Formulas and Tables

Number Sets

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. \(\left| a \right| – \left| b \right| \le \left| a \right| + \left| b \right|\)