> For the complete documentation index, see [llms.txt](https://johnblommers.gitbook.io/notes-on-prime-numbers/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://johnblommers.gitbook.io/notes-on-prime-numbers/the_golden_ratio.md).

# The Golden Ratio

**References**:

* [Golden Ratio](https://en.wikipedia.org/wiki/Golden_ratio) at the WikiPedia.

Two quantities are in their golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities:

```
(a + b)   a    (1 + sqrt(5))
------- = - =  -------------
   a      b         2
```

The Golden Ratio a.k.a. **Phi** is about **1.618**. Phi is derived from the Fibonacci sequence:

```
1, 1, 2, 3, 5, 8, 13, 21, 34 ...
```

Each number in it is the sum of the preceding two numbers:

```
1+1=2, 2+1=3, 2+3=5, 3+5=8, 5+8=13, 8+13=21, 13+21=34 ...
```

The quotients of successive adjacent terms:

```
2  3  5  8      34
-, -, -, -, ... --
1  2  3  5      21
```

converge on **Phi** as you get further out.