Compound Interest and the Number e

Simple interest pays you on your original deposit and nothing more. Compound interest pays you on your deposit and on the interest already earned — so the earnings themselves start earning. That single difference turns a straight line into an exponential curve, and pushing it to its limit produces one of the most important numbers in mathematics.

Concept The compound interest formula

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

P

— the principal, the amount you start with.

r

— the annual interest rate, written as a decimal: 8% is 0.08, not 8.

n

— how many times interest is compounded per year.

t

— the number of years.

A

— the final amount, principal plus all accumulated interest.

Read the structure and it explains itself. Each period adds a fraction $\frac{r}{n}$ of the current balance, and there are $nt$ such periods in total — so the multiplier $\left(1 + \frac{r}{n}\right)$ is applied $nt$ times over.

Concept Simple interest versus compound interest

Simple:

A = P(1 + rt)

. Interest is calculated only on the original principal, so growth is linear.

Compound: earnings are added back to the principal, so growth is exponential.

Over a year or two the gap is small. Over decades it becomes the whole story — which is why compound interest matters far more for long-term saving than for short-term loans.

Example Compounding more often

Invest 1000 for one year at 10%. What does the balance become as the compounding frequency rises?

Annually,

n = 1

1000(1 + 0.10)^1 = 1100.00

Twice a year,

n = 2

1000\left(1 + \frac{0.10}{2}\right)^{2} = 1102.50

Quarterly,

n = 4

1103.81

Monthly,

n = 12

1104.71

⟹ more frequent compounding always pays more, but the gains shrink each time

Notice what is happening: the amounts keep rising, yet they rise by less and less. They are converging on a ceiling rather than growing without limit.

Concept Continuous compounding and the number e

A = Pe^{rt}

Let the compounding frequency $n$ grow without bound — compounding every day, every second, continuously. The expression $\left(1 + \frac{r}{n}\right)^{nt}$ does not run off to infinity; it settles on $e^{rt}$ , where

e \approx 2.71828

This is exactly where Euler's number comes from. It is not an arbitrary constant someone chose — it is the limit that continuous growth converges to, which is why $e$ appears throughout finance, biology and physics wherever something grows in proportion to its current size.

Example Ten years, four compounding schemes

Invest 1000 at 8% for 10 years. Compare annual, monthly, daily and continuous compounding.

Annually:

1000(1 + 0.08)^{10} = 2158.92

Monthly:

1000\left(1 + \frac{0.08}{12}\right)^{120} = 2219.64

Daily:

2225.35

Continuously:

1000e^{0.8} = 2225.54

Simple interest for comparison:

1000(1 + 0.08 \times 10) = 1800

⟹ compounding beats simple interest by over 400, and daily is already within 0.20 of continuous

The difference between annual and continuous compounding is about 67 here. The difference between compounding at all and not compounding is far larger — the frequency matters much less than the principle.

Note Where this shows up

Finance — savings accounts and deposits, investment funds and retirement plans, car and property loans, long-term investment appraisal.

Science and nature — growth of bacteria and cells, radioactive decay, population growth, the spread of epidemics.

The same formula describes all of them, because in each case the rate of change is proportional to the amount currently present. Decay is simply the case where $r$ is negative.

Note Practical points worth remembering

Start early — time is the strongest variable in the formula, since it sits in the exponent.

Be patient — compounding needs years before its effect becomes visible.

Contribute regularly — steady small amounts outperform occasional large ones.

A higher rate matters — even one extra percent compounds into a large difference.

Avoid early withdrawals — breaking the accumulation cycle removes the earnings that would themselves have earned.

Note Mistakes to avoid

Entering the rate as a percentage instead of a decimal — 8% must go in as 0.08.

Getting

n

wrong: monthly compounding means

n = 12

, not

n = 1

Forgetting that the exponent is

nt

, not

t

alone.

Using the simple interest formula for a long-term calculation.

Ignoring inflation, which reduces what the final amount can actually buy.

Overlooking fees and taxes when appraising a real investment.

Summary

Compound interest pays on the principal and on interest already earned, so A = P(1 + r/n)^(nt).
Simple interest, A = P(1 + rt), grows linearly; compound interest grows exponentially.
Raising the compounding frequency always increases the total, but by ever smaller amounts.
As n grows without bound the formula converges to A = Pe^(rt), and that limit is exactly where e ≈ 2.71828 comes from.
Enter r as a decimal, count n correctly, and remember the exponent is nt.