The Mathematical Adventure of Natural Constant e

Bernoulli and the Discovery of Natural Constant

📜 Historical Background

In the late 17th century, Swiss mathematician Jacob Bernoulli (1654-1705) accidentally discovered the natural constant $e$ while studying compound interest problems in finance. This story begins with basic financial concepts.

💰 Principal, Interest, and Interest Rate

Before discussing compound interest, let’s understand some basic financial concepts:

Basic Financial Concepts

Principal: The initial amount of money invested or borrowed, denoted as $P$ . For example, if you deposit 100 yuan in a bank, that 100 yuan is the principal.
Interest: The compensation earned by the principal over a certain period of time when lent or deposited. For example, if the bank gives you an additional 2 yuan after one year, that 2 yuan is the interest.
Interest Rate: The ratio of interest to principal, usually expressed as a percentage. For example, an annual interest rate of 5% means that for every 100 yuan of principal, 5 yuan of interest is earned per year.

🔄 The Compound Interest Problem

In the 17th century, bankers and merchants already used compound interest to calculate interest in practice. Compound interest means adding the interest generated in each period to the principal, and then calculating interest together in the next period. This way, “interest begets interest,” and the sum of principal and interest grows faster than simple interest.

Bernoulli posed this classic question:

Problem: If you have 1 unit of principal with an annual interest rate of 100%, how much money will you have after one year?

1️⃣ Compounded Once per Year

If interest is calculated only once a year, the year-end sum is:

A_1 = 1 \times (1 + 1) = 2

2️⃣ Compounded Twice per Year

If interest is calculated twice a year, with each time at 50% rate:

Annual rate 100%, compounded twice, so each rate is:
$\text{Each rate} = \frac{100\%}{2} = 50\% = 0.5$
First calculation (after half year):
$1 \times (1 + 0.5) = 1.5$
Second calculation (after another half year):
$1.5 \times (1 + 0.5) = 2.25$
In formula form:
$\text{Sum} = 1 \times (1 + 0.5)^2 = 1.5^2 = 2.25$

3️⃣ Compounded n Times per Year

If interest is calculated $n$ times per year, with each rate at $1/n$ , the year-end sum is:

A_n = 1 \times \left(1 + \frac{1}{n}\right)^n

4️⃣ Compounding Frequency Approaches Infinity: Continuous Compounding

Bernoulli wondered: as the compounding frequency increases ( $n \to \infty$ ), what value does the sum approach? He calculated results for different $n$ values:

Compounding Frequency n	Sum (1+1/n)^n
1	2.0000
2	2.2500
10	2.5937
100	2.7048
1000	2.7169
10000	2.7181
∞	e ≈ 2.71828

As $n$ approaches infinity, the sum approaches a limit, which is:

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.71828

5️⃣ Historical Significance

In 1683, Bernoulli first published related calculations. Although the constant was not yet denoted by $e$ , this was the first appearance of $e$ .

In the 18th century, the great mathematician Leonhard Euler (1707-1783) systematically studied this constant and first used the letter $e$ to denote it. Euler proved the irrationality of $e$ and discovered its profound connections with trigonometric functions and complex numbers (the transcendence of $e$ was later proved by French mathematician Charles Hermite in 1873). $e$ gradually became an indispensable important constant in mathematical analysis, probability theory, number theory, and other fields.

Deep Understanding of Natural Constant e

📐 Two Definitions

Definition of Natural Constant e

The natural constant $e$ can be defined in the following two equivalent ways:

Method 1: Limit Definition (Compound Interest Limit)

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n

Method 2: Series Definition (Taylor Expansion)

e = \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots

符号说明

Symbol	Type	Pronunciation/Note	Meaning in this article
$e$	Mathematical constant	Natural constant	Euler’s number, approximately 2.71828
$n!$	Mathematical symbol	n factorial	$n! = n \times (n-1) \times \cdots \times 1$
$\sum$	Greek letter	Sigma	Summation symbol
$\lim$	Mathematical symbol	Limit	Limit operation

🔍 Rigorous Proof of Limit Definition

Proof Method 1: Binomial Expansion

Binomial Expansion Proof

We want to prove:

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n

Proof Process:

Using the binomial theorem:

\left(1 + \frac{1}{n}\right)^n = \sum_{k=0}^{n} \binom{n}{k} \left(\frac{1}{n}\right)^k

Where the binomial coefficient is:

\binom{n}{k} = \frac{n!}{k!(n-k)!} = \frac{n(n-1)\cdots(n-k+1)}{k!}

Substituting:

\left(1 + \frac{1}{n}\right)^n = \sum_{k=0}^{n} \frac{n(n-1)\cdots(n-k+1)}{k! \cdot n^k}

= \sum_{k=0}^{n} \frac{1 \cdot \left(1 - \frac{1}{n}\right) \cdots \left(1 - \frac{k-1}{n}\right)}{k!}

As $n \to \infty$ , for fixed $k$ , each factor $\left(1 - \frac{j}{n}\right) \to 1$ , so:

\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = \sum_{k=0}^{\infty} \frac{1}{k!} = e

Q.E.D.

证明

Proof Method 2: Squeeze Theorem

Squeeze Theorem Proof

We can prove the following inequality:

\left(1 + \frac{1}{n}\right)^n < e < \left(1 + \frac{1}{n}\right)^{n+1}

As $n$ approaches infinity, both sides approach the same limit $e$ .

Proof Process:

Let $a_n = \left(1 + \frac{1}{n}\right)^n$ . We can prove that the sequence $\{a_n\}$ is monotonically increasing and bounded above.

Monotonicity: $a_{n+1} > a_n$ (can be proved by AM-GM inequality)
Boundedness: $a_n < 3$ (can be proved by comparison with geometric series)

Therefore $\{a_n\}$ converges, and its limit is $e$ .

Q.E.D.

证明

🔢 Derivation of Series Definition

Starting from the limit definition, we can derive the series definition:

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n

Using binomial expansion and letting $n \to \infty$ , we get:

e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots = \sum_{k=0}^{\infty} \frac{1}{k!}

This series converges very quickly! Calculating the first few terms:

$1 + 1 = 2$
$+ \frac{1}{2} = 2.5$
$+ \frac{1}{6} \approx 2.6667$
$+ \frac{1}{24} \approx 2.7083$
$+ \frac{1}{120} \approx 2.7167$
$+ \frac{1}{720} \approx 2.7181$

With only 7 terms, we can achieve accuracy to 3 decimal places!

Amazing Properties of e

⭐ Core Properties

Core Properties of e

1. Base of Natural Logarithm

\ln x = \log_e x

Natural logarithm has a special position in calculus because its derivative form is the simplest.

2. Derivative Equals Itself

\frac{d}{dx}e^x = e^x

$e^x$ is the only real function whose derivative equals itself!

3. Irrational and Transcendental

$e$ is an irrational number (proved by Euler in 1737)
$e$ is a transcendental number (proved by Hermite in 1873)

Transcendental means $e$ is not a root of any polynomial with rational coefficients, which is fundamentally different from algebraic numbers (like $\sqrt{2}$ ).

4. Decimal Expansion

e = 2.7182818284590452353602874713527\ldots

The decimal part is infinite and non-repeating.

🎯 Why Choose e as the Base?

Comparison of Exponential Function Derivatives

For a general exponential function $f(x) = a^x$ , its derivative is:

\frac{d}{dx}a^x = a^x \ln a

Only when $a = e$ , $\ln e = 1$ , the derivative equals itself:

\frac{d}{dx}e^x = e^x \cdot \ln e = e^x \cdot 1 = e^x

This is why in calculus we prefer exponential functions with base $e$ !

🔗 Continued Fraction Representation

$e$ also has an elegant continued fraction representation:

e = 2 + \frac{1}{1 + \frac{1}{2 + \frac{1}{1 + \frac{1}{1 + \frac{1}{4 + \frac{1}{1 + \frac{1}{1 + \cdots}}}}}}}

The pattern is: $[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, \ldots]$

This pattern was discovered by Euler, revealing the internal structural beauty of $e$ .

Euler’s Formula: The Most Beautiful Formula in Mathematics

🌟 Discovery of Euler’s Formula

Euler's Formula

e^{ix} = \cos x + i\sin x

Where $i$ is the imaginary unit, $i^2 = -1$ .

证明

Proof (using Taylor series):

Expand $e^{ix}$ , $\cos x$ and $\sin x$ into Taylor series respectively:

e^{ix} = 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \cdots

= 1 + ix - \frac{x^2}{2!} - i\frac{x^3}{3!} + \frac{x^4}{4!} + i\frac{x^5}{5!} - \cdots

= \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\right) + i\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\right)

= \cos x + i\sin x

Q.E.D.

💎 Euler’s Identity

When $x = \pi$ , we get the famous Euler’s Identity:

e^{i\pi} + 1 = 0

This formula is known as “the most beautiful formula in mathematics” because it connects the five most important constants in mathematics:

Constant	Symbol	Meaning
1	1	Multiplicative identity
0	0	Additive identity
π	$\pi$	Pi, geometric constant
e	$e$	Natural constant, analytical constant
i	$i$	Imaginary unit, algebraic constant

Physicist Richard Feynman called it “the most remarkable formula in mathematics.”

Practical Applications of e

📈 Continuous Growth and Decay Models

Exponential Growth/Decay Model

P(t) = P_0 e^{kt}

$P_0$ : Initial value
$k$ : Growth rate ( $k>0$ ) or decay rate ( $k<0$ )
$t$ : Time

Radioactive Decay

Carbon-14 has a half-life of about 5730 years. The decay of radioactive substances follows:

N(t) = N_0 e^{-\lambda t}

Where $\lambda = \frac{\ln 2}{T_{1/2}}$ is the decay constant.

Bacterial Growth

Under ideal conditions, bacterial population grows exponentially:

N(t) = N_0 e^{rt}

Where $r$ is the growth rate.

🎲 Applications in Probability Theory

Poisson Distribution: $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$
Normal Distribution: Probability density function contains $e^{-x^2}$
Central Limit Theorem: $e$ appears in limit expressions

🏦 Financial Mathematics

Continuous Compounding: $A = Pe^{rt}$
Option Pricing: Black-Scholes formula uses $e$

🔬 Physics and Engineering

Newton’s Law of Cooling: $T(t) = T_{env} + (T_0 - T_{env})e^{-kt}$
RC Circuit Charging: $V(t) = V_0(1 - e^{-t/RC})$
Atmospheric Pressure: $P(h) = P_0 e^{-h/H}$

Comparison of e and π

Property	e	π
Definition Source	Analysis (limits, calculus)	Geometry (circle circumference/diameter ratio)
First Discovery	17th century (Bernoulli)	Ancient (Archimedes, etc.)
Mathematical Properties	Irrational, transcendental	Irrational, transcendental
Decimal Representation	2.71828…	3.14159…
Main Applications	Calculus, differential equations, probability	Geometry, trigonometry, physics
Series Expansion	$\sum_{n=0}^{\infty} \frac{1}{n!}$	$4\left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\right)$
Euler’s Formula	$e^{i\pi} + 1 = 0$	Same, both are unified here

Exercises

Exercise 1: Basic Concepts

Write the limit definition and series definition of $e$ .

Answer and Analysis (4 个标签)

natural constant e definition limit series

Limit Definition: $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

Series Definition: $e = \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots$

Exercise 2: Property Judgment

Determine whether $e$ is a rational number and explain why.

Answer and Analysis (3 个标签)

natural constant e irrational number transcendental number

$e$ is not a rational number.

Reason:

$e$ is an irrational number (proved by Euler in 1737)
Furthermore, $e$ is a transcendental number (proved by Hermite in 1873), meaning it is not a root of any polynomial with rational coefficients

A rational number can be expressed as a ratio of two integers, but $e$ cannot be expressed this way.

Exercise 3: Basic Calculation

Calculate the values of $e^0$ , $e^1$ , and $\ln e$ .

Answer and Analysis (3 个标签)

natural constant e exponential function logarithmic function

$e^0 = 1$ (any non-zero number to the power of 0 equals 1)
$e^1 = e$ (any number to the power of 1 equals itself)
$\ln e = 1$ (because $e^1 = e$ )

Exercise 4: Limit Identification (Graduate Exam Type)

Which of the following limits equals $e$ ? (A) $\lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n$ (B) $\lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^{2n}$ (C) $\lim_{n\to\infty} \left(1 + \frac{2}{n}\right)^n$ (D) $\lim_{n\to\infty} \left(1 + \frac{1}{2n}\right)^n$

Answer and Analysis (3 个标签)

natural constant e limit graduate exam

Answer: (A)

Analysis:

(A) $\lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n = e$ ✓
(B) $\lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^{2n} = [\lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n]^2 = e^2$ ✗
(C) $\lim_{n\to\infty} \left(1 + \frac{2}{n}\right)^n = \lim_{n\to\infty} \left[\left(1 + \frac{2}{n}\right)^{\frac{n}{2}}\right]^2 = e^2$ ✗
(D) $\lim_{n\to\infty} \left(1 + \frac{1}{2n}\right)^n = \lim_{n\to\infty} \left[\left(1 + \frac{1}{2n}\right)^{2n}\right]^{\frac{1}{2}} = e^{\frac{1}{2}} = \sqrt{e}$ ✗

Exercise 5: Transcendental Number Concept

Determine whether $e$ is a transcendental number and explain why.

Answer and Analysis (3 个标签)

natural constant e transcendental number graduate exam

$e$ is a transcendental number.

Reason: A transcendental number is a number that is not a root of any polynomial with rational coefficients.

In 1873, French mathematician Charles Hermite first proved that $e$ is a transcendental number
This means there is no polynomial $P(x)$ with integer coefficients such that $P(e) = 0$
This is a stronger property than being irrational: all transcendental numbers are irrational, but not all irrational numbers are transcendental (for example, $\sqrt{2}$ is irrational but not transcendental)

Exercise 6: Series Calculation

Use the series definition to calculate the approximate value of $e$ , accurate to 3 decimal places.

Answer and Analysis (3 个标签)

natural constant e series numerical calculation

Using the series $e = \sum_{n=0}^{\infty} \frac{1}{n!}$ :

n	1/n!	Cumulative Sum
0	1	1
1	1	2
2	1/2 = 0.5	2.5
3	1/6 ≈ 0.1667	2.6667
4	1/24 ≈ 0.0417	2.7083
5	1/120 ≈ 0.0083	2.7167
6	1/720 ≈ 0.0014	2.7181
7	1/5040 ≈ 0.0002	2.7183

Answer: $e \approx 2.718$ (accurate to 3 decimal places)

Exercise 7: Euler’s Formula

Use Euler’s formula to prove: $e^{i\pi} + 1 = 0$

Answer and Analysis (3 个标签)

natural constant e Euler formula complex number

According to Euler’s formula:

e^{ix} = \cos x + i\sin x

Let $x = \pi$ :

e^{i\pi} = \cos \pi + i\sin \pi

Since $\cos \pi = -1$ and $\sin \pi = 0$ :

e^{i\pi} = -1 + i \cdot 0 = -1

Rearranging:

e^{i\pi} + 1 = 0

Q.E.D. This is the famous Euler’s identity!

Exercise 8: Continuous Compounding

If the principal is 1000 yuan, the annual interest rate is 5%, calculated by continuous compounding, what is the sum after 10 years?

Answer and Analysis (3 个标签)

natural constant e continuous compounding application

Using the continuous compounding formula:

A = Pe^{rt}

Where:

$P = 1000$ (principal)
$r = 0.05$ (5% annual rate)
$t = 10$ (years)

Calculation:

A = 1000 \cdot e^{0.05 \times 10} = 1000 \cdot e^{0.5}

Since $e^{0.5} \approx 1.6487$ :

A \approx 1000 \times 1.6487 = 1648.72 \text{ yuan}

Answer: Approximately 1648.72 yuan

Exercise 9: Limit Transformation

Find the limit: $\lim_{x\to 0} (1 + 2x)^{\frac{1}{x}}$

Answer and Analysis (3 个标签)

natural constant e limit graduate exam

Solution: Let $t = \frac{1}{x}$ , when $x \to 0$ , $t \to \infty$

\lim_{x\to 0} (1 + 2x)^{\frac{1}{x}} = \lim_{t\to\infty} \left(1 + \frac{2}{t}\right)^t

= \lim_{t\to\infty} \left[\left(1 + \frac{2}{t}\right)^{\frac{t}{2}}\right]^2

= [e]^2 = e^2

Answer: $e^2$

Exercise 10: Differential Equation

Find the particular solution of the differential equation $\frac{dy}{dx} = y$ with $y(0) = 1$ .

Answer and Analysis (3 个标签)

natural constant e differential equation derivative

Solution: Separate variables:

\frac{dy}{y} = dx

Integrate both sides:

\ln y = x + C

From $y(0) = 1$ , we get $\ln 1 = 0 + C$ , so $C = 0$

Therefore:

\ln y = x \Rightarrow y = e^x

Answer: $y = e^x$

This also explains why the derivative of $e^x$ equals itself!

Summary

📚 Core Points Review

Historical Discovery: Discovered by Bernoulli while studying compound interest (1683), systematically studied and named by Euler (18th century)
Dual Definitions: Limit definition (compound interest) and series definition (Taylor expansion)
Core Properties: Derivative equals itself, base of natural logarithm, irrational number, transcendental number
Euler’s Formula: $e^{ix} = \cos x + i\sin x$ , connecting different mathematical fields
Wide Applications: Calculus, probability theory, physics, engineering, finance

🔗 Course Connections

This course is closely related to:

Exponential Functions (exponential-function.mdx): Foundation of $e^x$
Logarithmic Functions (logarithmic-function.mdx): Base of $\ln x$
Limits (limits/): Second important limit
Series (infinite-series/): Exponential series
Derivatives (differential-calculus/): $(e^x)' = e^x$
Integrals (integral-calculus/): $\int e^x dx = e^x + C$

🌟 Mathematical Quote

“God created the integers, all else is the work of man.” — Leopold Kronecker

And the discovery of $e$ tells us: even starting from human-made concepts (like compound interest), we can discover profound and beautiful truths in mathematics.

📖 Further Reading

“Euler’s Complete Works”: Understand Euler’s systematic research on $e$
“Mathematical Constants”: by Steven Finch, deep exploration of various mathematical constants
“Mathematical Thought from Ancient to Modern Times”: by Morris Kline, understand the context of mathematical history

Symbols in this Article

Symbol	Type	Pronunciation/Note	Meaning in this article
$e$	Mathematical constant	Natural constant	Euler’s number, approximately 2.71828
$\pi$	Greek letter	Pi	Circle constant, approximately 3.14159
$i$	Mathematical symbol	Imaginary unit	$i^2 = -1$
$\ln$	Mathematical symbol	Natural logarithm	Logarithm with base $e$
$\sum$	Greek letter	Sigma	Summation symbol
$\lim$	Mathematical symbol	Limit	Limit operation
$n!$	Mathematical symbol	n factorial	$n \times (n-1) \times \cdots \times 1$
$\infty$	Mathematical symbol	Infinity	Infinite quantity
$\int$	Mathematical symbol	Integral	Integral operation

Greek Letter Quick Reference

Upper	Lower	English Name	Chinese Pronunciation	Meaning in Text
Π	$\pi$	Pi	派	Circle constant (Pi)
Σ	$\sum$	Sigma	西格玛	Summation symbol
-	$\infty$	Infinity	无穷大	Infinity

Chinese-English Terminology

Chinese Term	English Term	Pronunciation	Description
自然常数	natural constant e	/ˈnætʃrəl ˈkɒnstənt iː/	Constant defined by continuous compounding and limits
欧拉数	Euler’s number	/ˈɔɪlər ˈnʌmbər/	Alternative name for natural constant e
复利	compound interest	/ˈkɒmpaʊnd ˈɪntrəst/	Interest calculation with interest added to principal
连续复利	continuous compounding	/kənˈtɪnjʊəs kəmˈpaʊndɪŋ/	Compounding frequency approaches infinity
超越数	transcendental number	/ˌtrænsɛndɛnˈtl ˈnʌmbər/	Number that is not a root of algebraic equations
欧拉公式	Euler’s formula	/ˈɔɪlər ˈfɔːrmjələ/	$e^{ix} = \cos x + i\sin x$
欧拉恒等式	Euler’s identity	/ˈɔɪlər aɪˈdɛntɪti/	$e^{i\pi} + 1 = 0$
自然对数	natural logarithm	/ˈnætʃrəl ˈlɔːɡərɪðəm/	Logarithm with base e
连分数	continued fraction	/kənˈtɪnjuːd ˈfrækʃən/	Special fractional representation
微分方程	differential equation	/ˌdɪfəˈrɛnʃəl ɪˈkweɪʒən/	Equation containing derivatives

Previous Elementary Functions at a Glance Power, exponential, logarithmic, trigonometric, and inverse trigonometric functions—key shapes, domains, and quick properties.

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