Euler’s number, denoted as $e$ is a mathematical constant approximately equal to $2.71828$. It is named after the Swiss mathematician Leonhard Euler who made significant contributions to many areas of mathematics, including calculus, number theory, and graph theory.

Euler’s number is an irrational and transcendental number, which means it cannot be expressed as a finite decimal or a fraction. Its decimal representation goes on indefinitely without repeating.

Euler’s number has various important applications in mathematics, science, and engineering. It arises naturally in many areas, including calculus, differential equations, compound interest calculations, exponential growth and decay, probability theory, and complex analysis.

The value of Euler’s number is commonly used as the base of the natural logarithm, denoted as $\ln$. It has a range of mathematical properties and relationships making it a fundamental constant in many mathematical formulas and equations.

The value of $e$ can be approximated using various methods, such as infinite series expansions, continued fractions, or as the limit of certain mathematical expressions. One common series representation is the infinite sum

$$ e = \sum_{n=0}^{\infty} \frac{1}{n!} = 1+ \frac{1}{1!} + \frac{1}{2!} + . . . $$

where $n!$ represents the factorial of $n$ (the product of all positive integers up to $n$).