Euler's Number

Euler's number, denoted e, is the mathematical constant approximately 2.71828 and the base of the natural logarithm. It is the limiting value of compound growth as compounding becomes continuous.

Natural Logarithm: The logarithm with base e, inverse to the exponential function eˣ. It is the logarithm whose base is this constant.
Compound Interest: Interest calculated on both the original principal and previously accumulated interest. Compounding more frequently approaches the growth factor e over one time unit.
John Napier: A Scottish mathematician who published influential logarithm tables in 1614. His logarithmic work provided historical groundwork for identifying the constant.
Exponential Growth: Growth in which a quantity's rate of increase is proportional to its current size. The natural model expresses proportional growth with powers of e.
Common Logarithm: A logarithm with base 10, widely used in decimal measurement and scientific notation. Unlike the natural logarithm, it uses a convenient decimal base rather than e.
Exponential Function: A function in which the variable appears in an exponent; the natural exponential is eˣ. The function eˣ is uniquely unchanged by differentiation.
Taylor Series: A power series that represents a function through its derivatives at a point. Expanding eˣ gives the series whose value at x = 1 defines e.
Jacob Bernoulli: A Swiss mathematician whose work helped establish early probability theory and the calculus of variations. He encountered the compound-interest limit that yields e.
Radioactive Decay: The statistical process by which unstable atomic nuclei transform, reducing their number exponentially over time. Its continuous-time decay law is commonly written using e.
Binary Logarithm: A logarithm with base 2, commonly used to measure information and algorithmic growth. Its base suits powers of two, while e is natural for continuous change.

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