We can approximate it using iterative improvement.
If we allow the approximation to be off by less than 1/50,000, we can find a golden ratio of 377/233. How?
Some algebra:
A/B | = | (A + B)/A |
= | 1 + B/A | |
= | 1 + 1/(A/B) |
Call the golden ratio A/B = φ. Then
φ | = | 1 + 1/φ |