Set of natural numbers: \(\mathbb{N}\)

Set of integers: \(\mathbb{Z}\)

Set of positive integers: \(\mathbb{Z^ + }\)

Set of negative integers: \(\mathbb{Z^ – }\)

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 \(a:\,\)\(\left| a \right|\)

Absolute value of the number \(a:\,\)\(\left| a \right|\)

- Positive integers \(\mathbb{Z^ + } = \mathbb{N} = \left\{ {1,2,3, \ldots } \right\}\)
- Negative integers \(\mathbb{Z^ – } = \left\{ { \ldots , – 3, – 2, – 1} \right\}\)
- 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 \(\left\{ 0 \right\}\).

\(\mathbb{Z} = \mathbb{Z^ – } \cup \left\{ 0 \right\} \cup \mathbb{Z^ + } =\) \( \left\{ { \ldots , – 3, – 2, – 1,0,1,2,3, \ldots } \right\}\) - The sum, difference, or product of two integers is also an integer.
- Commutativity of addition \(a + b = b + a\)
- Associativity of addition \(a + \left( {b + c} \right) =\) \( \left( {a + b} \right) + c\)
- Existence of an additive identity element \(a + 0 = a\)
- Subtraction \(a – b = a + \left( { – b} \right)\)
- \(a – 0 = a\)
- \(0 – a = -a\)
- \(a + \left( { – a} \right) = 0\)
- Commutativity of multiplication \(a \cdot b = b \cdot a\)
- Associativity of multiplication \(a \cdot \left( {b \cdot c} \right) = \left( {a \cdot b} \right) \cdot c\)
- Distributivity of multiplication over addition \(a \cdot \left( {b + c} \right) =\) \( a \cdot b + a \cdot c\)
- Existence of a multiplicative identity element \(a \cdot 1 = a\)
- \(a \cdot 0 = 0\)
- If \(a \lt b\) and \(c \lt d\), then \(a + c \lt b + d\)
- If \(a \lt b\) and \(c \gt 0\), then \(ac \lt bc\)
- If \(a \lt b\) and \(c \lt 0\), then \(ac \gt bc\)
- Absolute value of a number \(\left| a \right| = \begin{cases} a, & \text{if} \;\;a \gt 0 \\ 0, & \text{if} \;\;a = 0 \\ -a, & \text{if} \;\;a \lt 0 \end{cases}\)
- \(\left| a \right| \ge 0\)
- \(\left| { – a} \right| = \left| a \right|\)
- \(a \le \left| a \right|\)
- \( – \left| a \right| \le a\)
- Triangle inequality \(\left| {a + b} \right| \le \left| a \right| + \left| b \right|\)
- \(\left| a \right| – \left| b \right| \le \left| a \right| + \left| b \right|\)