Number Theory Section Summary: 2.1

The Division Algorithm

``...the foundation stone upon which our whole development rests.'' (p. 17)

  1. Theorems

    Division Algorithm: Given integers a and b, with b>0, there exist unique integers q and r satisfying

    displaymath183

    with tex2html_wrap_inline197 . q is called the quotient, and r is called the remainder.

    If a>0 as well, then this is an obvious extension of the Archimedean property: if any positive b can be added to itself enough times to exceed any positive a, then clearly there will come a point at which tex2html_wrap_inline209 and (q+1)b > a. r just represents the amount by which qb is short (if any!).

    (Proof using well-ordering and contradiction.)

    Corollary: Given integers a and b, with tex2html_wrap_inline221 , there exist unique integers q and r satisfying

    displaymath183

    with tex2html_wrap_inline227 .

  2. Notes

    The author shows a couple of interesting properties immediately:

  3. Summary

    Burton comments that the focus will fall on the applications of the division algorithm: ``...it allows us to prove assertions about all the integers by considering only a finite number of cases.'' (p. 19)




Tue Jan 17 17:21:34 EST 2006