Certain numbers hold celebrity status in mathematics. One such number is pi, the ratio of the circumference to the diameter in any circle. Like other celebrities, pi has become a part of our popular culture. In cartoons, the symbol is often used to indicate the presence of smart people, or even as an icon for mathematics itself. Indeed, many people who know very little else about mathematics are able to recite, from memory, several of the digits of pi's infinite, aperiodic, decimal expansion, pi = 3.1415926536...

Ancient and venerable, pi has the aura of a very important number.

And yet, even in your most generous recollections, you've probably experienced only an occasional need to compute the circumference of a circle from its diameter, or vice versa. Why is pi so important?

The answer comes from pi's central place in the study of trigonometry. Trigonometry is used to model all types of repeating, periodic phenomena in nature, and the archetype of such repeating, periodic phenomena is going around in a circle. If you haven't studied trigonometry, you may have mistaken pi's cultural ubiquity for a kind of importance, as we often do with celebrities.

If you go on from trigonometry to study calculus, there is another irrational number whose role is equal in importance to that of pi, though it is much less widely known in the popular culture. Among the mathematics in-crowd, however, this other celebrated constant is regarded with a special fascination. A comparative youngster to the grandfatherly pi, there is a calculated mystery about it.

The number is e, with a decimal expansion that begins:
e = 2.71828182845904523536...
The number was first introduced by the mathematician John Napier, who used it in the development of his theory of logarithms in the early 1600s. His version of natural logarithms were abandoned almost immediately, however, in favor of common logarithms with a base of ten, and it was left to Leonhard Euler(1707-1783) to discover many of the number's remarkable properties. Euler was the first to use the symbol e . Despite appearances, it is unlikely that Euler named the number after himself, even though it is still sometimes referred to as Euler's number. ( This term can be confusing: There is another, unrelated number usually called Euler's constant. ) As we will see in the first example, the number has a rightful claim to being the base for exponential functions, and so it has been suggested that Euler meant e to stand for exponential. The truth may be even more prosaic: Euler was using the letter a in some of his other mathematical work, and e was the next vowel.

The MNEMONIC to remember the value of 'e':-
To
express
e
remember
to
memorize
a
sentence
to
simplify
this.

There are a lot of interesting properties of "e", as there are with Pi. Some of them are pretty advanced, and some of them are easy to write down but hard to prove.

"e" is a number, just like Pi, and it has the value 2.718281828459045235306.... So e is not the same as Pi, which is about 3.14159265358979323846.... But while lots of people know Pi, not so many know about e.

Now, what exactly is so special about e? Does it have a geometric meaning like Pi (which is the ratio of the circumference of a circle to its diameter)?

Why does e=2.71828... instead of, say, 17.391526381? Or -592? Or Pi/2 (which it isn't)? Perhaps you've been told that the decimal digits of Pi never end, and they never repeat like 22/7 = 3.142857142857142857.... Well, e is just like Pi in this respect; if you keep going, it never stops, and it never repeats.

What sets them apart from fractions like 3/4 and -65/11, is just this fact, which is more commonly known as irrationality. Fractions are *rational*, while numbers like the square root of 2, Pi, and e are *irrational*. Still, we haven't really talked about what e is useful for. To get to this, we need to talk about logarithms. Say you have 2^x = 64 (2^x is my way of saying "2 raised to the x power"), and you want to solve for x. Well, we know that 64 = 2*32 = 2*(2*16) = 2*2*2*8 = 2*2*2*2*2*2 = 2^6, so x = 6. How about 2^x = 5? There we're stuck, because 2^2 = 4 and 2^3 = 8 so x should be somewhere between 2 and 3 to make 2^x = 5. There's another way of saying what x should be, and this is called the logarithm of 5 to the base 2. That is, x = log[2](5); (the 2 should be a subscript, like a power but typed a bit below the log, so it isn't log 10). In general, the solution to b^x = n for some given b and n is x = log[b](n). b is called the *base* of the logarithm.

Logarithms are useful, but there is a particular kind of logarithm that is used the most often: the *natural* logarithm. This is just the logarithm to the base [e]. In fact, the natural logarithm is so useful that people often say "ln(n)" instead of log[e](n). Now, why is all this important? It's hard to say without going into a lot of details, but here's a little hint of the interesting things about e and ln(n):
Think about (1+1/n)^n for some value n. For n=1, this is 2. For n=2, this is 2.25. For n=5, this is 2.48832. For n=10, this is 2.5937.... For n=100, this is 2.7048.... For n=10000, this is 2.7169.... Can you guess what happens to (1+1/n)^n as n gets larger and larger? In fact, it becomes e. A way of expressing this in mathematical notation is lim (1+1/n)^n = e. n->infinity
(the "lim" stands for "limit"; we say "the limit as n goes to infinity of the quantity one plus one over n to the nth power is e.)
Another thing to think about: If you've graphed equations, look at the graph of y=1/x. If you look at the region enclosed by y=1/x, the line y=0 (the x-axis), and the lines x=1 and x=e, it looks like a rectangle but with one curved side. What is the area of this shape? In fact, it is exactly 1. Mathematically speaking, we say "the area under the curve y=1/x from 1 to e is 1," or even better, "the *integral* of 1/x from 1 to e is 1." This is because if we replaced the line x=e with some line x=b for some b>1, the area of the region is the natural logarithm of b. Note the natural logarithm of e is 1, because e^1 = e; that is, 1 is the exponent for which the base (e) is equal to e.