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Math 2.0

Number theory Object

Golden ratio

Description

The golden ratio is the real number $\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887.$ It is the positive root of the polynomial $x^2 - x - 1$ , so $\varphi^2 = \varphi + 1$ . The other root is $\psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = -\frac{1}{\varphi}$ .

Remarks

Two positive numbers $a > b$ are in the golden ratio when $\frac{a + b}{a} = \frac{a}{b}$ . Then $\frac{a}{b} = \varphi$ . The number $\varphi$ is irrational, because $\sqrt{5}$ is irrational. The ratios of consecutive Fibonacci numbers tend to $\varphi$ . Binet's formula uses $\varphi$ and $\psi$ .

Where this leads

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Further reading

Web https://en.wikipedia.org/wiki/Golden_ratio, Calculation

About this page

Tag	`0SC8T7548D`
URI	`m2://0SC8T7548D`
Kind	object
Writing	W1 stated
MSC	11B39
Same as	Wikidata Q41690 Wikipedia Golden ratio

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section Invariants: a table of the main invariants (0 of 1)
edge instance_of out (0 of 1)

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