bonjour
nombre d'or : φ = (1 + √5)/2
propriété A
φ² = (1 + √5)²/4 = (1 + 2√5 + 5) / 4
= (6 + 2√5)/4
= (4 + 2 + 2√5)/4
= 4/4 + 2(1 +√5)/4
= 1 + (1 + √5)/2 = 1 + φ
propriété B
1/φ = 1/[(1 + √5)/2] = 1 x 2/(1 + √5)
= 2/(1 + √5)
= 2(1 - √5) / (1 + √5)(1 - √5)
= 2(1 - √5)/ (1 - 5)
= 2(1 - √5) / -4
= -(1 - √5) / 2
= (√5 - 1)/ 2
= [(√5 + 1) - 2] /2
= (√5 + 1)/2 - 2/2
= φ - 1
propriété C
φ³ = φ² x φ = (1 + φ) x φ [ φ² = 1 + φ ]
= φ + φ²
= φ + (1 + φ) = 2φ + 1
