Rational Numbers

Introduction

We work with basic math in our daily life like counting your fingers and toes, calculating your age, finding temperatures of a cold winter day in Michigan which would be below 0 degree, dividing a birthday cake among twelve of your friends, and many more. In each of these circumstances, we use different types of numbers. The types of numbers that we already know, are natural numbers, whole numbers and integers.


When you were studying fractions, you were just learning the way of writing a numerical quantity, with a numerator and a denominator.  You also learned about using arithmetic operations with the fractions and about representing them as decimals. Now you are growing older, and are ready to dig mathematics a little deeper.  You are going to learn about the numbers themselves and not just the way you write them on a piece of paper. Let us walk through a new type of number called rational numbers.

Standard Form of Rational Numbers  

A rational number is not just a fraction. It means that any number, be it natural number or integers, can be written in the form of \[\;\frac{p}{q}\] such that q is never equal to 0. But we cannot write every number in that way.

A rational number is said to be in a standard form if and only if

a) The denominator is an integer greater than 0 

b) The only common divisor between the numerator and the denominator is 1.

Steps to express a rational number in the standard form:

a) Check if the denominator is positive or negative. If the denominator is negative, change it to positive by multiplying both numerator and denominator by -1. 

b) Take the absolute values of the numerator and denominator and find their greatest common divisor.

c) Perform the division of each of the numerator and denominator by the obtained greatest common divisor. The resulting rational number is the standard form.

Positive and Negative Rational Numbers

A positive rational number is one which has the same signs (either positive or negative) in both its numerator and denominator.

Eg : \[\frac{{ - 5}}{{ - 7}},\frac{{10}}{{13}}\]

A negative rational number is one which has opposite signs in both its numerator and denominator.

Eg : \[\;\frac{{ - 5}}{7},\;\frac{{10}}{{-13}}\]

Properties of Rational Numbers  

Useful Properties-

  1. If \[\;\frac{p}{q}\] is a rational number and r is any non-zero integer, then

\[\frac{p}{q} = \;\frac{{p\times r}}{{q \times r}}\].

This implies that the rational number remains the same if we multiply both the numerator and denominator by the same integer provided the integer is not equal to 0.

  1.  If \[\;\;\frac{p}{q}\] and \[\frac{r}{s}\] are any two rational numbers and \[\frac{p}{q}\; = \frac{r}{s}\] then, \[p \times s = q \times r\]and vice-versa.

Closure Property-

  • Addition

The closure property of rational numbers under addition states that if \[\;\;\frac{p}{q}\] and \[\frac{r}{s}\] are any two rational numbers, \[\frac{p}{q}\; + \frac{r}{s}\]is also a rational number. 

 

  • Subtraction

The closure property of rational numbers under subtraction states that if \[\;\;\frac{p}{q}\] and \[\frac{r}{s}\] are any two rational numbers, \[\frac{p}{q} - \frac{r}{s}\]is also a rational number.

 

  • Multiplication

The closure property of rational numbers under multiplication states that if \[\;\;\frac{p}{q}\] and \[\frac{r}{s}\] are any two rational numbers, \[\frac{p}{q} \times \frac{r}{s}\]is also a rational number

 

  • Division

The closure property of rational numbers under division states that if \[\;\;\frac{p}{q}\] and \[\frac{r}{s}\] are any two rational numbers, \[\frac{p}{q} \div \frac{r}{s}= \frac{p}{q} \times \frac{s}{r}\]is also a rational number


Commutative Property-

  • Addition

The commutative property of rational numbers under addition states that if \[\;\;\frac{p}{q}\] and \[\frac{r}{s}\]are any two rational numbers, \[\frac{p}{q} + \frac{r}{s} = \frac{r}{s} + \frac{p}{q}\]. 

 

  • Subtraction

The commutative property of rational numbers under subtraction states that if \[\;\;\frac{p}{q}\] and \[\frac{r}{s}\] are any two rational numbers, \[\frac{p}{q} - \frac{r}{s}\ne\frac{r}{s} - \frac{p}{q}\]. Hence, the commutative property does not hold for the subtraction of rational numbers.

 

  • Multiplication

The commutative property of rational numbers under multiplication states that if \[\;\;\frac{p}{q}\] and \[\frac{r}{s}\] are any two rational numbers, \[\frac{p}{q} \times \frac{r}{s} = \frac{r}{s} \times \frac{p}{q}\]. 

 

  • Division

The commutative property of rational numbers under division states that if \[\;\;\frac{p}{q}\] and \[\frac{r}{s}\] are any two rational numbers, \[\frac{p}{q} \div \frac{r}{s}\ne\frac{r}{s} \div \frac{p}{q}\]. Hence, the commutative property does not hold for division of any two rational numbers


Associative Property-

  • Addition

The associative property of rational numbers under addition states that if \[\frac{p}{q},\frac{r}{s}\] and \[\frac{u}{t}\] are any three rational numbers, \[\frac{p}{q} + \left( {\frac{r}{s} + \frac{u}{t}} \right) = \left( {\frac{p}{q} + \frac{r}{s}} \right) + \frac{u}{t}\]. 


  • Subtraction

The associative property of rational numbers under subtraction states that if \[\frac{p}{q},\frac{r}{s}\] and \[\frac{u}{t}\] are any three rational numbers, \[\frac{p}{q} - \left( {\frac{r}{s} - \frac{u}{t}} \right){ = }\left( {\frac{p}{q} - \frac{r}{s}} \right) - \frac{u}{t}\]. Hence, the associative property does not hold for the subtraction of rational numbers.

 

  • Multiplication

The associative property of rational numbers under multiplication states that if \[\frac{p}{q},\frac{r}{s}\] and \[\frac{u}{t}\] are any three rational numbers, \[\frac{p}{q} \times \left( {\frac{r}{s} \times \frac{u}{t}} \right) = \left( {\frac{p}{q} \times \frac{r}{s}} \right) \times \frac{u}{t}\].  

 

  • Division

The associative property of rational numbers under division states that if \[\frac{p}{q},\frac{r}{s}\] and \[\frac{u}{t}\] are any three rational numbers, \[\frac{p}{q} \div \left( {\frac{r}{s} \div \frac{u}{t}} \right){ = }\left( {\frac{p}{q} \div \frac{r}{s}} \right) \div \frac{u}{t}\]. Hence, the associative property does not hold for division of any three or more rational numbers.


 

Identity Property-

 

  • Addition

The identity property of rational numbers under addition states that if \[\;\frac{p}{q}\] is a rational number, then  \[\frac{p}{q} + 0 = 0 + \frac{p}{q} = \frac{p}{q}\]. Hence 0 is called the additive identity for rational numbers.


  • Multiplication

The identity property of rational numbers under multiplication states that if \[\;\frac{p}{q}\] is a rational number, then  \[\frac{p}{q} \times 1 = 1 \times \;\frac{p}{q} = \;\frac{p}{q}\]. Hence 1 is called the multiplicative identity for rational numbers.  


Inverse Property-

  • Addition

The inverse property of rational numbers under addition states that if \[\;\frac{p}{q}\] is a rational number, then there exists a rational number \[ - \;\frac{p}{q}\]  such that\[\frac{p}{q} + \left( { - \frac{p}{q}} \right) = \left( { - \frac{p}{q}} \right) + \frac{p}{q} = 0\]. Hence \[ - \;\frac{p}{q}\] is called the additive inverse of \[\;\frac{p}{q}\]. 


  • Multiplication

The inverse property of rational numbers under multiplication states that if \[\;\frac{p}{q}\] is a rational number, then there exists a rational number \[\;\frac{q}{p}\]  such that\[\frac{p}{q} \times \frac{q}{p} = \frac{q}{p} \times \frac{p}{q} = 1\]. Hence \[\frac{q}{p}\]is called the multiplicative inverse or reciprocal of \[\;\frac{p}{q}\].


Distributive Property of Multiplication over Addition

If \[\frac{p}{q},\frac{r}{s}\] and \[\frac{u}{t}\] are any three rational numbers, \[\frac{p}{q} \times \left( {\frac{r}{s} + \frac{u}{t}} \right) = \frac{p}{q} \times \frac{r}{s} + \frac{p}{q} \times \frac{u}{t}\]. The distributive property of multiplication over addition holds good for all rational numbers.


Distributive Property of Multiplication over Subtraction

If \[\frac{p}{q},\frac{r}{s}\] and \[\frac{u}{t}\] are any three rational numbers, \[\frac{p}{q} \times \left( {\frac{r}{s} - \frac{u}{t}} \right) = \frac{p}{q} \times \frac{r}{s} - \frac{p}{q} \times \frac{u}{t}\]. The distributive property of multiplication over subtraction holds good for all rational numbers.


Difference Between Rational and Irrational Numbers

S.No

Rational Numbers

Irrational Numbers

1

These numbers can be expressed in the form of a fraction \[\frac{p}{q}\] where \[q{ = }0\]

These numbers cannot be expressed in the form of a fraction \[\frac{p}{q}\]

2

These numbers include numbers that are finite or recurring in nature.

These numbers include numbers that are non-terminating and non-repeating in nature.

3

Perfect squares are always rational numbers

Surds such as \[\sqrt 2 ,\sqrt 3 ,\sqrt 5 \]are always irrational numbers

 

Rational Numbers Examples

  • \[\frac{2}{7}\]– Both the numerator 2 and the denominator 7 are integers.

  • 5 – Can be expressed as \[\frac{5}{1}\], wherein 5 is the quotient when we divide the integer 5 by the integer 1.

  • \[\sqrt {49} \]– As the square root of 49 can be simplified to 7, which is also the quotient obtained when the integer 7 is divided by 1

  • 0.5 – Can be written as \[\frac{5}{{10}}\]or \[\frac{1}{2}\]. Thus, all terminating decimals are rational numbers.

  • 0.3333333333 – All recurring decimals are rational numbers.