The Number "e"
The number e, often called the "natural base", is an important constant, just like . Its value is 2.718 to three decimal places. It is called the natural base because as the base of an exponential function, it models natural growth very well. Here's an app that should give you a little insight into why "e" is what it is.
Imagine we have a colony of organisms, maybe people or bacteria. At time t=0, we will assume the population is "1", which could mean 1 or 100 or 1 million. After a time interval of 1 (minute, year, millennium), we want this population to double. Our first model has two points: (0,1) and (1, 2). All the action happens in one big burst at time t=1.
This is not very realistic, since real organisms in large populations tend to reproduce essentially continuously. So let's start moving toward something more continuous. Instead of doubling all at once at t=1, let's increase by half, twice. We'll do the first increase at time t=1/2, when we multiply the population by 1.5. Then, at t=1, we multiply by 1.5 again for the other half. But wait - this means our population increased not by a factor of 2 but by 1.5 x 1.5 = 2.25. Interesting.
Let's try increasing three times, by 1/3 each time. Now the population increases by 4/3 x 4/3 x 4/3 or about 2.37 times. We can extend this idea to larger numbers of multiplications, at shorter time intervals. Our model becomes
As n approaches infinity, the factor asymptotically approaches a value - the number "e"!
On the app, we start with the two points (0,1) and (1, 2). By moving the n slider to larger values, more points appear, representing the increasing number of diminishing increases. The number in the box labeled "m" is the maximum value of n, so you chan change that to larger and larger numbers as you please.
Notice that the points move toward a curve that ultimately (as n grows toward infinity) becomes the graph of . You can verify this by checking the "Show f(x) = e^x" box. The horizontal red dashed line is at , and so the rightmost point will eventually end up at (1, e), as predicts.