# Integers

Set of natural numbers: $$\mathbb{N}$$
Set of integers: $$\mathbb{Z}$$
Set of positive integers: $$\mathbb{Z^ + }$$
Set of negative integers: $$\mathbb{Z^ – }$$
Integers: $$a$$, $$b$$, $$c$$, $$d$$
Absolute value of the number $$a:\,$$$$\left| a \right|$$
1. Positive integers $$\mathbb{Z^ + } = \mathbb{N} = \left\{ {1,2,3, \ldots } \right\}$$
2. Negative integers $$\mathbb{Z^ – } = \left\{ { \ldots , – 3, – 2, – 1} \right\}$$
3. The set of integers consists of natural numbers $$\left\{ {1,2,3, \ldots } \right\},$$ opposite natural numbers (i.e. with a negative sign) $$\left\{ { \ldots , – 3, – 2, – 1} \right\}$$, and zero $$\left\{ 0 \right\}$$.
$$\mathbb{Z} = \mathbb{Z^ – } \cup \left\{ 0 \right\} \cup \mathbb{Z^ + } =$$ $$\left\{ { \ldots , – 3, – 2, – 1,0,1,2,3, \ldots } \right\}$$
4. The sum, difference, or product of two integers is also an integer.
5. Commutativity of addition $$a + b = b + a$$
6. Associativity of addition $$a + \left( {b + c} \right) =$$ $$\left( {a + b} \right) + c$$
7. Existence of an additive identity element $$a + 0 = a$$
8. Subtraction $$a – b = a + \left( { – b} \right)$$
9. $$a – 0 = a$$
10. $$0 – a = -a$$
11. $$a + \left( { – a} \right) = 0$$
12. Commutativity of multiplication $$a \cdot b = b \cdot a$$
13. Associativity of multiplication $$a \cdot \left( {b \cdot c} \right) = \left( {a \cdot b} \right) \cdot c$$
14. Distributivity of multiplication over addition $$a \cdot \left( {b + c} \right) =$$ $$a \cdot b + a \cdot c$$
15. Existence of a multiplicative identity element $$a \cdot 1 = a$$
16. $$a \cdot 0 = 0$$
17. If $$a \lt b$$ and $$c \lt d$$, then $$a + c \lt b + d$$
18. If $$a \lt b$$ and $$c \gt 0$$, then $$ac \lt bc$$
19. If $$a \lt b$$ and $$c \lt 0$$, then $$ac \gt bc$$
20. Absolute value of a number $$\left| a \right| = \begin{cases} a, & \text{if} \;\;a \gt 0 \\ 0, & \text{if} \;\;a = 0 \\ -a, & \text{if} \;\;a \lt 0 \end{cases}$$
21. $$\left| a \right| \ge 0$$
22. $$\left| { – a} \right| = \left| a \right|$$
23. $$a \le \left| a \right|$$
24. $$– \left| a \right| \le a$$
25. Triangle inequality $$\left| {a + b} \right| \le \left| a \right| + \left| b \right|$$
26. $$\left| a \right| – \left| b \right| \le \left| a \right| + \left| b \right|$$