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 when the bowels of the earth were sought,
And men her golden entrails did espy,
This mischief then into the world was brought,
This framed the mint which coined our misery.
...
And thus began th’exordium of our woes,
The fatal dumb-show of our misery;
Here sprang the tree on which our mischief grows,
The dreary subject of world’s tragedy.”
—Michael Drayton (1563–1631)
“A magazine or a newspaper is a shop. Each is an experiment and represents a new focus, a new ratio between commerce and intellect.”
—John Jay Chapman (1862–1933)
“Adolescence is a time when children are supposed to move away from parents who are holding firm and protective behind them. When the parents disconnect, the children have no base to move away from or return to. They aren’t ready to face the world alone. With divorce, adolescents feel abandoned, and they are outraged at that abandonment. They are angry at both parents for letting them down. Often they feel that their parents broke the rules and so now they can too.”
—Mary Pipher (20th century)