A colleague of mine once asked me if I knew of any way to give a simple intuition for the value of e, the base of the natural logarithms. He was teaching a class to non-technical students, and he said he had no problem giving them an intuition for the value of π. After all, π is just the ratio between circumference and diameter: How far it takes to travel around a circle (or around the rim of a glass of beer) divided by how far it takes to travel across the same circle (or the same glass of beer).
“I wish,” he said, “there were some way to explain the value of e that was so simple my grandmother could understand it.”
As it happens, I saw my friend Josh the next day, and posed the question to him. “So you want your grandmother to understand the value of e? No problem,” Josh said. “Here’s a way that’s guaranteed to work.”
“One thing grandmothers understand is saving their money in the bank,” he continued. “You tell your grandmother `Look Grandma, suppose you put a dollar into a bank at 100% interest. At the end of a year you will have two dollars.'”
Grandmothers, he assured me, understand this kind of thing.
Now suppose there’s another bank down the block that also gives 100% interest, but which compounds every six months. If you invest your dollar there, then at the end of six months you have $1.50, and at the end of a year you have 3/2 as much again, or $2.25. Chopping the year into two pieces definitely makes for a better investment.
Further down the block is another bank that offers to compound once a month, chopping the year into 12 pieces, giving you an even higher return. A fourth bank compounds every day, chopping the year into 365 little pieces, and the returns just keep getting better.
Finally, you come to a bank that compounds continually. At every moment it’s giving you (1 + 1/n) of your money, where the year is chopped into n vanishingly small pieces. That bank gives you the best value for your money. In fact, at the end of the year, your one U.S. dollar has turned into two dollars and 71.828183… cents, or e.
A formal definition of e is actually the limit of (1 + 1/n)n as n goes to infinity. And that’s what you just explained to your grandmother.