Euler's number

It's everywhere we look

Suppose you deposit $1 in a savings account paying 100% interest per year. At the end of a year you will have $2.

You will, that is, if your account pays simple interest, uncompounded.

Now suppose your account pays interest semi-annually. And suppose the interest is deposited back into the account at the same rate. At 6 months you will receive 50% of $1, or $.50. This is added to the $1 principal. At 12 months you receive 50% of $1.50. This is $0.75. At the end of the year you will have $2.25.

You like this compounding idea and ask that you be paid quarterly. The bank agrees. At 3 months you are paid 25% of $1, boosting your balance to $1.25. At 6 months you are paid 25% of $1.25, giving you a balance of $1.56. At 9 months your balance becomes $1.95. At 12 months it's $2.44.

This is more than when you compounded twice per year. But you notice that the bonus is smaller than the previous bonus. You do this again with a similar result. If you compound monthly, you end up with $2.61. If you compound weekly, $2.69. If daily, $2.71.

The increments are getting smaller and smaller. What if you compounded every second? Every millisecond? Every nanosecond?

Compound as often as you like, and you will never reach $2.72. The precise limit is $2.718281828459045…. (The digits go on indefinitely, but with no pattern.)

This number is associated with the 18th-century Swiss mathematician Leonhard Euler and is denoted by e.

As Euler and others discovered, e appears in mathematical descriptions of all sorts of physical phenomena involving growth. In the natural world growth is a continuous process, equivalent to compounding interest every nanosecond. The general formula for the amount A at time t is

A = Pe^rt

where P is the beginning amount (principal) and r is the rate of growth per unit of time.

Whenever growth is described as exponential, this formula is what is referred to. An illustration of the power of exponential growth cites the mythical origins of the game of chess. The Persian emperor at the time was so pleased with the new game that he said the inventor could have anything he wanted. The inventor said he wanted one grain of wheat placed on the first square of a chess board. He wanted two grains on the second square, four grains on the third square, eight grains on the fourth square and so on. The emperor said this was too modest a gift for the creator of such a wonderful game. The inventor said it would be enough. In fact, long before the payoff reached the 64th and final square of the chess board, the inventor would have received more grains of wheat than would be harvested in all of human history should humans last a million years.

The growth formula works in reverse as well. If r is negative, it describes decay, such as radioactive decay. The half-life of a radioactive element is the length of time required for half of it to decay. If the half-life is a thousand years, after a thousand years only half the original amount still exists. After another thousand years, half of that — a quarter of the original — is left. Another thousand years knocks it down to an eighth. Just as your bank balance never quite reached $2.72, the remaining amount of the radioactive element will never quite reach zero.

It's fair to say that modern approaches to nearly every branch of science would be impossible without Euler's number. Population studies, climatology, linguistics, biology, virology, oncology, optics, economics, thermodynamics and dozens of other disciplines rest on his formula for growth.

In the running debate over whether mathematics is invented or discovered, Euler's number is one of the strongest arguments for the discovery position. It appears everywhere in nature.

And not just in the nature we humans experience. Suppose a civilization exists in another galaxy. Suppose the individuals in that civilization look nothing like humans. Maybe their biology isn't even based on carbon. Maybe their brains work entirely differently than ours. Yet when studying thermodynamics or radioactive decay, they would see what we see, as these phenomena behave similarly across the universe, as far as we know. If they are sufficiently clever, they will arrive at a formula functionally equivalent to Euler's growth formula.

In this case at least, Shakespeare's Cassius had it wrong when he told Brutus that fault was not in the stars but in themselves. Here the fault, or the credit, really is in the stars.