The mathematical constant e (≈2.71828) is a cornerstone of
modern mathematics, especially in calculus, exponential growth, and compound
interest. Its history spans centuries of discovery and refinement. Here’s an
overview:
What is e?
Ø e is
an irrational number (cannot be expressed as a simple fraction) and a transcendental
number (not the root of any polynomial with rational coefficients).
Ø It
is the base of the natural logarithm and arises naturally in problems
involving exponential growth, decay, and calculus.
Historical Milestones
1. Early Beginnings: Compound Interest
Ø Jacob
Bernoulli (1683):
ü
The origins of e trace back to problems in compound
interest.
ü
Bernoulli studied the growth of money under
continuous compounding. He calculated: $$\lim_{n \to \infty} \left(1 +
\frac{1}{n}\right)^n$$
- This
limit equals e, showing how it arises naturally in financial
mathematics.
2. Development of Logarithms
Ø John
Napier (1614):
ü
Napier developed logarithms to simplify
calculations, laying the groundwork for e, though he did not explicitly define e.
ü
His logarithmic tables were implicitly based on
a number close to e.
3. Euler’s Work (1727–1737)
Ø Leonhard
Euler was the first to formally define e as the base of natural logarithms.
Ø Euler
explored its properties in detail and introduced the exponential function \(e^x\).
Ø He
also discovered its connection to the famous Euler’s identity: $$e^{i\pi}
+ 1 = 0$$
4. Exponential Functions in Calculus
Ø Isaac
Newton and Gottfried Wilhelm Leibniz (Late 1600s):
ü
Both mathematicians used exponential growth and
logarithmic functions in their development of calculus.
ü
e naturally emerged as the unique number where
the derivative of \(e^x\) equals \(e^x\).
5. Growth and Decay in Real Life
Ø By
the 18th century, mathematicians realized that e models:
ü
Population growth.
ü
Radioactive decay.
ü
Heat transfer.
ü
Spread of diseases.
ü Trigonometric calculation using infinite series.
Mathematical Definitions of e
e can be defined in several equivalent ways:
- Limit
Definition:
$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$
- Infinite
Series:
$$e = \sum_{n=0}^\infty \frac{1}{n!} = 1 + \frac{1}{1!} +
\frac{1}{2!} + \frac{1}{3!} + \dots$$
- Differential
Equation: e is the unique number for which:
$$\frac{d}{dx}e^x = e^x$$
Applications of e
- Compound
Interest:
ü
Continuous compounding of interest uses e: \(A =
P e^{rt}\) Where A is the amount, P is the principal, r is the rate, and t is
time.
- Calculus:
ü
e appears in derivatives and integrals of
exponential and logarithmic functions.
- Physics
and Engineering:
ü
Used in natural growth, decay, waveforms, and
electrical circuits.
- Statistics:
ü
The normal distribution curve involves e in its
probability density function.
- Computer
Science:
ü
Used in algorithms, cryptography, and complexity
analysis.
Modern Computation of e
Ø With
the advent of computers, millions of digits of e have been calculated.
Ø Like
π, e's digits continue infinitely without repeating.
The value of e≈2.71828 is derived from mathematical
definitions and computations. Here's how we arrive at this number step by step:
1. Using the Limit Definition of e
The most intuitive way to compute e is through the limit
definition:
$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$
To approximate e, compute the expression for large values of
n:
|
|
As n→∞n , the value converges to e≈2.71828.
2. Using the Infinite Series Definition
Another method is summing the infinite series:
$$e = \sum_{n=0}^\infty \frac{1}{n!} = 1 + \frac{1}{1!} +
\frac{1}{2!} + \frac{1}{3!} + \dots$$
To approximate e, sum the first few terms:
|
|
|
|
n |
Terms \(\frac{1}{n!}\) | \(e=e^1\) |
|
0 |
\(\frac{1}{0!}\) |
1.0000000000 |
|
1 |
\(\frac{1}{0!}+\frac{1}{1!}\) |
2.0000000000 |
|
2 |
\(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}\) |
2.5000000000 |
|
3 |
\(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}\) |
2.6666666666 |
|
4 |
\(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}\) |
2.7083333333 |
|
5 |
\(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}\) |
2.7166666666 |
|
6 |
\(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}\) |
2.7180555555 |
|
7 |
\(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}+\frac{1}{7!}\) |
2.7182539682 |
|
8 |
\(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}+\frac{1}{7!}+\frac{1}{8!}\) |
2.71827877698 |
Summing more terms gives better approximations of e.
Find out trigonometric equation from infinite series
We converted this formula to \(e^x\) terms formula
\(e^x=\frac{1}{0!}+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\frac{x^8}{8!}+ \dots\)
Now we know
\(sin(x)=\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+ \dots\)
\(cos(x)=\frac{1}{0!}-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}+ \dots\)
\( sin(x)+cos(x)=\frac{1}{0!}+\frac{x}{1!}-\frac{x^2}{2!}-\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}-\frac{x^6}{6!}-\frac{x^7}{7!}+\frac{x^8}{8!}+ \dots\)
Both are identical; however, certain areas are positive
while others are negative.
We have now adopted a different approach.
we converted this formula to \(e^{ix}\) terms formula
\(e^{ix}=\frac{1}{0!}+\frac{(ix)}{1!}+\frac{(ix)^2}{2!}+\frac{(ix)^3}{3!}+\frac{(ix)^4}{4!}+\frac{(ix)^5}{5!}+\frac{(ix)^6}{6!}+\frac{(ix)^7}{7!}+\frac{(ix)^8}{8!}+ \dots\)
where \(i=\sqrt{-1}\)
\(i^2=-1\)
\(i^3=i^2*i\)
\(i^4=i^2*i^2\)
\(e^{ix}=\frac{1}{0!}+\frac{(ix)}{1!}+\frac{(ix)^2}{2!}+\frac{(ix)^3}{3!}+\frac{(x)^4}{4!}+\frac{(ix)^5}{5!}+\frac{(x)^6}{6!}+\frac{(ix)^7}{7!}+\frac{(ix)^8}{8!}+ \dots\)
$$e^{ix}=\frac{1}{0!}+\frac{(ix)}{1!}+\frac{(ix)^2}{2!}+\frac{(ix)^3}{3!}+\frac{(x)^4}{4!}+\frac{(ix)^5}{5!}+\frac{(x)^6}{6!}+\frac{(ix)^7}{7!}+\frac{(ix)^8}{8!}+ \dots$$ $$= (\frac{1}{0!}-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}+ \dots)+(\frac{ix}{1!}-\frac{ix^3}{3!}+\frac{ix^5}{5!}-\frac{ix^7}{7!}+ \dots)$$ $$= (\frac{1}{0!}-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}+ \dots)+i(\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+ \dots)$$ $$=cos(x)+isin(x)$$
When x = π
Then
\(e^{i\pi} =cos(\pi)+i.sin(\pi)=-1+0\)
\(\bbox[yellow]{e^{i\pi} +1=0}\)
3. Using Continued Fractions
The value of e can also be approximated using its continued
fraction representation:
$$e = 2 + \frac{1}{1 + \frac{1}{2 + \frac{1}{1 + \frac{1}{1
+ \frac{1}{4 + \dots}}}}}$$
Computing this continued fraction iteratively also converges
to e.
4. Using Numerical Computation (Programming)
Programming languages like Java can compute e to high
precision. Here's how:
java:
|
public class
ComputeE { public static void main(String[] args) { double r = 0.1; // Interest rate double ePowerR = Math.exp(r); //
Calculate e^r double principal = 1000; double amount = principal * ePowerR; System.out.printf("Amount with
continuous compounding: $%.2f%n", amount); } } |
Custom Exponential Java class:
|
public class
CustomMathExp { public static double customExp(double r,
int terms) { double result = 1.0; // The sum
starts with the first term: 1 double term = 1.0; // Each term in the series for (int i = 1; i <= terms; i++) { term *= r / i; // Calculate the next term: (r^i / i!) result += term; // Add the term
to the result } return result; } public static void main(String[] args) { double r = 1.0; // Example input int terms = 20; // Number of terms
for approximation // Using custom implementation double customResult = customExp(r,
terms); // Using built-in Math.exp for
comparison double mathResult = Math.exp(r); System.out.printf("Custom
Math.exp(%f): %.15f%n", r, customResult); System.out.printf("Built-in
Math.exp(%f): %.15f%n", r, mathResult); System.out.printf("Difference:
%.15f%n", Math.abs(customResult - mathResult)); } } |
Output Example
For r=1.0r = 1.0r=1.0 and terms=20terms = 20terms=20:
|
Custom Math.exp(1.000000):
2.718281828459045 Built-in
Math.exp(1.000000): 2.718281828459045 Difference: 0.000000000000000 |
5. Using a Calculator
Many scientific calculators have a dedicated e key.
Alternatively, you can compute \(e^1\) or evaluate the series/limit definitions
to approximate the value.
The story of compound interest involves calculating
how investments grow when interest is calculated and added to the principal at
regular intervals. The frequency of compounding—whether it's secondly,
minutely, hourly, daily, weekly, monthly, or yearly—determines how much
interest accrues over a given period. Here's an explanation of how compound
interest works and how to calculate the maximum possible interest in
different scenarios.
Compound Interest Formula
The standard formula for compound interest is:
$$A = P \left(1 + \frac{r}{n}\right)^{n \cdot t}$$
Where:
Ø A =
Final amount (principal + interest)
Ø P =
Principal amount (initial investment)
Ø r =
Annual interest rate (in decimal form)
Ø n =
Number of times interest is compounded per year
Ø t =
Time in years
Scenarios for Different Compounding Frequencies
As n (the frequency of compounding) increases, the final
amount A approaches a maximum value. This happens because as compounding
becomes more frequent, the growth mimics continuous compounding,
governed by the mathematical constant e.
1. Yearly Compounding (n=1)
Ø Interest
is added once per year.
$$A = P \left(1 + r\right)^t$$
2. Monthly Compounding (n=12)
Ø Interest
is added 12 times a year.
$$A = P \left(1 + \frac{r}{12}\right)^{12 \cdot t}$$
3. Weekly Compounding (n=52)
Ø Interest
is added 52 times a year.
$$A = P \left(1 + \frac{r}{52}\right)^{52 \cdot t}$$
4. Daily Compounding (n=365)
Ø Interest
is added daily (assuming no leap years).
$$A = P \left(1 + \frac{r}{365}\right)^{365 \cdot t}$$
5. Hourly Compounding (n=8760)
Ø Interest
is added every hour (24 hours × 365 days).
$$A = P \left(1 + \frac{r}{8760}\right)^{8760 \cdot t}$$
6. Minutely Compounding (n=525600)
Ø Interest
is added every minute (60 minutes × 24 hours × 365 days).
$$A = P \left(1 + \frac{r}{525600}\right)^{525600 \cdot t}$$
7. Secondly Compounding (n=31536000)
Ø Interest
is added every second (60 seconds × 60 minutes × 24 hours × 365 days).
$$A = P \left(1 + \frac{r}{31536000}\right)^{31536000 \cdot
t}$$
Continuous Compounding (Maximum Interest)
As \(n \to \infty\), the compound interest formula becomes:
$$A = P \cdot e^{r \cdot t}$$
Here:
Ø e≈2.71828
(the base of natural logarithms).
Ø This
represents the theoretical maximum interest for a given annual rate r
and time t.
Comparison of Different Frequencies
Assume:
Ø Principal
(P) = $1,000
Ø Annual
interest rate ® = 10% (0.1)
Ø Time
(t) = 1 year
|
Compounding Frequency |
Amount ($) |
Extra Interest Over Yearly ($) |
|
Yearly (n=1) |
1,100.00 |
0 |
|
Monthly (n=12) |
1,104.71 |
4.71 |
|
Weekly (n=52) |
1,105.06 |
5.06 |
|
Daily (n=365) |
1,105.16 |
5.16 |
|
Hourly (n=8760) |
1,105.17 |
5.17 |
|
Minutely (n=525600) |
1,105.17 |
5.17 |
|
Secondly (n=31536000) |
1,105.17 |
5.17 |
|
Continuous (n→∞n) |
1,105.17 |
5.17 |
Insights
- Diminishing
Returns:
Ø
As compounding frequency increases, the added
interest becomes negligible after daily compounding.
Ø
Continuous compounding represents the
theoretical maximum.
- Practical
Limits:
Ø
Financial institutions typically offer daily or
monthly compounding. Second-by-second or continuous compounding is rarely
implemented in practice.
- Real-Life
Applications:
Ø
Continuous compounding is used in advanced
financial modeling, physics (exponential decay/growth), and biology.
Conclusion
The story of compound interest shows how increasing
compounding frequency results in more interest, but the benefits diminish as
the intervals shorten. Continuous compounding, driven by the natural constant e,
provides the theoretical upper limit for growth, reflecting nature’s most
efficient way to grow exponentially.
Key Takeaway
The constant e represents the natural growth rate and is
deeply tied to real-world phenomena, from finance to biology. Its discovery and
exploration have shaped calculus and mathematics as we know it today, making it
one of the most important constants in mathematics.
Historical Note
Ø The
concept of e was introduced in the context of compound interest and exponential
growth.
Ø Jacob
Bernoulli discovered the limit definition while studying compound interest in
1683.
Ø Leonhard
Euler, who formalized e, showed its connection to logarithms, calculus, and
series.
Thus, e=2.71828 represents one of the most fundamental
constants in mathematics and nature.
 |
| The value of e |
For Math information, visit:
Ø
Calculating
the area of a triangle
Ø
The
method for calculating the area of a rectangle in Java
Ø
The value of π = 3.14159 in
java [Click
Here]
For Design information, visit:
Ø
Mastering
Design Patterns Practical Implementation Tips
Ø
How
to draw sequence diagram and other diagrams using plantuml
Ø
Time
Analysis for Congested Routes Using Shortest Path Algorithms
Ø Passive
Infrared (PIR) sensors understand and implementation by java
For Custom information, visit:
Ø
Custom
ArrayList By Java Source code
Ø
Custom
SinglyLinkList By java code
Ø
Custom
Doubly LinkList By java
Ø
Custom
Stack using an Array in Java code
Ø
Custom
Combination and permutation program
Ø
Custom
middle Element of array
Ø
Find
Middle & Reverse a Custom Linked List Using Iteration
Ø
Detect
& Handle Infinite Loops and Cycles in Linked Lists
Ø
Custom
Palindrome of a link list
Ø
Creating
a custom HashMap in Java
Ø
Custom
Combination and permutation program
For Security information, visit:
Ø
Asymmetric
Encryption: Public Key for Encryption, Private for Decryption
Ø
Symmetric:
Encryption and decryption by same key
Ø
Asynchronous
Encryption and decryption without file only key pass
Ø
public
key encryption and private key decryption with keystore file
Ø
OWASP
(Open Web Application Security Project)
Ø
To
securely obtain employee information utilizing TLS 1.3 or TLS 1.2
0 Comments