The Division Algorithm
``...the foundation stone upon which our whole development rests.'' (p. 17)
Division Algorithm: Given integers a and b, with b>0, there exist unique integers q and r satisfying
with . 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 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 , there exist unique integers q and r satisfying
with .
The author shows a couple of interesting properties immediately:
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)