The square root of 2
The square root of 2 is the length of the diagonal of a unit square, the first number shown to be irrational. Its value is 1.41421356237309504880168872420969807856967187537694…, irrational (an algebraic number).
How to work it out
- 1
What it is: the length of the diagonal of a unit square, the first number shown to be irrational.
- 2
The first 100 digits, computed exactly with big-integer arithmetic:
- 3
1.41421356 2373095048 8016887242 0969807856 9671875376 9480731766 7973799073 2478462107 0388503875 3432764157 27
- 4
Its continued fraction is [1; 2; 2; 2; 2; 2; 2; 2; 2; 2, …]. Cutting it off gives the best fractions for a given size of denominator, such as 3/2, 7/5, 17/12.
LaTeX source of the formulas on this page
\[ \sqrt{2} = 1.414213562373\ldots \]
\[ \sqrt{2} \]How often each digit appears in the first 100 digits (computed)
| Digit | Count |
|---|---|
| 0 | 10 |
| 1 | 8 |
| 2 | 8 |
| 3 | 11 |
| 4 | 9 |
| 5 | 7 |
| 6 | 10 |
| 7 | 17 |
| 8 | 12 |
| 9 | 8 |
Best fractions from the continued fraction (computed)
| Fraction | Decimal | Error |
|---|---|---|
| 1/1 | 1.00000000000 | 4.14e-1 |
| 3/2 | 1.50000000000 | 8.58e-2 |
| 7/5 | 1.40000000000 | 1.42e-2 |
| 17/12 | 1.41666666667 | 2.45e-3 |
| 41/29 | 1.41379310345 | 4.20e-4 |
| 99/70 | 1.41428571429 | 7.22e-5 |
| 239/169 | 1.41420118343 | 1.24e-5 |
| 577/408 | 1.41421568627 | 2.12e-6 |
Questions
What is the value of The square root of 2?
1.41421356237309504880168872420969807856967187537694…
Is The square root of 2 irrational?
It is irrational (an algebraic number).
What are the first 10 digits of The square root of 2?
1.4142135623.
What is a good fraction for The square root of 2?
3/2 is simple, and 17/12 is accurate to about 2.5e-3.