Number Types

Introduction

Numbers are often classified based on their uses. For example, natural numbers are used for counting, negative numbers are used to describe debts or below-zero temperatures, rational numbers are used to describe fractions such as half of an orange and irrational numbers are used for certain distances like the diagonal of a 1 ft side square that cannot be expressed with a fraction

Integers

The set of numbers $\text{{}\dots \text{,-3, -2, -1, 0, 1, 2, 3,}\dots \text{}}$ is called the set of integers and is denoted with the letter . Within this set are various subsets. The set $\text{{}\text{1, 2, 3,}\dots \text{}}$ is called the set of positive integers. The set $\text{{}\text{-1, -2, -3,}\dots \text{}}$ is called the set of negative integers. Note that the number 0, although it is an integer, is neither positive nor negative.

The Natural Numbers

The set of natural numbers is denoted with the letter and is defined as the set of positive integers.

$=\text{{}\text{1, 2, 3,}\dots \text{}}$

Rational Numbers

The rational numbers are the set of numbers that can be represented as a fraction using two integers. That is to say, the rational numbers are those numbers that can be expressed as $\frac{m}{n}$, where m and n are integers and n≠0. $\frac{1}{3},\frac{-5}{4},30=\frac{30}{1},0=\frac{0}{5},0.25=\frac{1}{4}$

Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. The most common irrational numbers are $\pi$ and expressions that contain a root that can not be eliminated.

Example: $\sqrt{3},\sqrt{5},\sqrt[3]{2},$$\sqrt{3},\frac{-3}{{\pi }^2}$

Real Numbers

The set of all real numbers is usually denoted by the symbol ? and can be considered the set of all numbers that exist on a number line. This includes integers, rational numbers and irrational numbers.

Non-Real Numbers

The set of non-real numbers can be considered the set of all numbers that do not exist on a number line.

What Is Not A Real Number?

Non-Real Number	Explanation	Examples
$\sqrt{-a}$, a > 0	The argument to a square root can not be negative.	$\sqrt{-2}$is not a real number.
$\frac{a}{\text{0}}\text{,\hspace{0.28em}}a\text{\hspace{0.28em}is real.}$	The denominator of a fraction can't be zero.	$\frac{\text{0}}{\text{0}}and\frac{3}{0}$are not real.

Decimal Expansions and Rational Numbers

Any finite decimal can be expressed as a fraction. For examples: 0.23 $=\frac{\mathrm{23}}{\mathrm{100}}$, 5.235 $=\frac{\mathrm{5235}}{\mathrm{1000}}$ Some fractions have infinite decimal expansions but they repeat $\frac{1}{3}=0.33333$. Hence, finite decimals or infinite decimals that repeat are rational numbers.

It can be shown that irrational numbers such as $\sqrt{3}\text{or}\pi$ are infinite decimals that do not repeat. Hence, irrational numbers can also be considered as numbers with infinite decimal expansions that do not repeat.