Golden ratio base is a noninteger 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 baseφ, golden mean base, phibase, or, colloquially, phinary. Any nonnegative real number can be represented as a baseφ numeral using only the digits 0 and 1, and avoiding the digit sequence "11"  this is called a standard form. A baseφ 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 = φ^{2}. For instance, 11_{φ} = 100_{φ}.
Despite using an irrational number base, all nonnegative integers have a unique representation as a terminating (finite) baseφ expansion, but only if in the standard form. Other numbers have standard representations in baseφ, with rational numbers having recurring representations. These representations are unique, except that numbers with a terminating expansion also have a nonterminating expansion, as they do in base10; for example, 1=0.99999….
Contents
Examples
Writing golden ratio base numbers in standard form
211.01_{φ} is not a standard baseφ 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: 011_{φ} = 100_{φ}, 0200_{φ} = 1001_{φ} and 010_{φ} = 101_{φ}. 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 ▸
