How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

The square in the picture is called the inscribed square; each of its corners just touches the circle, but it doesn’t extend beyond the circle’s boundary. Why do this? Because circles are mysterious and intimidating, and squares are easy. If you have before you a square whose side has length X, its area is X times X—indeed, that’s why we call the operation of multiplying a number by itself squaring! A basic rule of mathematical life: if the universe hands you a hard problem, try to solve an easier one instead, and hope the simple version is close enough to the original problem that the universe doesn’t object. The inscribed square breaks up into four triangles, each of which is none other than the isosceles triangle we just drew.* So the square’s area is four times the area of the triangle. That triangle, in turn, is what you get when you take a 1 x 1 square and cut it diagonally in half like a tuna fish sandwich.