Number Systems

This page discusses the types of number systems found in real analysis and introduction to proving classes.

Number System
is a set of numbers together with one or more operations.
Symbol Name Set
$\mathbb{N}$ Natural Numbers {1, 2, 3, ...}
$\mathbb{Z}$ Integers {...,-3, -2, -1, 0, 1, 2, 3, ...}
$\mathbb{Q}$ Rational Numbers $\{\frac{a}{b}| a,b \in \mathbb{Z}\}$. All fractions.
$\mathbb{R} - \mathbb{Q}$ Irrational Numbers Numbers which cannot be expressed in the form of a fraction.
$\mathbb{R}$ Real Numbers Satisfies the Real Number Axioms. All finite and infinite decimal numbers.
$\mathbb{C}$ Complex Numbers $\{a+bi \ \big| \ a,b \in \mathbb{R} \text{ and } i = \sqrt{-1}\}$

An agreement has not been reached whether the natural numbers begin with 0 or 1. The symbol for the integers, $\mathbb{Z}$, comes from the German word, zahlen, meaning 'numbers.' $\sqrt{2}$ is example of an irrational number whose decimal representation does not terminate and can not be expressed as a fraction. The real numbers, $\mathbb{R}$, are the union of the irrational and rational numbers. Hence, $\sqrt{2}$ is a real number. The real numbers are a subset of the complex numbers, $\mathbb{C}$. Complex numbers have the imaginary root, $\sqrt{-1}$ which has many uses in electrical engineering. The relationship between the number systems can be expressed as follows: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$.