Concepts · 3 min read
What Is Euler's Number e? Compound Growth and Series
From compound interest
Put 1 unit in an account paying 100% per year. Compounding once gives 2. Compounding more often gives more, but not without limit:
| Compoundings n | (1 + 1/n)ⁿ |
|---|---|
| 1 | 2 |
| 2 | 2.25 |
| 4 | 2.44140625 |
| 12 | 2.61303529 |
| 365 | 2.71456748 |
| 1,000 | 2.71692393 |
| 1,000,000 | 2.71828047 |
The values creep up to e. See compound interest for the practical side.
The factorial series
e = 1 + 1/1! + 1/2! + 1/3! + …, which converges extremely fast:
| Terms | Sum |
|---|---|
| 3 | 2.5 |
| 5 | 2.7083333333 |
| 8 | 2.7182539683 |
| 10 | 2.7182815256 |
| 15 | 2.7182818285 |
Why e is special
- The function eˣ is its own derivative.
- The natural logarithm ln is the logarithm with base e.
- Euler’s identity connects it to π and i: e^(iπ) + 1 = 0.
ln(20)
- Natural log ln(20) = 2.995732
- Log base 10 = 1.30103
Step-by-step working 3 steps
- 1Meaning
log_e(20) = y means e^y = 20
A logarithm answers: “what power of the base gives x?”
- 2Change of base
y = ln(20) / ln(e) = 2.99573227 / 1 = 2.99573227
- 3Check
e^2.99573227 ≈ 20
Raising the base to the answer returns x (to rounding).
Try it
Evaluate eˣ in the scientific calculator, or study growth with the exponent calculator and logarithms explained.
Frequently asked questions
Is e rational?
No; e is irrational and transcendental.
Who discovered e?
Jacob Bernoulli studied the compound-interest limit; Euler named and developed it.
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