Integer Arithmetic
Why
What are addition and multiplication for integers? What are the identity elements?
Definition
We call the operation of forming integer sums integer addition. We call the operation of forming integer products integer multiplication.
Results
It is easy to see the following.1
The additive identity for $\Z $ is $\eqc{(0,
0)}$.
The multiplicative identity for $\Z $ is
$\eqc{(1, 0)}$.
Notation
We denote the additive identity of $\Z $ by $0_{\Z }$ and the multiplicative identity by $1_{\Z }$. When it is clear from context, we call $0_{\Z }$ “zero” and we call $1_{\Z }$ “one”.
Distributive
For integers $x, y, z \in \Z $, $x \cdot (y
+ z) = x \cdot y + x \cdot z$.2