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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. $$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$