By Murray Bourne, 07 Feb 2010

"e" is one of those amazing numbers that arises naturally in the scheme of things.

Today, 2/7, is celebrated as "e" Day, since the first 2 digits in the number "e" are 2 and 7.

Other special numbers include "pi" (π = 3.141592653..., celebrated on Mar 14th), which is the circumference of any circle divided by its diameter; and "phi" φ = 1.6180339887..., which is the so-called "beauty ratio". These numbers are irrational (that is, their decimals go on forever and never repeat).

e is also an irrational number and it has value:

e = 2.718281828459...

The number e was "discovered" by several mathematicians (Oughtred, Huygens, Jacob Bernoulli, Mercator and Leibniz) but they didn't quite know they had stumbled on it and didn't necessarily know its significance at first.

There are some curious properties of e, one of which is that it's the limiting value of this expression:

What does this mean?

Let's substitute some values to see what this limit expression means.

We start with n = 1 (we cannot start with 0, because that would give a fraction with 0 in the denominator).

Next, we substitute n = 2:

One more: n = 3

It's easier to see what's going on in a graph.

I've plotted the values after substituting n = 1 up to n = 20. We can see it's tending (getting closer to) some value. That value, marked as "e" in red on the graph, is 2.718281828459...

This function converges slowly. If I substitute in n = 1000, I only get 2 decimal place accuracy, and even if I go up to n = 100,000, I get 2.718268, which is only accurate to 4 decimal places.

A better way to find e

The value of "e" can also be found by adding the infinite sum:

The "factorial" exclamation mark, "!", means "multiply by all positive integers smaller than the number given". So

2! = 2 × 1 = 2

and

3! = 3 × 2 × 1 = 6.

We can write our infinite sum using summation notation as:

I'll show the first few terms of this sum. When n = 0, using the convention 0! = 1, we have:

When n = 1:

When n = 2:

And one more, when n = 3:

This second infinite sum is a more efficient way of finding "e", since we only need to add 9 terms and we have 6 decimal place accuracy.

Why is "e" so important?

So what is e good for? See Exponential and Logarithmic Functions.

It is used extensively in logarithms (which was the only way to do difficult calculations for hundreds of years before calculators came along), exponential growth (of populations, money or drug concentrations over time) and complex numbers (which were used to design the computer or mobile device you are reading this on).

So happy "e" day (February 7th, or 2/7).

[For more information on e, see the MacTutor history.]

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