The hypotenuse and √2

In a right triangle, the side opposite the right angle is the hypotenuse. Take an isosceles right triangle (two equal sides) whose equal sides are 1 unit. How long is its hypotenuse?

You can't measure your way to an exact answer, but you can reason to one. A square of side 1 is made of two such triangles, cut along its diagonal.

And the square built on that diagonal has twice the area of the original, which is what the last two lessons showed.

So the square on the hypotenuse has area 2 × 1 = 2 square units. We don't yet know the hypotenuse's length, but we know the area of the square on it.

Call the hypotenuse c. The area of a square is side × side, so c × c = c² = 2. Therefore c = √2 (the number that, multiplied by itself, gives 2).

Now, what number is √2? Start by trapping it. A square of side 1 has area 1, and a square of side √2 has area 2, so 1 < √2. A square of side 2 has area 4, which is more than 2, so √2 < 2. Hence 1 < √2 < 2, with 1 a lower bound and 2 an upper bound.

Squeeze harder by squaring decimals:

Each step pins down another digit, getting closer and closer to 2, and, as the next lesson shows, the steps never end.