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MAThe number e is a mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted γ. Alternatively, e can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.
The number e is of great importance in mathematics, alongside 0, 1, π, and i. All five appear in one formulation of Euler's identity e^(iπ) + 1 = 0 and play important and recurring roles across mathematics. Like the constant π, e is irrational, meaning that it cannot be represented as a ratio of integers. Moreover, it is transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficients.
The Taylor series for the exponential function can be deduced from the facts that the exponential function is its own derivative and that it equals 1 when evaluated at 0: eˣ = ∑ₙ₌᪲₀ xⁿ/n! . Setting x=1 recovers the definition of e as the sum of an infinite series.
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