Golden ratio base

Golden ratio base is a <a href="/facts/Non-integer_representation/Yp7ASnTj">non-integer positional numeral system</a> that uses the <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a> (the irrational number 
 
 
 
 
 
 
 1
 +
 
 
 5
 
 
 
 2
 
 
 
 
 {\textstyle {\frac {1+{\sqrt {5}}}{2}}}
 
 ≈ 1.61803399 symbolized by the <a href="/facts/Greek_alphabet/1cHwB8Dg">Greek letter</a> <a href="/facts/Phi/pUzerkKk">φ</a>) as its <a href="/facts/Radix/J5RGWRJF">base</a>. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary. Any <a href="/facts/Non-negative/z0xollg1">non-negative</a> <a href="/facts/Real_number/R02gw5Pb">real number</a> 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 φn + φn−1 = φn+1. For instance, 11φ = 100φ.
Despite using an <a href="/facts/Irrational_number/Psv19TET">irrational number</a> base, when using standard form, all non-negative <a href="/facts/Integer/PUwwrawx">integers</a> have a unique representation as a terminating (finite) base-φ expansion. The set of numbers which possess a finite base-φ representation is the <a href="/facts/Ring_(algebra)/JnCUXz63">ring</a> <a href="/facts/Quadratic_integer/uGsLh5kL">Z[
 
 
 
 
 
 
 1
 +
 
 
 5
 
 
 
 2
 
 
 
 
 {\textstyle {\frac {1+{\sqrt {5}}}{2}}}
 
]</a>; it plays the same role in this numeral systems as <a href="/facts/Dyadic_rational/LPwRYKaD">dyadic rationals</a> play in <a href="/facts/Binary_number/a6JDSRPs">binary numbers</a>, providing a possibility to <a href="/facts/Multiplication/VtcUjpNv">multiply</a>.
Other numbers have standard representations in base-φ, with <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> having recurring representations. These representations are unique, except that numbers with a terminating expansion also have a non-terminating expansion. For example, 1 = 0.1010101… in base-φ just as <a href="/facts/0.999.../fdtw8eBL">1 = 0.99999…</a> in <a href="/facts/Decimal/7omfVNIM">decimal</a>.

Golden ratio base open-in-new

Golden ratio base