mathlib3 documentation

data.real.golden_ratio

The golden ratio and its conjugate #

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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.

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@[reducible]
noncomputable def golden_ratio  :
ℝ

The golden ratio φ := (1 + √5)/2.

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@[reducible]
noncomputable def golden_conj  :
ℝ

The conjugate of the golden ratio ψ := (1 - √5)/2.

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theorem inv_gold  :
golden_ratio⁻¹ = -golden_conj

The inverse of the golden ratio is the opposite of its conjugate.

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theorem inv_gold_conj  :
golden_conj⁻¹ = -golden_ratio

The opposite of the golden ratio is the inverse of its conjugate.

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@[simp]
theorem gold_mul_gold_conj  :
golden_ratio * golden_conj = -1
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@[simp]
theorem gold_conj_mul_gold  :
golden_conj * golden_ratio = -1
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@[simp]
theorem gold_add_gold_conj  :
golden_ratio + golden_conj = 1
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theorem one_sub_gold_conj  :
1 - golden_ratio = golden_conj
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theorem one_sub_gold  :
1 - golden_conj = golden_ratio
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@[simp]
theorem gold_sub_gold_conj  :
golden_ratio - golden_conj = √ 5
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@[simp]
theorem gold_sq  :
golden_ratio ^ 2 = golden_ratio + 1
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@[simp]
theorem gold_conj_sq  :
golden_conj ^ 2 = golden_conj + 1
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theorem gold_pos  :
0 < golden_ratio
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theorem gold_ne_zero  :
golden_ratio ≠ 0
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theorem one_lt_gold  :
1 < golden_ratio
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theorem gold_conj_neg  :
golden_conj < 0
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theorem gold_conj_ne_zero  :
golden_conj ≠ 0
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theorem neg_one_lt_gold_conj  :
-1 < golden_conj

Irrationality #

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theorem gold_irrational  :
irrational golden_ratio

The golden ratio is irrational.

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theorem gold_conj_irrational  :
irrational golden_conj

The conjugate of the golden ratio is irrational.

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def fib_rec {α : Type u_1} [comm_semiring α] :
linear_recurrence α

The recurrence relation satisfied by the Fibonacci sequence.

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theorem fib_rec_char_poly_eq {β : Type u_1} [comm_ring β] :
fib_rec.char_poly = polynomial.X ^ 2 - (polynomial.X + 1)

The characteristic polynomial of fib_rec is X² - (X + 1).

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theorem fib_is_sol_fib_rec {α : Type u_1} [comm_semiring α] :
fib_rec.is_solution (λ (x : ℕ), ↑(nat.fib x))

As expected, the Fibonacci sequence is a solution of fib_rec.

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theorem geom_gold_is_sol_fib_rec  :
fib_rec.is_solution (has_pow.pow golden_ratio)

The geometric sequence λ n, φ^n is a solution of fib_rec.

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theorem geom_gold_conj_is_sol_fib_rec  :
fib_rec.is_solution (has_pow.pow golden_conj)

The geometric sequence λ n, ψ^n is a solution of fib_rec.

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theorem real.coe_fib_eq'  :
(λ (n : ℕ), ↑(nat.fib n)) = λ (n : ℕ), (golden_ratio ^ n - golden_conj ^ n) / √ 5

Binet's formula as a function equality.

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theorem real.coe_fib_eq (n : ℕ) :
↑(nat.fib n) = (golden_ratio ^ n - golden_conj ^ n) / √ 5

Binet's formula as a dependent equality.