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:
“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)
“Imperialism is capitalism at that stage of development at which the dominance of monopolies and finance capitalism is established; in which the export of capital has acquired pronounced importance; in which the division of the world among the international trusts has begun, in which the division of all territories of the globe among the biggest capitalist powers has been completed.”
—Vladimir Ilyich Lenin (1870–1924)
“Affection, indulgence, and humor alike are powerless against the instinct of children to rebel. It is essential to their minds and their wills as exercise is to their bodies. If they have no reasons, they will invent them, like nations bound on war. It is hard to imagine families limp enough always to be at peace. Wherever there is character there will be conflict. The best that children and parents can hope for is that the wounds of their conflict may not be too deep or too lasting.”
—New York State Division of Youth Newsletter (20th century)