Golden ratio base

related topics
{math, number, function}
{language, word, form}
{rate, high, increase}

Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number (1+√5)/2 ≈ 1.61803399... symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary. Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence "11" - this is called a standard form. A base-φ numeral that includes the digit sequence "11" can always be rewritten in standard form, using the algebraic properties of the base φ — most notably that φ+1 = φ2. For instance, 11φ = 100φ.

Despite using an irrational number base, all non-negative integers have a unique representation as a terminating (finite) base-φ expansion, but only if in the standard form. Other numbers have standard representations in base-φ, with rational numbers having recurring representations. These representations are unique, except that numbers with a terminating expansion also have a non-terminating expansion, as they do in base-10; for example, 1=0.99999….



Writing golden ratio base numbers in standard form

211.01φ is not a standard base-φ numeral, since it contains a "11" and a "2", which isn't a "0" or "1", and contains a 1=-1, which isn't a "0" or "1" either.

To "standardize" a numeral, we can use the following substitutions: 011φ = 100φ, 0200φ = 1001φ and 010φ = 101φ. We can apply the substitutions in any order we like, as the result is the same. Below, the substitutions used are on the right, the resulting number on the left.

Full article ▸

related documents
Automata theory
Borel algebra
Least common multiple
Special linear group
Operator overloading
Column space
Cotangent space
Linear subspace
Monotone convergence theorem
Theory of computation
Product topology
Spectrum of a ring
Generating trigonometric tables
Mean value theorem
De Morgan's laws
Associative algebra
Measure (mathematics)
1 (number)
Cauchy-Riemann equations
Stirling's approximation
Isomorphism theorem
Sylow theorems
Differential topology
Tree (data structure)
Knight's tour