The golden ratio and its conjugate #
This file defines the golden ratio φ := (1 + √5)/2
and its conjugate
ψ := (1 - √5)/2
, which are the two real roots of X² - X - 1
.
Along with various computational facts about them, we prove their irrationality, and we link them to the Fibonacci sequence by proving Binet's formula.
The conjugate of the golden ratio ψ := (1 - √5)/2
.
Instances For
The inverse of the golden ratio is the opposite of its conjugate.
The opposite of the golden ratio is the inverse of its conjugate.
Irrationality #
The conjugate of the golden ratio is irrational.
Links with Fibonacci sequence #
The recurrence relation satisfied by the Fibonacci sequence.
Instances For
The characteristic polynomial of fibRec
is X² - (X + 1)
.
As expected, the Fibonacci sequence is a solution of fibRec
.
The geometric sequence fun n ↦ φ^n
is a solution of fibRec
.
The geometric sequence fun n ↦ ψ^n
is a solution of fibRec
.
Binet's formula as a function equality.
Binet's formula as a dependent equality.