Defining the Number e

Now that we have the natural logarithm defined, we can use that function to define the number \(e .\)

To put it another way, the area under the curve \(y = 1 / t\) between \(t = 1\) and \(t = e\) is \(1\) (Figure 6.77). The proof that such a number exists and is unique is left to you. (Hint: Use the Intermediate Value Theorem to prove existence and the fact that \(\text{ln} x\) is increasing to prove uniqueness.)

Figure 6.77 The area under the curve from \(1\) to \(e\) is equal to one.

The number \(e\) can be shown to be irrational, although we won’t do so here (see the Student Project in Taylor and Maclaurin Series). Its approximate value is given by