# 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

### Note: Adding *the square root of 5* to -φ gives you φ - 1 and subtracting *the square root of 5* from φ gives you -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:

Note: You can multiply all 3 numbers in the fraction __individually__ by a non-zero number(the exact same multiplier for each of the 3 numbers) to get the exact same ratio!

In this function, y will always be equal to φ as long as x ≠ 0!

Also:

P.S.: If the numbers **φ**^{3} & 1 swap places with the minus sign in-between them, then you'll get **-φ** because subtracting bigger numbers from smaller numbers makes the difference __negative!__ (**φ**^{3} > 1)

__Even more information:__

Note: With a *minus sign* between the **1 & √(5)**, with **√(5)** on the *right* & the **1** on the *left*, the algebraic fraction is equal to **-φ + 1!** (If it's *vice versa,* then the algebraic fraction is equal to **φ - 1.**)

Here's another fact about Phi: **(φ + 1)**^{x} = φ^{2x}.

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