Wednesday, September 17, 2008
Wednesday Math, Vol. 39: e, the base of natural logs
After pi, the number e is the most important and useful transcendental number. Remember that a transcendental number is not only irrational, but it is not the n-th root of any rational number. Unlike pi, defining e is not a simple thing, easily demonstrated with a visual idea. Just think of any circle, and pi is the ratio of the circumference to the diameter.
The number e is very useful in calculus because the derivative of the function e^x is e^x. Defining the number can be done in several different ways. It is the infinite sum of the reciprocals of the factorial numbers, where 0! = 1, 1! = 1, 2! = 2 x 1 = 2, 3! = 3 x 2 x 1 = 6 and n! is the product of all the whole numbers from 1 to n.
Another way to think of e is compound interest, and the number of compounding. If an investment promised 100% return on your money in a year, that would be pretty cool. On the other hand, it would be even better if it was 100% a year compounded twice a year. In six months, you would have 1.5 times your original investment, and at the end of the year you would get 1.5x1.5 = 2.25 times your original investment. If there were three compounding periods, you would do even better, four would be better still.
What if there were infinitely many compounding periods? In math we would call compounded continuously. Would you get an infinite amount of money? No, each time you increase the number of compounding periods, you get an improvement, but those improvements get smaller and smaller, and the limit of this compounding advantage is that 100% a year compounded continuously would mean you would have e times your original investment in a year.
The number e is named for my personal hero Leonhard Euler, who I tagged as My Favorite Lenny just last year, the 300th anniversary of his birth.