PROOF:
A Definition of the Number e

In algebra and precalculus classes I tell my students that there is a "special" number in mathematics called e, the exponential number. This number is of fundamental importance for learning higher mathematics and it is a "real" number that appears in many practical, everyday, real-world situations. However, for many students this is the first time they have ever heard of this number. If the number is so important, why have they not seen it before? The reason is that it is a tricky number to define, requiring skills learned over several mathematics courses.

To aid students in understanding it, I begin by discussing the different sets of numbers:

The natural numbers = those numbers that people "naturally" first started using. These are the numbers in the pattern 1, 2, 3, and so on.

The integers = all the natural numbers, along with their negative counterparts, and 0. Thus the integers are { … , -3, -2, -1, 0, 1, 2, 3, … }.

The rational numbers = all the numbers that can be written as the ratio of two integers (all the fractions).

The irrational numbers = all those numbers that cannot be written as fractions.

It is with the irrational numbers that most students begin to question from where the numbers originate. Nearly all have heard of p, with most having been told the decimals for p are a never ending string of numbers, for which there is no repeating pattern. The reason for this involves some really hard mathematics, but most accept the fact that was taught to them. Other irrationals include many square roots, such as the sqr(2), sqr(3), sqr(5), sqr(6), and so on. Any square root of a non-perfect square will be irrational. We discuss the proof that the square root of 2 is irrational, and many students are convinced that the proof is a convincing argument.

By the time the number e is presented, though, students often seem to have had enough. What is going on? Is math full of these irrational, "weird" numbers? The answer is, "Yes it is!" There are, in fact, more irrational numbers than there are rational numbers. The reason we do not recognize them in daily life is that they are often very hard to define.

Such is the case with e. The number e has several definitions, all involving a process that extends out infinitely many steps. The one I use most often is:

This is read: "e equals the limit of the expression '1 plus 1 divided by x,' all raised to the exponential power of x, as x is allowed to approach infinity."

The math required to work out this problem is pretty difficult (at least it is for me). For algebra and precalculus students, it is beyond their current skill level. Therefore, to convince students in these course that this equation is indeed true, we look at a table of values, letting x get really large (that is the meaning of "the limit as x approaches infinity"), as well as some graphs.

The y-values seem to be approaching a limit, a certain value toward which they creep closer and closer to as the x-values move toward infinity. As all mathematicians know, these numerical and graphical demonstrations do not prove that the limit converges on some decimal expansion (which we call e). For algebra and precalculus students, though, it is a pretty good argument.

However, every so often some curious students will ask why the values for y converge as x gets larger and larger. One proof that I know involves a tool mathematicians call L'Hôpital's Rule, which involves taking derivatives (a calculus procedure). Here is the proof:

and begin by taking the natural logarithm of both sides of this equation.

If at this point we attempted to evaluate the limit by direct substitution, the form would be 0/0. This is an indeterminant form (we cannot determine what the limit is at this point). However, we are allowed to apply L'Hôpital's Rule to attempt to find the limit by taking the derivative of the numerator and denominator and finding the limit of that ratio.

Thus, we see that ln y = 1. This implies y = e (by definitions of logarithms). Since we let y = the original limit expression, then

This completes the proof.

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Some Links regarding the number e:

• e - MathWorld.com
• Leonhard Euler [ ... first proved the irrationality of e]
• FAQ ... About e [From Ask Dr. Math]

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