The golden ratio φ
The golden ratio φ is (1 + √5)/2, the ratio of successive Fibonacci numbers in the limit. Its value is 1.61803398874989484820458683436563811772030917980576…, irrational (an algebraic number).
How to work it out
- 1
What it is: (1 + √5)/2, the ratio of successive Fibonacci numbers in the limit.
- 2
The first 100 digits, computed exactly with big-integer arithmetic:
- 3
1.61803398 8749894848 2045868343 6563811772 0309179805 7628621354 4862270526 0462818902 4497072072 0418939113 74
- 4
Its continued fraction is [1; 1; 1; 1; 1; 1; 1; 1; 1; 1, …]. Cutting it off gives the best fractions for a given size of denominator, such as 2/1, 3/2, 5/3.
LaTeX source of the formulas on this page
\[ \varphi = 1.618033988749\ldots \]
\[ \varphi = \frac{1 + \sqrt{5}}{2} \]How often each digit appears in the first 100 digits (computed)
| Digit | Count |
|---|---|
| 0 | 11 |
| 1 | 10 |
| 2 | 11 |
| 3 | 9 |
| 4 | 11 |
| 5 | 5 |
| 6 | 9 |
| 7 | 10 |
| 8 | 15 |
| 9 | 9 |
Best fractions from the continued fraction (computed)
| Fraction | Decimal | Error |
|---|---|---|
| 1/1 | 1.00000000000 | 6.18e-1 |
| 2/1 | 2.00000000000 | 3.82e-1 |
| 3/2 | 1.50000000000 | 1.18e-1 |
| 5/3 | 1.66666666667 | 4.86e-2 |
| 8/5 | 1.60000000000 | 1.80e-2 |
| 13/8 | 1.62500000000 | 6.97e-3 |
| 21/13 | 1.61538461538 | 2.65e-3 |
| 34/21 | 1.61904761905 | 1.01e-3 |
Questions
What is the value of The golden ratio φ?
1.61803398874989484820458683436563811772030917980576…
Is The golden ratio φ irrational?
It is irrational (an algebraic number).
What are the first 10 digits of The golden ratio φ?
1.6180339887.
What is a good fraction for The golden ratio φ?
2/1 is simple, and 5/3 is accurate to about 4.9e-2.