Golden Ratio Base

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, when using standard form, all non-negative integers have a unique representation as a terminating (finite) base-φ expansion. 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….

Read more about Golden Ratio Base:  Examples, Writing Golden Ratio Base Numbers in Standard Form, Representing Integers As Golden Ratio Base Numbers, Representing Rational Numbers As Golden Ratio Base Numbers, Representing Irrational Numbers of Note As Golden Ratio Base Numbers, Addition, Subtraction, and Multiplication, Division, Relationship With Fibonacci Coding

Famous quotes containing the words golden, ratio and/or base:

    But if that Golden Age would come again,
    And Charles here rule as he before did reign;
    Robert Herrick (1591–1674)

    Personal rights, universally the same, demand a government framed on the ratio of the census: property demands a government framed on the ratio of owners and of owning.
    Ralph Waldo Emerson (1803–1882)

    Yet herein will I imitate the sun,
    Who doth permit the base contagious clouds
    To smother up his beauty from the world,
    That when he please again to be himself,
    Being wanted, he may be more wondered at
    By breaking through the foul and ugly mists
    Of vapors that did seem to strangle him
    William Shakespeare (1564–1616)