Tucked away inside yesterday’s blog was one of the most important innovations in the entire history of mathematics — the ability to deal with infinitely small things. This innovation was independently developed by Newton and Leibniz in the mid-seventeenth century — the former working in England, the latter in Germany.
Infinity is a slippery thing, and naive questions like “What’s the value of one divided by zero” can just result in nonsense. What both Newton and Leibniz realized is that you can sneak up on infinity — by starting with two finite things, and then sending them both off to infinity together.
And that’s exactly what yesterday’s discussion about e was doing: A naive statement like “Something infinitely close to one raised to an infinitely big power” — would produce no sensible answer. So instead we put a throttle on both parts of this statement, by sending several finite things off to infinity and watching where the result heads off to.
It’s a little like sending a rocket ship into space. You can’t see the rocket ship after it has left the solar system, but if you know enough about its launch trajectory, you can predict where it will be in the sky even after it has become too small to see.
In the case of yesterday’s discussion, (1 + 1/n) gets closer and closer to 1, as n gets bigger. Meanwhile we are raising to a progressively higher power. The result (1 + 1/n)n indeed converges to a single number — e = 271828… — as n becomes ever larger, and our little rocket ship disappears into the night sky.
So many aspects of our modern life — the fruits of research into physics, biology, chemistry, electronics, economics, manufacturing, computers, vehicle design, and much more — depend on this astonishing innovation from over three hundred years ago. It really is a kind of infinite revolution.