Zero: The Biography of a Dangerous Idea

by Charles Seife

L’Hôpital’s rule took the first crack at the troubling 0/0 expressions that were popping up throughout calculus. The rule provided a way to figure out the true value of a mathematical function that goes to 0/0 at a point. L’Hôpital’s rule states that the value of the fraction was equal to the derivative of the top expression divided by the derivative of the bottom expression. For instance, consider the expression x/(sin x) when x=0; x=0, as does sin x, so the expression is equal to 0/0. Using L’Hopital’s rule, we see that the expression goes to 1/(cos x), as 1 is the derivative of x and cos x is the derivative of sin x. Cos x=1 when x=0, so the whole expression equals 1/1=1. Clever manipulations could also bring l’Hopital’s rule to resolve other odd expressions: ∞/∞, 00, 0∞, and ∞0. All of these expressions, but especially 0/0, could take on any value you desire them to have, depending on the functions you put in the numerator and denominator. This is why 0/0 is dubbed indeterminate. It was no longer a complete mystery; mathematicians could extract some information about 0/0 if they approached it very carefully. Zero was no longer an enemy to be avoided; it was an enigma to be studied.