The square root of 5
The square root of 5 is the diagonal of a 1 by 2 rectangle, and the number inside the golden ratio. Its value is 2.23606797749978969640917366873127623544061835961152…, irrational (an algebraic number).
How to work it out
- 1
What it is: the diagonal of a 1 by 2 rectangle, and the number inside the golden ratio.
- 2
The first 100 digits, computed exactly with big-integer arithmetic:
- 3
2.23606797 7499789696 4091736687 3127623544 0618359611 5257242708 9724541052 0925637804 8994144144 0837878227 49
- 4
Its continued fraction is [2; 4; 4; 4; 4; 4; 4; 4; 4; 4, …]. Cutting it off gives the best fractions for a given size of denominator, such as 9/4, 38/17, 161/72.
LaTeX source of the formulas on this page
\[ \sqrt{5} = 2.236067977499\ldots \]
\[ \sqrt{5} \]How often each digit appears in the first 100 digits (computed)
| Digit | Count |
|---|---|
| 0 | 8 |
| 1 | 8 |
| 2 | 12 |
| 3 | 7 |
| 4 | 14 |
| 5 | 7 |
| 6 | 10 |
| 7 | 14 |
| 8 | 9 |
| 9 | 11 |
Best fractions from the continued fraction (computed)
| Fraction | Decimal | Error |
|---|---|---|
| 2/1 | 2.00000000000 | 2.36e-1 |
| 9/4 | 2.25000000000 | 1.39e-2 |
| 38/17 | 2.23529411765 | 7.74e-4 |
| 161/72 | 2.23611111111 | 4.31e-5 |
| 682/305 | 2.23606557377 | 2.40e-6 |
| 2889/1292 | 2.23606811146 | 1.34e-7 |
| 12238/5473 | 2.23606797003 | 7.47e-9 |
| 51841/23184 | 2.23606797792 | 4.16e-10 |
Questions
What is the value of The square root of 5?
2.23606797749978969640917366873127623544061835961152…
Is The square root of 5 irrational?
It is irrational (an algebraic number).
What are the first 10 digits of The square root of 5?
2.2360679774.
What is a good fraction for The square root of 5?
9/4 is simple, and 161/72 is accurate to about 4.3e-5.