Summary: Chapter 3
The Integers and Beyond
God made the integers; all the rest is
the work of man.
- Leopold Kronecker
3.1 Properties of the Integers
We begin with a technical property that is useful in
proofs. While the principle applies to the integers it
is given in a more general context that includes the
Proposition 3.1 The trichotomy principle: A
real number r is positive, negative, or equal to zero.
We now move on to divisibility, one of the most in-
teresting properties of the integers.
Definition 3.1 We say that an integer m divides
an integer n if there is an integer q so that
n = qm
If m divides n we use the notation m|n. If m fails to
divide n we say m n. If m|n we also say that n is a
multiple of m.