Formulas and Tables

Number Sets

Sets of Numbers

Natural numbers: \(\mathbb{N}\)
Whole numbers: \(\mathbb{N_0}\)
Integers: \(\mathbb{Z}\)
Positive integers: \(\mathbb{Z^+}\)
Negative integers: \(\mathbb{Z^-}\)

Rational numbers: \(\mathbb{Q}\)
Real numbers: \(\mathbb{R}\)
Complex numbers: \(\mathbb{C}\)

  1. The natural numbers are those used for counting and ordering: \(\mathbb{N} = \left\{ {1,2,3, \ldots } \right\}\)
  2. The natural numbers including zero (or whole numbers) are those used for indicating the number of objects: \(\mathbb{N_0} = \left\{ {0,1,2,3, \ldots } \right\}\)
  3. The integers include the natural numbers, the negatives of the natural numbers and zero.
    Positive integers: \(\mathbb{Z^+} = \mathbb{N} = \left\{ {1,2,3, \ldots } \right\}\)
    Negative integers: \(\mathbb{Z^-} = \left\{ { \ldots , – 3, – 2, – 1} \right\}\)
    \(\mathbb{Z} = \mathbb{Z^-} \cup \left\{ 0 \right\} \cup \mathbb{Z^+} =\) \( \left\{ { \ldots , – 3, – 2, – 1,0,1,2,3, \ldots } \right\}\)
  4. The rational numbers are those that can be represented as a fraction \(a/b\) where \(a\) and \(b\) are integers and \(b \ne 0.\)
    \(\mathbb{Q} = \big\{ {x \mid x = a/b,\;a \in \mathbb{Z},}\) \({b \in \mathbb{Z},\;b \ne 0} \big\}\)
    The decimal expansion of a rational number either terminates after a finite number of digits (i.e. is a finite decimal) or becomes an infinite periodic decimal.
  5. The irrational numbers are those that may be represented as an infinite non-repeating decimal.
  6. The set of real numbers is the union of rational and irrational numbers: \(R\)
  7. Complex numbers \(\mathbb{C} = \big\{ {x + iy \mid x \in \mathbb{R}}\) \({\text{and}\;y \in \mathbb{R}} \big\}\), where \(i\) is the imaginary unit.
  8. \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}\)
Nested sets of numbers