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Golden φ

On this date, we need to join in the celebration of the number Phi = φ = [1+SQRT(5)]/2 = 1.618033989.... More specifically, celebrate on January 6 at 6:03 pm!

So, I will offer two well-known oddities using the form ...that tend to not be well-known! In both, use the form of φ = [1+SQRT(5)]/2.

First, start with φ. Square it. Subtract 1. What do you get?

Second, start with φ. Subtract 1. what do you get?

In the first case, you should get φ again. In the second, you should get the reciprocal of φ, or 1/φ.

In both instances, φ is the only number that fits these patterns:

  • φ2 = φ + 1
  • φ - 1 = 1/φ
Can you prove both of these claims? That is, use algebra to solve x2 = x + 1 or x - 1 = 1/x.

Amazing φ!