In the last lesson, the ratios of consecutive Fibonacci numbers kept drifting towards the same value.
1.6180339887…
This number is called the golden ratio. Mathematicians usually write it with the Greek letter φ (phi).
A rectangle that copies itself
Imagine a rectangle whose long side has length φ and short side has length 1. Remove a 1×1 square. If the smaller rectangle left behind has exactly the same proportions as the original, then:
φ = 1 + 1/φ
Multiply by φ:
φ² = φ + 1
It is the positive number that satisfies φ = 1 + 1/φ. Solving φ² − φ − 1 = 0 gives φ = (1 + √5)/2.
Why Fibonacci ratios approach φ
Suppose the ratio between consecutive Fibonacci numbers settles towards some number L. Since Fₙ₊₁ = Fₙ + Fₙ₋₁, divide through by Fₙ:
Fₙ₊₁/Fₙ = 1 + Fₙ₋₁/Fₙ
If the ratios settle, then the left side approaches L while the final fraction approaches 1/L. So L must satisfy:
L = 1 + 1/L
That is exactly the golden-ratio equation.
If φ² = φ + 1, simplify φ³ without using a decimal approximation.
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Reveal answer
φ³ = φ·φ² = φ(φ + 1) = φ² + φ = (φ + 1) + φ = 2φ + 1.
A surprising Fibonacci formula
There is even a direct formula for the nth Fibonacci number:
Fₙ = (φⁿ − ψⁿ)/√5, where ψ = (1 − √5)/2
This is called Binet’s formula. It is surprising because Fibonacci numbers are whole numbers, while the formula uses irrational numbers involving √5.
Compare 13/8, 21/13, 34/21 and 55/34 with φ ≈ 1.618034. Which is closest?
Reveal answer
55/34 ≈ 1.617647, which is the closest of these four. In general the Fibonacci ratios get closer and closer to φ.
What have we learned?
- φ = (1 + √5)/2 ≈ 1.618.
- φ satisfies φ = 1 + 1/φ and φ² = φ + 1.
- Ratios of consecutive Fibonacci numbers approach φ.
- The same number links recursion, geometry and algebra.
Next we meet numbers that cannot be written as ordinary fractions at all.