What is Euler's Number e? Natural Growth Explained

Euler's number e is approximately 2.71828182845.... It is the base of the natural logarithm and appears wherever continuous growth or decay occurs.

The unique property

The function e^x is the only function that is its own derivative. Its slope at every point equals its value. This makes e the natural base for calculus.

Where e comes from

Imagine a bank account earning 100% interest per year. Compounded more and more frequently, the result converges to e. Formally: e = lim(n→∞) (1 + 1/n)^n

Where e appears

• Compound interest: continuous growth at rate r gives e^(rt)
• Radioactive decay: half-life formulas use e
• Normal distribution: the bell curve is e^(-x²/2) / √(2π)
• Euler's identity: e^(iπ) = -1

Key facts

Fact	Value
Decimal value	2.71828182845...
Discovered	Jacob Bernoulli, 1683
Named by	Leonhard Euler, 1731
ln(e)	1
e is irrational	Proved 1737

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