A fundamental mathematical constant (approximately 2.71828) that serves as the base of natural logarithms and appears naturally in descriptions of growth, decay, and continuous compound interest.

Euler's Number (e)

Euler's number, denoted as 'e', stands as one of the most important mathematical constants in mathematics, alongside pi and the imaginary unit. Discovered by Swiss mathematician Leonhard Euler, this irrational and transcendental number emerges naturally in countless mathematical and scientific contexts.

Definition and Properties

The number e can be defined in several equivalent ways:

As the limit of (1 + 1/n)^n as n approaches infinity
As the sum of the infinite series: 1 + 1/1! + 1/2! + 1/3! + ...
As the unique number whose natural logarithm equals 1

Its approximate value is:

e ≈ 2.71828 18284 59045 23536...

Mathematical Significance

Calculus and Analysis

The exponential function e^x is its own derivative, making it fundamental in differential equations
Forms the basis for natural logarithms
Appears in the complex analysis expression e^(iπ) + 1 = 0, known as Euler's Identity

Applications

Growth and Decay
Scientific Applications

Historical Development

The number e was first studied in the context of compound interest calculations by Jacob Bernoulli in 1683. He investigated the limit of (1 + 1/n)^n when calculating interest compounded n times per year. Later, Leonhard Euler extensively studied its properties and established its significance in calculus.

Properties in Nature

The number e appears naturally in various phenomena:

Computational Methods

Modern calculations of e employ various techniques:

Taylor series expansions
Numerical methods for high-precision computation
Computer algebra systems for symbolic manipulation

Cultural Impact

While less well-known to the general public than π, Euler's number has inspired:

Mathematical art and poetry
Popular science writings
Educational demonstrations and puzzles

The ubiquity of e in natural phenomena has led some mathematicians to argue that it is even more fundamental than π in describing the natural world.

Euler's Number