Euler's Identity


Eddie sez:

Euler's identity unifies five of the fundamental ingredients of mathematics: 0, 1, π e, and i.

The value e is the foundation of the natural logarithm. But even the most technically astute engineer or mathematician can be forgiven for not recognizing it:

e i π + 1 = 0

Solving, you get e = 2.718281828459...

(It's one of those numbers that goes on forever.)

You may say this is one of those tedious math things that means nothing to a pilot and perhaps that's true. But it is one of those things that comes up now and then so I keep it in my notes. If you really want to see it in action, the best explanation I've ever found appeared in the book The Simpsons and Their Mathematical Secrets, by Simon Singh.

Everything here is from the references shown below, with a few comments in an alternate color.

Last revision:



[Singh, pp. 132-135]

  • The number e was discovered when mathematicians began to study a fascinating question about the usually tedious subject of bank interest. Imagine a simple investment scenario, in which one invests $1.00 in an extraordinarily convenient and generous bank account that offers 100 percent interest per year. At the end of the year that $1.00 would have accrued $1.00 interest, giving a total of $2.00.
  • Now, instead of 100 percent interest after one year, consider a scenario in which the interest is halved, but calculated twice. In other words, the investor receives 50 percent interest after both six and twelve months. Thus, after the first six months, the $1.00 would have accrued $0.50 interest give a total of $1.50. During the second six months, interest is gathered on both the $1.00 and the additional $0.50 interest that has already accrued. Therefore the additional interest added after twelve months is 50 percent which equals $0.75, resulting in an overall total of $2.25 at the end of the year. This is known as compound interest.
  • As you can see, the good news is that this half-year compound interest is more profitable than simple annual interest. The bank balance could have been even higher if the compound interest had been calculated more frequently. For instance, if it had been calculated quarterly (25 percent every three months), then the total would have been $1.25 at the end of March, $1.56 at the end of June, $1.95 at the end of September, and $2.44 at the end of the year.
  • By the time compound interest is calculated on a weekly basis, we are almost $0.70 better off than if we had been earning only simple annual interest. However, after this point, calculating the compound interest even more frequently achieves only one or two more pennies. This leads us to the fascinating question that began to obsess mathematicians: If the compound interest could be calculated not just every hour, not just every second, not just every microsecond, but at every moment, what would be the final sum at the end of the year?
  • The answer turns out to be $2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427.... As you can probably guess, the decimal places continue to infinity, so this is an irrational number, and it is the number that we call e.
  • 2.718... was named e because it relates to exponential growth, which describes the surprising rate of growth experienced when money gathers interest year after year, or when anything repeatedly grows by a fixed rate again and again.

Singh, Simon, The Simpsons and Their Mathematical Secrets, 2013, Thomson-Shore Inc., Dexter, Michigan.