As we saw earlier, the amount earned on an account increases as the compounding frequency increases. The table below 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.

Frequency | [latex]A\left(t\right)={\left(1+\frac{1}{n}\right)}^{n}[/latex] | Value |
---|---|---|

Annually | [latex]{\left(1+\frac{1}{1}\right)}^{1}[/latex] | $2 |

Semiannually | [latex]{\left(1+\frac{1}{2}\right)}^{2}[/latex] | $2.25 |

Quarterly | [latex]{\left(1+\frac{1}{4}\right)}^{4}[/latex] | $2.441406 |

Monthly | [latex]{\left(1+\frac{1}{12}\right)}^{12}[/latex] | $2.613035 |

Daily | [latex]{\left(1+\frac{1}{365}\right)}^{365}[/latex] | $2.714567 |

Hourly | [latex]{\left(1+\frac{1}{\text{8766}}\right)}^{\text{8766}}[/latex] | $2.718127 |

Once per minute | [latex]{\left(1+\frac{1}{\text{525960}}\right)}^{\text{525960}}[/latex] | $2.718279 |

Once per second | [latex]{\left(1+\frac{1}{31557600}\right)}^{31557600}[/latex] | $2.718282 |

These values appear to be approaching a limit as *n* increases without bound. In fact, as *n* gets larger and larger, the expression [latex]{\left(1+\frac{1}{n}\right)}^{n}[/latex] approaches a number used so frequently in mathematics that it has its own name: the letter [latex]e[/latex]. 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.

### A General Note: The Number *e*

The letter *e* represents the irrational number

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

### Example 7: Using a Calculator to Find Powers of *e*

Calculate [latex]{e}^{3.14}[/latex]. Round to five decimal places.

### Solution

On a calculator, press the button labeled [latex]\left[{e}^{x}\right][/latex]. The window shows [*e*^(]. Type 3.14 and then close parenthesis, (]). Press [ENTER]. Rounding to 5 decimal places, [latex]{e}^{3.14}\approx 23.10387[/latex]. 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*.

## Investigating Continuous Growth

So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, *e *is used as the base for exponential functions. Exponential models that use *e* as the base are called *continuous growth or decay models*. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.

### A General Note: The Continuous Growth/Decay Formula

For all real numbers *t*, and all positive numbers *a* and *r*, continuous growth or decay is represented by the formula

where

*a*is the initial value,*r*is the continuous growth rate per unit time,- and
*t*is the elapsed time.

If *r *> 0, then the formula represents continuous growth. If *r *< 0, then the formula represents continuous decay.

For business applications, the continuous growth formula is called the continuous compounding formula and takes the form

where

*P*is the principal or the initial invested,*r*is the growth or interest rate per unit time,- and
*t*is the period or term of the investment.

### How To: Given the initial value, rate of growth or decay, and time *t*, solve a continuous growth or decay function.

- Use the information in the problem to determine
*a*, the initial value of the function. - Use the information in the problem to determine the growth rate
*r*.- If the problem refers to continuous growth, then
*r*> 0. - If the problem refers to continuous decay, then
*r*< 0.

- If the problem refers to continuous growth, then
- Use the information in the problem to determine the time
*t*. - Substitute the given information into the continuous growth formula and solve for
*A*(*t*).

### Example 8: Calculating Continuous Growth

A person invested $1,000 in an account earning a nominal 10% per year compounded continuously. How much was in the account at the end of one year?

### Solution

Since the account is growing in value, this is a continuous compounding problem with growth rate *r *= 0.10. The initial investment was $1,000, so *P *= 1000. We use the continuous compounding formula to find the value after *t *= 1 year:

The account is worth $1,105.17 after one year.

### Try It 10

A person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years?

### Example 9: Calculating Continuous Decay

Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?

### Solution

Since the substance is decaying, the rate, 17.3%, is negative. So, *r *= –0.173. The initial amount of radon-222 was 100 mg, so *a *= 100. We use the continuous decay formula to find the value after *t *= 3 days:

So 59.5115 mg of radon-222 will remain.