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Mathematics

Continuous Compounding

The Number e, Exponential Growth, and the Limit Behind Modern Finance — A TLDR Primer

Compound interest shows up on every precalculus exam, every AP Calculus free-response section, and in nearly every personal finance conversation — yet most students have never seen where the formula actually comes from. If you can follow the algebra but the number *e* still feels like magic, this guide is for you.

**Continuous Compounding: The Number e, Exponential Growth, and the Limit Behind Modern Finance** builds the whole story from scratch. It starts with simple interest and annual compounding — concrete dollar figures, no hand-waving — then shows what happens as compounding frequency increases without bound. That limit is where *e* lives, and the guide derives it step by step rather than just announcing it. From there it moves to the continuous compounding formula $A = Pe^{rt}$, works through varied examples, and then flips the problem around: using the natural log to solve for unknown time or rate. The final section ties everything to the differential equation $dy/dt = ky$, connecting compound interest to population growth, radioactive decay, and the calculus you are either taking now or about to take.

Written for high school students in precalculus or calculus and early college students who want a clean, concise explanation without the bloat of a doorstop textbook. Parents helping a student through exponential growth formula homework and tutors prepping a session will find it equally direct and useful.

If you want to understand *e* — not just use it — pick this up.

What you'll learn
  • Distinguish simple interest, discrete compounding, and continuous compounding
  • Derive and use the formula A = Pe^(rt) for continuously compounded growth
  • Understand why e arises as the limit of (1 + 1/n)^n
  • Convert between nominal rates, effective annual rates, and continuous rates
  • Apply continuous compounding to finance, population growth, and radioactive decay
  • Solve for time, rate, or principal using natural logarithms
What's inside
  1. 1. From Simple Interest to Compounding
    Sets up the problem by comparing simple interest, annual compounding, and more frequent compounding with concrete dollar figures.
  2. 2. The Limit That Gives Us e
    Shows how pushing the compounding frequency to infinity produces the number e, and defines e as a limit.
  3. 3. The Continuous Compounding Formula
    Derives A = Pe^(rt) from the discrete formula and works through canonical examples with varying P, r, and t.
  4. 4. Solving for Time and Rate with Natural Logs
    Uses ln to invert the exponential and solve real problems where time, rate, or principal is unknown.
  5. 5. Why It Matters: Growth, Decay, and the Calculus Connection
    Connects continuous compounding to the differential equation dy/dt = ky and applies it to population growth, radioactive decay, and finance.
Published by Solid State Press
Continuous Compounding cover
TLDR STUDY GUIDES

Continuous Compounding

The Number e, Exponential Growth, and the Limit Behind Modern Finance — A TLDR Primer
Solid State Press

Continuous Compounding

The Number e, Exponential Growth, and the Limit Behind Modern Finance — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 From Simple Interest to Compounding
  2. 2 The Limit That Gives Us e
  3. 3 The Continuous Compounding Formula
  4. 4 Solving for Time and Rate with Natural Logs
  5. 5 Why It Matters: Growth, Decay, and the Calculus Connection
Chapter 1

From Simple Interest to Compounding

Borrow $1,000 from a bank, and the bank will charge you for the privilege. Lend \$1,000 to a bank by opening a savings account, and the bank will pay you for the same reason. Either way, there is a core quantity called principal — the initial amount of money — and an interest rate — the percentage of that principal charged or paid per unit of time, almost always expressed as an annual percentage.

The simplest possible rule is to pay interest only on the original principal, every year, forever. That rule is called simple interest. If you deposit $P$ dollars at an annual rate $r$ (written as a decimal, so 6% becomes $r = 0.06$), then after $t$ years you have:

$A = P(1 + rt)$

The total amount $A$ grows, but it grows in a straight line — every year adds exactly the same number of dollars, $P \cdot r$.

Example. You deposit $1,000 at 6% simple interest. How much do you have after 5 years?

Solution. $A = 1000(1 + 0.06 \times 5) = 1000(1.30) = \$1{,}300.00$

That $300 in interest sounds reasonable. But notice: in year 1 you earned \$60 on your original $1,000. In year 2 you *still* earned only \$60, even though you now had $1,060 sitting there. The extra \$60 from year 1 earned nothing. That feels wrong — and banks and investors agreed. The alternative is compound interest: once interest is earned, it is added to the principal, and future interest is calculated on the new, larger balance. Interest earns interest.

Annual Compounding

Under annual compounding, at the end of each year the bank calculates 6% of whatever the current balance is and adds it. After one year: $1000 \times 1.06 = \$1{,}060$. After two years: $1060 \times 1.06 = $1{,}123.60$. Each year the multiplier $1.06$ is applied again, so the growth is multiplicative rather than additive. After $t$ years:

$A = P(1 + r)^t$

About This Book

If you are staring at the $Pe^{rt}$ formula in a precalculus or calculus class and have no idea where it came from, this book is for you. It is also for the student who got through compound interest in Algebra 2 but never connected it to the number $e$, for anyone prepping for the AP Calculus AB or BC exam, and for the dual-enrollment or early-college student who needs a solid grip on exponential functions before the pace picks up.

This guide builds from simple interest all the way through continuous compounding interest, explained simply and in order. You will see what the number $e$ actually is, work through the exponential growth formula step by step, and learn natural log solving for time and rate in finance and math contexts. The calculus exponential decay connection gets its own treatment, making this a genuine precalculus-to-calculus bridge. Short by design, no filler.

Read straight through once to get the full picture, then work each example as you hit it. The problem set at the end is your real test — attempt it before checking the solutions.

Keep reading

You've read the first half of Chapter 1. The complete book covers 5 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon