What is the value of?

Euler's Number (e)

Euler's number, often denoted as e, is a mathematical constant that is the base of the natural logarithm. It's an incredibly important number that appears in many areas of mathematics and physics.

• Value: Approximately equal to 2.71828. It is an irrational number, meaning its decimal representation never ends or repeats, and it cannot be expressed as a simple fraction.

• Definition: e can be defined in several ways:

  • As the limit of (1 + 1/n)^n as n approaches infinity: lim (n→∞) (1 + 1/n)^n
  • As the sum of the infinite series: 1 + 1/1! + 1/2! + 1/3! + ... = Σ (1/n!) from n=0 to infinity. Here, "!" denotes the factorial operation.

• Importance:

  • Calculus: e is crucial in calculus because the derivative of the exponential function e^x is equal to itself. This makes it the basis for many natural growth and decay models. Related concepts include exponential functions and logarithms.
  • Natural Logarithm: The natural logarithm (ln) is the logarithm to the base e.
  • Compound Interest: e appears in the formula for continuous compound interest.
  • Probability and Statistics: e is used in probability distributions, such as the normal distribution, and in statistical calculations.
  • Complex Analysis: e is central to Euler's formula (e^(ix) = cos(x) + i sin(x)), connecting it to complex numbers and trigonometric functions.
  • Differential Equations: It is fundamental in solving many types of differential equations which are used to model physical phenomena.