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:
“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.
“O, if you raise this house against this house
It will the woefullest division prove
That ever fell upon this cursed earth.”
—William Shakespeare (1564–1616)
“The glory of the farmer is that, in the division of labors, it is his part to create. All trade rests at last on his primitive activity.”
—Ralph Waldo Emerson (1803–1882)