Euler's number e
Euler's number e is the base of natural logarithms; the limit of (1 + 1/n)ⁿ and the sum 1 + 1 + 1/2! + 1/6 + …. Its value is 2.71828182845904523536028747135266249775724709369995…, irrational and transcendental.
How to work it out
- 1
What it is: the base of natural logarithms; the limit of (1 + 1/n)ⁿ and the sum 1 + 1 + 1/2! + 1/6 + ….
- 2
The first 100 digits, computed exactly with big-integer arithmetic:
- 3
2.71828182 8459045235 3602874713 5266249775 7247093699 9595749669 6762772407 6630353547 5945713821 7852516642 74
- 4
Its continued fraction is [2; 1; 2; 1; 1; 4; 1; 1; 6; 1, …]. Cutting it off gives the best fractions for a given size of denominator, such as 3/1, 8/3, 11/4.
How often each digit appears in the first 100 digits (computed)
| Digit | Count |
|---|---|
| 0 | 5 |
| 1 | 6 |
| 2 | 13 |
| 3 | 8 |
| 4 | 10 |
| 5 | 13 |
| 6 | 12 |
| 7 | 16 |
| 8 | 7 |
| 9 | 10 |
Best fractions from the continued fraction (computed)
| Fraction | Decimal | Error |
|---|---|---|
| 2/1 | 2.00000000000 | 7.18e-1 |
| 3/1 | 3.00000000000 | 2.82e-1 |
| 8/3 | 2.66666666667 | 5.16e-2 |
| 11/4 | 2.75000000000 | 3.17e-2 |
| 19/7 | 2.71428571429 | 4.00e-3 |
| 87/32 | 2.71875000000 | 4.68e-4 |
| 106/39 | 2.71794871795 | 3.33e-4 |
| 193/71 | 2.71830985915 | 2.80e-5 |
Questions
What is the value of Euler's number e?
2.71828182845904523536028747135266249775724709369995…
Is Euler's number e irrational?
It is irrational and transcendental.
What are the first 10 digits of Euler's number e?
2.7182818284.
What is a good fraction for Euler's number e?
3/1 is simple, and 11/4 is accurate to about 3.2e-2.