Euler’s Number (e): The Foundation of Natural Logarithms and Compound Interest

Among the many fascinating mathematical constants, Euler’s number (e ≈ 2.71828) stands out as a cornerstone of modern mathematics. Known as the base of the natural logarithm, e is deeply connected to growth, decay, probability, and finance. From describing how populations grow to explaining compound interest in banking, Euler’s number plays a vital role in both theoretical mathematics and real-life applications.

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The Origin and Discovery of e

The history of e can be traced back to the study of compound interest in the 17th century. Mathematicians were intrigued by the question: What happens when interest is compounded more and more frequently?

• Jacob Bernoulli (1683): While studying compound interest, Bernoulli discovered the limit:

$$\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n=e$$

This formula showed how e naturally arises when money grows continuously.

• Leonhard Euler (1707–1783): The Swiss mathematician Leonhard Euler gave the constant its symbol e and expanded its importance beyond finance, showing that it underlies exponential functions, logarithms, and calculus.

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The Mathematical Definition of e

Euler’s number is defined as the limit:

$$e=\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n$$

Alternatively, it can be expressed as an infinite series:

$$e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots$$

This definition connects e to calculus, probability, and infinite processes.

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e and Natural Logarithms

One of the most profound roles of e is as the base of the natural logarithm (ln).

• The natural logarithm, ln(x), is the inverse of the exponential function e^x.
• The derivative of e^x is unique because it equals itself:

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

This property makes e central to calculus, enabling the modeling of continuous growth and decay in science, biology, and physics.

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e and Compound Interest

Perhaps the most relatable application of e is in finance. When money grows under continuous compounding, the formula is:

$$A=P{e}^{rt}$$

Where:

• A = final amount
• P = principal (initial investment)
• r = interest rate
• t = time

This formula explains why continuous compounding grows faster than traditional yearly or monthly compounding, making e a natural part of modern banking and economics.

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e in Science and Nature

Euler’s number is not limited to mathematics and finance. It appears in many natural processes:

• Population Growth: The exponential growth of populations often follows equations involving e.
• Radioactive Decay: The half-life of elements is modeled with exponential decay based on e.
• Probability Theory: The constant arises in problems such as the “hat-check problem,” where the probability of no match approaches 1/e.
• Physics and Engineering: Natural processes such as cooling, diffusion, and wave behavior are all described by exponential functions involving e.

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Conclusion

Euler’s number (e) is more than just a mathematical curiosity—it is a universal constant that connects natural logarithms, compound interest, calculus, and real-world processes. From finance to physics, e is the mathematical key that unlocks the mysteries of growth and decay.

Its infinite presence across disciplines proves that mathematics is not only abstract but also deeply tied to the way the world works. Like π and ϕ, e is a symbol of the profound harmony between numbers, nature, and human understanding.