Fibonacci Sequence: Lesson 5

Phi Arithmetic

In the previous lesson, you saw how the ratio of consecutive Fibonacci terms converge on the Golden Ratio phi, symbolized by ϕ. We then solve this proportion:

And we got this value for ϕ from solving the proportion.

In this lesson, we’ll look at some special properties of ϕ.

The Square of ϕ

Let’s multiply ϕ by itself to see what the product is.

So ϕ has this interesting property:

The Reciprocal of ϕ

Now let’s investigate the reciprocal of ϕ.

So ϕ has this property for the reciprocal:

The Product of ϕ and Its Reciprocal

You know from your study of arithmetic that for non-zero real number the product of the number and its reciprocal is 1. Let’s see what happens when we multiply Phi by its reciprocal.

As you can see the result is still 1.