Algebra and Trigonometry, 5th Edition

The Number e

We now consider a very special number in mathematics. In 1741, Leonhard Euler named this number e. Though you may not have encountered it before, you will see here and in future mathematics courses that it has many important applications. To explain this number, we use the compound interest formula $A=P{(1+r/n)}^{nt}$ discussed in Example 4. Suppose that $1 is invested at 100% interest for 1 year. Since $P=1$, $r=100\%=1$, and $t=1$, the formula above becomes a function A defined in terms of the number of compounding periods n:

$$A=P{(1+\frac{r}{n})}^{nt}=1{(1+\frac{1}{n})}^{n·1}={(1+\frac{1}{n})}^n.$$

Let’s visualize this function using its graph, shown at left, and explore the values ...