Meet π’s cousin, φ

So what’s that symbol, you ask? It’s the Greek letter Phi (pronounced FEE).

We’ve all heard of the (in)famous π, or 3.14159265358… Our math teachers pounded this into our heads every year, and the nerds would always hold competitions to see who could recite the most digits of pi. So far, computers have sputtered out over 5 trillion digits of pi.

But another number, just as beautiful as pi, is relatively unknown to the masses. That number is phi, or 1.61803399… It is the number that satisfies the following algebraic equation: (a+b)/a = a/b.

There are many ways of expressing this golden number:

As an infinite series,

As a continued fraction,

As a continued square root, which bears striking resemblance to the above:

But wait, there’s even more:

φ^2 = φ + 1,

and even more surprisingly,

1/φ = φ – 1. Wow!

If that surprised you, this will truly give you a shock:

φ to the nth power is equal to the sum of the previous two integer powers of φ. Check this out:

  • φ^3 = φ^2 + φ^1
  • φ^4 = φ^3 + φ^2
  • φ^n = φ^(n-1) + φ^(n-2)

The same thing goes with negative powers! Try it out!

Thought it couldn’t get even more fascinating? WRONG!

Compare the digits after the decimal point in this table:

n phin phi-n
1 1.61803398875 0.61803398875
3 4.23606797750 0.23606797750
5 11.09016994375 0.09016994375
7 29.03444185375 0.03444185375
9 76.01315561752 0.01315561752

Lo and behold, the decimal parts in the latter 2 columns are exactly the same! Also notable: this relationship does not exist if n is an even number.

For those of you who are still reading, there is a special link between phi and the Fibonacci series:

Let F(n) = 1, 1, 2, 3, 5, 8, 13, …

WHOA! Who knew?