# Rational Number

Pronunciation: /ˈræʃ.nl ˈnʌm.bər/ Explain

A rational number is a real number that can be expressed exactly as the ratio of two integers.

An integer is always a rational number. This is because integers can be expressed as a ratio of themselves and 1. For example, the number 5 can be written as 5 / 1.

In math, numbers can be represented in ways that mask their true identity. For example, the square root of 4 can be reduced to 2, and so is a rational number, even if it is represented by using a square root.

 5.2 All finite decimals are rational numbers. Why? All fractions with a rational numerator and denominator are rational numbers. Why? 3 All integers are rational numbers. Why? Any representation of a number that can be simplified to a rational number is also a rational number. 3.420742074207… Any repeating decimal can be represented as a fraction with integer numerator and denominator. So any repeating decimal is a rational number. Table 1: Representations of rational numbers

 π π has been proven to be irrational. Any square root that can not be simplified to a rational number is irrational. Table 2: Representations of irrational numbers

## Why?

#### All finite decimals are rational numbers.

Any finite decimal can be represented by a fraction of integers. Using the definition of a decimal number, the number 5.2 can be represented as #### All fractions with a rational numerator and denominator are rational numbers.

Since all rational numbers can be represented as the ratio of two integers, the fraction can be written as where and . Using the properties of multiplication, Since a1, a2, b1 and b2 are integers, a1b2 and a2b1 are also integers, is a rational number.

#### All integers are rational numbers.

Start with the fact that anything divided by one remains unchanged. So Since both 3 and 1 are integers, is a rational number, so 3 must also be a rational number.

#### Any square root that can be simplified to a rational number is a rational number.

The definition of a rational number is a number that can be represented as the ratio of two integers. If a square root can be simplified to a rational number, then that square root represents a rational number. Since , represents a rational number.

#### Any repeating decimal is a rational number.

The repeating decimal 3.420742074207… can be written as . Since a repeating decimal can be written as the ratio of two integers, all repeating decimals are rational number.

### Properties of Rational Numbers

PropertyDescription
Associativity The set of rational numbers is associative with respect to addition, subtraction, multiplication and division. Example: a + (b + c) = (a + b) + c.
Commutativity The set of rational numbers is commutative with respect to addition and multiplication. The set of rational numbers is not commutative with respect to subtraction or division. Example: a + b = b + a.
Additive identity The additive identity for rational numbers is 0. Example: a + 0 = 0 + a = a.
Multiplicative identity The multiplicative identity for rational numbers is 1. Example: a · 1 = 1 · a = a.
Closure The set of rational numbers is closed with respect to addition, subtraction, multiplication, and division. Example: if a and b are rational numbers then a + b is also a rational number.
Discrete The set of rational numbers is a discrete (not continuous) set.
Cardinality The cardinality of the set of rational numbers is 0.
Table 1: Properties of the rational numbers.

1. McAdams, David E.. All Math Words Dictionary, rational number. 2nd Classroom edition 20150108-4799968. pg 151. Life is a Story Problem LLC. January 8, 2015. Buy the book

### Cite this article as:

McAdams, David E. Rational Number. 5/2/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/r/rationalnumber.html.

### Revision History

5/2/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/5/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
12/19/2009: Added "References". (McAdams, David E.)
12/31/2008: Changed equations from hot_eqn to images. (McAdams, David E.)
9/4/2008: Added Hot_Eqn, added 'More Information', and added 'Why?' section. (McAdams, David E.)
3/20/2008: Corrected examples of irrational numbers. (McAdams, David E.)
2/27/2008: Change Javascript vocabulary hot links to HTML. (McAdams, David E.)
7/12/2007: Initial version. (McAdams, David E.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2018 Life is a Story Problem LLC. All rights reserved.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License