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:
“The division between the useful arts and the fine arts must not be understood in too absolute a manner. In the humblest work of the craftsmen, if art is there, there is a concern for beauty, through a kind of indirect repercussion that the requirements of the creativity of the spirit exercise upon the production of an object to serve human needs.”
—Jacques Maritain (1882–1973)
“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)
“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)