# The Irrational Number Phi

### This number is also known as the Golden Ratio. It's approximately equal to 1.618 and it is written as a Greek letter, just like Pi.

## Phi looks like this: φ

## Pi looks like this: π

### In fact, they're both Greek letters, but this page is about the irrational number Phi! What's so interesting about this number? Squaring this special number is same as adding 1 to it! 1 divided by this number or raising it to the power of -1 is the same as subtracting 1 from it!

## φ + 1 = φ^{2}

## 1/φ = φ^{-1} = φ - 1

### However, if Phi is negative...

## 1/-φ = (-φ)^{-1} = -φ + 1

### This unique number is the intersection of these 2 functions:

## Y = X^{2}

## Y = X + 1

### Actually, these functions intersect twice; but at each intersection, X = -φ + 1 or φ. Here's how Phi looks as an algebraic fraction:

Also:

One more fact about Phi: φ^{x} = φ^{x-1} + φ^{x-2}. Also, φ^{x} = φ^{x+2} - φ^{x+1}.

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© Derek Cumberbatch