The golden mean or golden ratio, denoted by φ, is the ratio a / b of the lengths of two sides of the rectangle below:

The definition of φ says that the ratio of sides of the smaller rectangle to the left must be the same as that of the large rectangle: (a + b) / a = a / b = φ. Set b = 1 to get (a + 1) / a = a = φ. Then φ is the positive root of a² - a - 1 = 0, which is

Formulas involving φ are:

Also, F_n+1 / F_n converges to φ as n tends to infinity, and F_n / F_n+1 converges to 1 / φ as n tends to infinity.

Recall from a previous page that Newton's Method is used to find successive approximations to the roots of a function. If the function is y = f(x) and x₀ is close to a root, then we usually expect the formula below to give x₁ as a better approximation. Then you plug the x₁ back in as x₀ and iterate.

In this case we are using the equation for 1 / φ, namely x² + x - 1 = 0. (Everything below would also work if we started with the equation for φ instead.)

If we plug x₀ = 1 into the above equation, we get x₁ = 2 / 3. Now iterate with x₀ = 2 / 3.

There are a lot of Fibonacci numbers floating around here. Try one more iteration:

In general, start with p / q, where these are successive Fibonacci numbers, F_n-1 / F_n.

So the same formulas for Fibonacci numbers that we got earlier from matrices are falling out from Newton's method: