What number most exceeds its square?

If we were to graph the number it would be:

y = x

If we were to graph the square it would be:

y = x²

The difference would be:

f(x) = x - x²

You want to maximize this difference, so take the derivative:

f'(x) = 1 - 2x

Then set it to zero:

0 = 1 - 2x

Add 2x to both sides:

2x = 1

Divide both sides by 2:

x = ½

Answer: ½ is the number that most exceeds its square.