• Home
    • A Brief Note on Accuracy
  • The Myth
    • A Brief History of Phi
    • Architectural and Aesthetic Approximations
    • The Human Body
  • The Mathematics
    • Mathematical Definitions of Phi
    • Mathematical Properties of Phi
    • Geometric Constructions Involving Phi
    • The Fibonacci Sequence
  • The Science
    • The Logarithmic Spiral
    • Phyllotaxis: The Fibonacci Sequence in Nature
    • Phi in Chemistry and Physics
  • The Anthropology
    • Phi in Psychology
    • Rationalizing Phi: The Golden Ratio as a Western Phenomenon
  • Works Cited
  • The Author
The Myth of the Golden Ratio

Mathematical Properties of Phi

The Golden Ratio is very interesting in mathematics for multiple reasons, first and foremost because its square and reciprocal have some rather peculiar properties (Tung 7). Note that when we square phi (that is, multiply it by itself), we see that the result is equal to phi plus one: 
Picture
Derivation of Phi Squared
Also, the reciprocal of phi (1/phi) is equal to phi minus one:
Picture
Derivation of Reciprocal Phi
Both of these properties can be deduced from the various definitions of phi found here. 
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