Infinite Powers: The Story of Calculus - The Language of the Universe

by Steven H. Strogatz

For those who can’t resist and want to understand why infinity lurks in the shadows, imagine dividing 6 by a number that’s small and getting close to zero, but that isn’t quite zero, say something like 0.1. There’s nothing taboo about that. The answer to 6 divided by 0.1 is 60, a fairly sizable number. Divide 6 by an even smaller number, say 0.01, and the answer grows bigger; now it’s 600. If we dare to divide 6 by a number much closer to zero, say 0.0000001, the answer gets much bigger; instead of 60 or 600, now it’s 60,000,000. The trend is clear. The smaller the divisor, the bigger the answer. In the limit as the divisor approaches zero, the answer approaches infinity. That’s the real reason why we can’t divide by zero. The faint of heart say the answer is undefined, but the truth is it’s infinite. All of this can be visualized as follows. Imagine dividing a 6-centimeter line into pieces that are each 0.1 centimeter long. Those 60 pieces laid end to end make up the original.