Everyday Usage and Use in Mathematics

Compound in everyday language typically refers to something made up of two or more parts, like a compound word ("notebook") or a compound substance (water is a compound of hydrogen and oxygen). The word interest refers to money paid in exchange for a loan.

In mathematics and finance, compound interest is a method of calculating interest where the interest earned over time is added to the principal, and future interest calculations are based on this increased amount. Each time the interest is caclulated and paid out, the interest is said to have been compounded.

The term compound interest emphasizes that the interest is paid both for the principal and on the interest that has already been earned.

Importance in mathematics

Compound interest contrasts with simple interest, which is paid only on the principal and not on the interest.

The formula for compound interest is:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

where
• A is the amount of money accumulated after t years, including interest.
• P is the principal amount.
• r is the annual interest rate (as a decimal).
• n is the number of times interest is compounded per year.
• t is the time the money is invested or borrowed for, in years.

This concept illustrates the exponential growth of investments or debts over time due to the repeated application of interest on an ever-growing principal. When applying the compound interest formula, $t$ should always be a multiple of $1/n$, when the interest is compounded.

\[ A = P e^{rt}\]

• A is the amount of money accumulated after $t$ years, including interest
• P is the principal amount.
• r is the annual interest rate (as a decimal).
• t is the time the money is invested or borrowed for, in years.

When applying the continuous compound interest formula, $t$ can be any positive real number, since the interest is compounded at every moment.