This preview shows page 1. Sign up to view the full content.
1.7. MODULAR ARITHMETIC
43
What are the objects
Z
n
actually good for? First, they are devices for
studying the integers. Congruence modulo
n
is a fundamental relation in
the integers, and any statement concerning congruence modulo
n
is equiv
alent to a statement about
Z
n
. Sometimes it is easier, or it provides better
insight, to study a statement about congruence in the integers in terms of
Z
n
.
Second, the objects
Z
n
are fundamental building blocks in several gen
eral algebraic theories, as we shall see later. For example, all ﬁnite ﬁelds
are constructed using
Z
p
for some prime
p
.
Third, although the algebraic systems
Z
n
were ﬁrst studied in the nine
teenth century without any view toward practical applications, simply be
cause they had a natural and necessary role to play in the development
of algebra, they are now absolutely fundamental in modern digital engi
neering. Algorithms for digital communication, for error detection and
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Congruence, Integers

Click to edit the document details