Evaluating Functions with Base e

As we saw earlier, the amount earned on an account increases as the compounding frequency increases. Table 5 shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.

Examine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies, listed in Table 5.

Frequency A(n)=(1+ \frac{1}{n})^n Value
Annually (1+ \frac{1}{1})^1 $2
Semiannually (1+ \frac{1}{2})^2 $2.25
Quarterly (1+ \frac{1}{4})^4 $2.441406
Monthly (1+ \frac{1}{12})^{12} $2.613035
Daily (1+ \frac{1}{365})^{365} $2.714567
Hourly (1+ \frac{1}{8760})^{8760} $2.718127
Once per minute (1+ \frac{1}{525600})^{525600} $2.718279
Once per second (1+ \frac{1}{31536000})^{31536000} $2.718282

Table 5

These values appear to be approaching a limit as n increases without bound. In fact, as n gets larger and larger, the expression (1+ \frac{1}{n})^n approaches a number used so frequently in mathematics that it has its own name: the letter e. This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.


THE NUMBER e

The letter e represents the irrational number

(1+ \frac{1}{n})^n, as n increases without bound

The letter e is used as a base for many real-world exponential models. To work with base e, we use the approximation, e \approx 2.718282. The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.


EXAMPLE 10

Using a Calculator to Find Powers of e

Calculate e^{3.14}. Round to five decimal places.


Solution

On a calculator, press the button labeled \left[e^{x}\right]. The window shows \left[e^{\wedge}(]\right.. Type 3.14 and then close parenthesis, [)]. Press [ENTER]. Rounding to 5 decimal places, e^{3.14} \approx 23.10387. Caution: Many scientific calculators have an "Exp" button, which is used to enter numbers in scientific notation. It is not used to find powers of e.


TRY IT #10

Use a calculator to find e^{−0.5}. Round to five decimal places.