# Identities of Phi

From Portal

- Phi is the Greek letter, Φ upper, and φ lower case, used frequently in mathematics to refer to the golden ratio.

- Phi equals the quantity five raised to the power of 1/2, multiplied by 1/2, and added to 1/2, or

- Φ
**=**5^{0.5}× 0.5 + 0.5**=**1.6180339887498948482045868343656…

- Φ

- More traditionally,
**Φ**and**φ**are notationally expressed as:

- Φ
**=**(√5 + 1)/2**=**1.6180339887498948482045868343656…

- Φ

- And:

- φ
**=**(√5 - 1)/2**=**0.6180339887498948482045868343656…

- φ

## Contents

- Only a small sample follows.

## See

## Connection of Adjacent Degrees of Phi

Φ^{n} = Φ^{n-1} + Φ^{n-2}

- Where
- φ = Φ - 1 = 0.618…
- n is the ordinal position in the Fibonacci sequence

- Source:
*Mathematics of Harmony*^{[1]}

## Representation of the n-th Degree of Phi by Fibonacci and Lucas Numbers

Φ^{n} = (L_{n} + F_{n}√5 ) / 2

- Where
- φ = Φ - 1 = 0.618…
- n is the ordinal position in the Fibonacci sequence
- L
_{n}refers to a number of the Lucas sequence (2, 1, 3, 4, 7, 11, 18...) at position n. - F
_{n}refers to a number of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...) at position n.

- Source:
*Mathematics of Harmony*^{[2]}

- ↑
*Mathematics of Harmony*Alexey Stakhov, Pg. 91 (2.56) World Scientific Publishing Co. Pte. Ltd., Singapore 2009 - ↑
*Mathematics of Harmony*Pg. 91 (2.60)