Golden Mean is Irrational

Theorem

The golden mean $\phi$ is irrational.

Proof

By definition of golden mean:

$\phi = \dfrac {1 + \sqrt 5} 2$

By Square Root of Prime is Irrational:

$\sqrt 5$ is irrational.

By Rational Number plus Irrational Number is Irrational:

$1 + \sqrt 5$ is irrational.

By Irrational Number divided by Rational Number is Irrational:

$\dfrac {1 + \sqrt 5} 2$ is irrational.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms