Given any set , where
we define two operations:
and
.
and
Modular Arithmetic is a binary operation. That is, a binary operation is where there are two inputs and one output.
Example in :
Theorem 2.6
If
and
in
, then
and
.
Example Modular Arithmetic table in :
Modular Arithmetic has several different types of properties.
Theorem 2.7
Powers
We can further this by putting a congruence set to a power. Say
. Jut as with exponents with integers, there are n number of
being multiplied.
A Power Example
For we have
.
General Problems
Now for :
. Every x in
is a solution of
.
Does there exist in
such that
?
No.
has no solution, yet
, if $x=[7]$,
Uses of Modular Arithmetic
- Encryption
- Clocks
Note: An analogous setting will reappear when looking at ‘rings’.