorite square number, !

This diagram shows that the sum of the first n odd numbers is always a square number – each odd number fits perfectly onto the last square number, to make the next square number.

Or, in symbols,

n2 + (2n + 1) = (n + 1)2

Which do you find more convincing, the diagram, or the symbols? (Or the diagram plus the symbols?)

Circles and sinusoids

Here's something strange: a row of clocks with the hand going around: 12pm, 3pm, 6pm, 9pm. As we increase the number of circles, we'll have the hand go around slower, and squeeze the circles horizontally so they fit. See what happens:

By the time we get to 25 circles or so, a curve is forming. It's called a sine curve. A sine curve tracks the vertical (or horizontal) position of something moving around a circle, like our clock hand.