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# Phi numeral system

(Redirected from Golden ratio base)

The phi numeral system (golden ratio base, golden section base, golden mean base, ϕ -base, base ϕ, phinary, phigital) uses the the golden ratio (symbolized by the Greek letter ϕ) as the base for a non-integer base positional numeral system. Although it is an irrational base, it is not only algebraic, but quadratic, and it is the number with the simplest continued fraction expansion (expressed with all ones) and nested radicals expansion (again, expressed with all ones).

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, the representation thus obtained is unique. 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 = ϕ 2
. For instance,
 11ϕ = 100ϕ
. Despite using an irrational number base, all integers have a unique representation as a terminating (finite) base
 ϕ
expansion, but only if in the standard form. Nonintegers also 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, 0.99999….

## Powers of ϕ in terms of ϕ and Fibonacci numbers

The following table expresses the powers of the Golden ratio
 ϕ
in terms of
 ϕ
itself and Fibonacci numbers, where
 ϕ
is
ϕ  =
 1 + 2√  5 2
.

Powers of
 ϕ

 n
 ϕ n = Fn  − 1 + Fn ϕ
 ϕ   − n + ϕ n = L2n

A005248(n), n   ≥   0
6 5 + 8 ϕ 18
5 3 + 5 ϕ
4 2 + 3 ϕ 7
3 1 + 2 ϕ
2 1 + 1 ϕ 3
1 0 + 1 ϕ
0 1 + 0 ϕ 2
−1 ( −1) + 1 ϕ
−2 2 + ( −1) ϕ 3
−3 ( −3) + 2 ϕ
− 4 5 + ( −3) ϕ 7
− 5 ( −8) + 5 ϕ
− 6 13 + ( −8) ϕ 18

A005248 Bisection of Lucas numbers: a (n) = L (2 n) = A000032 (2 n).

{2, 3, 7, 18, 47, 123, 322, 843, 2207, 5778, 15127, 39603, 103682, 271443, 710647, 1860498, 4870847, 12752043, 33385282, 87403803, 228826127, 599074578, 1568397607, ...}

## Unique representation of integers as a sum of non-consecutive powers of ϕ

Base 10 Sum of non-consecutive
powers of
 ϕ
Base
 ϕ
1 ϕ 0 1
2 ϕ 1 + ϕ −2 10.01
3 ϕ 2 + ϕ −2 100.01
4 ϕ 2 + ϕ 0 + ϕ −2 101.01
5 ϕ 3 + ϕ −1 + ϕ − 4 1000.1001
6 ϕ 3 + ϕ 1 + ϕ − 4 1010.0001
7 ϕ 4 + ϕ − 4 10000.0001
8 ϕ 4 + ϕ 0 + ϕ − 4 10001.0001
9 ϕ 4 + ϕ 1 + ϕ −2 + ϕ − 4 10010.0101
10 ϕ 4 + ϕ 2 + ϕ −2 + ϕ − 4 10100.0101

## Phigital representation of some interesting numbers

π = 100.0100101010010001010101000001010...ϕ (A102243)
e = 100.0000100001001000000001000...ϕ (A105165)