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:
“Is it that mens frayle eyes, which gaze too bold,
She may entangle in that golden snare:”
—Edmund Spenser (1552?1599)
“People are lucky and unlucky not according to what they get absolutely, but according to the ratio between what they get and what they have been led to expect.”
—Samuel Butler (18351902)
“It is a base thing for a man among the people not to obey those in command. Never in a state can the laws be well administered when fear does not stand firm.”
—Sophocles (497406/5 B.C.)