Division
No non-integer rational number can be represented as a finite base-φ number. In other words, all finitely representable base-φ numbers are either integers or (more likely) an irrational in the field Q. Due to long division having only a finite number of possible remainders, a division of two integers (or other numbers with finite base-φ representation) will have a recurring expansion, as demonstrated above.
Read more about this topic: Golden Ratio Base
Famous quotes containing the word division:
“If the technology cannot shoulder the entire burden of strategic change, it nevertheless can set into motion a series of dynamics that present an important challenge to imperative control and the industrial division of labor. The more blurred the distinction between what workers know and what managers know, the more fragile and pointless any traditional relationships of domination and subordination between them will become.”
—Shoshana Zuboff (b. 1951)
“Slow, slow, fresh fount, keep time with my salt tears;
Yet slower yet, oh faintly gentle springs:
List to the heavy part the music bears,
“Woe weeps out her division when she sings.”
Droop herbs and flowers;
Fall grief in showers;
“Our beauties are not ours”:
Oh, I could still,
Like melting snow upon some craggy hill,
Drop, drop, drop, drop,
Since nature’s pride is, now, a withered daffodil.”
—Ben Jonson (1572–1637)
“For in the division of the nations of the whole earth he set a ruler over every people; but Israel is the Lord’s portion: whom, being his firstborn, he nourisheth with discipline, and giving him the light of his love doth not forsake him. Therefore all their works are as the sun before him, and his eyes are continually upon their ways.”
—Apocrypha. Ecclesiasticus 17:17-9.