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.
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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.
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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.
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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.