Mathematics July 22, 2026 · 11 min read

Euler's Number e: The Mathematical Constant That Shapes Our World

Explore Euler's number e ≈ 2.71828. Learn its definition, properties, and applications in calculus, finance, and natural growth.

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The number e ≈ 2.71828 is one of the most important constants in mathematics. Like π, it is irrational and transcendental. It is the base of the natural logarithm and appears in formulas across calculus, physics, finance, and biology.

Definition

e = lim(n—∞) (1 + 1/n)^n ≈ 2.71828

Taylor Series

e^x = 1 + x + x²/2! + x³/3! + ...

Euler\'s Identity

e^(iπ) + 1 = 0

This equation connects five fundamental mathematical constants in one beautiful formula.

Applications

  • Compound interest: Continuous growth: A = Pe^(rt)
  • Calculus: d/dx[e^x] = e^x (its own derivative!)
  • Probability: Normal distribution, Poisson distribution
  • Physics: Radioactive decay, cooling, charging capacitors