Relation To The Golden Ratio and Fibonacci Numbers
This golden ratio φ is the arithmetic mean of 1 and the square root of 5. The algebraic relationship between the square root of 5, the golden ratio and the conjugate of the golden ratio are expressed in the following formulae:
(See section below for their geometrical interpretation as decompositions of a root-5 rectangle.)
The square root of 5 then naturally figures in the closed form expression for the Fibonacci numbers, a formula which is usually written in terms of the golden ratio:
The quotient of √5 and φ (or the product of √5 and Φ), and its reciprocal, provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the Lucas numbers:
The series of convergents to these values feature the series of Fibonacci numbers and the series of Lucas numbers as numerators and denominators, and viceversa, respectively:
Read more about this topic: Square Root Of 5
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