Rational Numbers

Rational numbers are the types of real numbers that can be written in the form of p/q, where p and q are integers and q ≠ 0. Thus, rational numbers themselves include natural numbers, whole numbers, integers, and others. The collection of all rational numbers is denoted by Q.

A Rational number is a number of the form a / b, where a and b are integers and b ≠ 0.

Examples: 1/4 , 3/7 , (-11)/(-6)

• All natural numbers, whole numbers, integers, and fractions are rational numbers.
• Every rational number can be represented on a number line.
• 0 is neither a positive nor a negative rational number.

What is a Rational Number?

Rational numbers can be defined as numbers represented in the form of p/q where q ≠ 0. Also, any fraction where the denominator is not equal to zero is considered to be a rational number. Thus, rational numbers are a group of numbers that includes fractions, decimals, whole numbers, and natural numbers.

How to identify rational numbers?

To identify a rational number use the following conditions:

• Rational numbers are represented in the form of p/q, where q≠0.
• Ratio p/q can be further simplified in simple form or decimal form
• Non-terminating decimals with repeating decimal values are also considered rational numbers as they can be represented in the form of p/q.

Difference Between Fractions and Rational Numbers

Fractions are the real numbers represented in the form of a/b where both a and b are whole numbers whereas rational numbers are the real numbers represented in the form of a/b where both a and b are integers. However, in both cases, the denominator should not be equal to 0. Thus, we can say that all fractions are rational numbers but vice-versa is not true.

Examples of Rational Numbers

Various types of numbers come under rational numbers. Some examples of rational numbers are as follows:

• 0 (which can be written as 0/1)
• 19  (which can be written as 19/1)
• 2/9
• √(64) which gives 4 or 4/1
• -6/7
• 0.333333 = 1/3
• -0.9 = -9/10

To know about, Is π a rational or irrational number? Click here,

Properties of Rational Numbers

Closure Property

• Closure property for Addition: For any two rational numbers a and b, the sum a + b is also a rational number.
• Closure property for Multiplication:  For any two rational numbers a and b, their sum ab is also a rational number

Example:

Take a = 3 / 4 and b = (-1) / 2

Now, 

  a + b = 3 / 4 + (-1) / 2 

           = 3 / 4 + (-2) / 4 

           = (3 – 2) / 4 

           = 1 / 4 is in Q
 Also, 
   a × b = 3 / 4 × (-1) / 2 

           = (-3) / 8 is in Q

Commutative Property

• Commutative property for Addition: For any two rational numbers a and b, a + b = b + a.
• Commutative property for Multiplication:  For any two rational numbers a and b, ab = ba.

Example:

Take a = (-7) / 8 and b = 3 / 5
Now, 

 a + b = (-7) / 8 + 3 / 5 

          = -7 x 5 + 3 x 8 / 40

          = (-35 + 24) / 40 

          = (-11) / 40
 Also   
 b + a = 3 / 5 + (-7) / 8 

          = 3 x 8 + (-7) x 5 / 40

          = (24 – 35) / 40 

          =  (-11) / 40 

 Hence addition is Commutative.
 Further,

     ab = (-7) / 8 x 3 / 5 

          = (-7 x 3) / (8 x 5)

          = (-21) / 40
Also,

     ba = 3 / 5 x (-7) / 8 

          = (3 x 7 ) / (5 x 8) 

          = (-21) / 40

  Hence multiplication is Commutative.

Associative Property

• Associative property for Addition: For any three rational numbers a, b, and c, a + (b + c) = (a + b) + c
• Associative property for Multiplication: For any three rational numbers a, b, and c, a (b c) = (a b)c

Example:

Take rational numbers a,b,c as a = -1 / 2, b = 3 / 5, c = -7 / 10
 Now, 

     a + b = -1 / 2 + 3 / 5 

              = -5 / 10 + 6 / 10 
              = -5 + 6 / 10 

              = 1 / 10

(a + b) + c = 1 / 10 + (-7) / 10 

                  = 1 – 7 / 10 

                  = -6 / 10

                  = -3 / 5………………………………………………….( 1 )

Also,

        b + c = 3 / 5 + (-7) / 10

                 = 6 / 10 + (-7) / 10 

                 = 6 – 7 / 10 

                 = -1 / 10

  a + (b + c) = -1 / 2 + (-1) / 10 

                    = -5 / 10 + (-1) / 10

                    = -5 – 1 / 10 

                    = -6 / 10 

                    = -3 / 5 ………………………………………………..( 2 )

 (1) and (2) shows that (a + b) + c = a + (b + c) is true for rational numbers.

Similarly,

          a × b = -1 / 2 × 3 / 5 

                   = -3 / 10

    (a × b)× c = -3 / 10 × -7 / 10 

                   = -3× (-7) / 100 

                   = 21 / 100  …………………………………………( 3 )

Also,

            b∗ c = 3 / 5 × (-7) / 10 

                   = -21 / 50
 a × ( b ×c ) = -1 / 2 × (-21) / 50

                   = 21 / 100     …………………………………….( 4 )

( 3 ) and ( 4 ) shows that (a× b)× c = a × ( b × c ) is true for rational numbers. Thus,the associative property is true for addition and multiplication of rational numbers. 

Identity Property

• Identity property for Addition: For any rational number a, there exists a unique rational numbers 0 such that 0 + a = a = a + 0.
• Identity property for Multiplication: For any rational number a, there exists a unique rational number 1 such that a × 1 = a = a × 1.

Example:

Take a = 3 / (-7) that is a = -3 / 7 

Now,
 -3 / 7 + 0 = -3 / 7 = 0 + (-3) / 7

 Hence,0 is the additive identity for -3 / 7 

Also,

 -3 / 7 × 1 = -3 / 7 = 1 × 3 / 7

 Hence,1 is the multiplicative identity for -3 / 7. 

Inverse Property

• Additive Inverse property: For any rational number a, there exist a unique rational number -a such that a + (-a) = (-a) +  a = 0.Here, 0 is the additive identity.
• Multiplicative Inverse property: For any rational number b, there exist a unique rational number 1/b such that b × 1 / b = 1 / b × b = 1. Here, 1 is the multiplicative identity.

Example:

Take a = -11 / 23 

Now, -a = -(-11) / 23 

             = 11 / 23
 So,

  a + (-a) = -11 / 23 + 11 / 23 

               = -11 + 11 / 23

               = 0 / 23  

               =0
Also,
(-a) + a  = 11 / 23 + (-11) / 23 

              = 11-11 / 23 

              = 0 / 23 

              = 0

Hence a + (-a) = (-a) + a = 0 is true.
Also,

Take b = -17/29 

Now,

1 / b = 29 / (-17) = -29 / 17
b × 1 / b = -17 / 29 × -29 / 17 = 1
Also,
1 / b × b = 29 / 17 × -17 / 29 = 1
Hence, b × 1 / b = 1 / b × b = 1 is true. 

Distributive Property

Multiplicative is distributive over addition for the collection of rational numbers. For any three rational numbers a, b, and c the distributive law is  a × (b +c) = (a× b) + (a × c)

Example:

Take rational number a, b, c as a = -7 / 9, b = 11 / 18 and c = -14 / 27

Now,
b + c = 11 / 18 + (-14) / 27 

         = 33 / 54 + (-28) / 54

         = 33 – 28 /54 

         = 5 / 54

a × ( b + c ) = -7 / 9 × 5 / 54 

                  = (-7) × 5 / 9 × 54

                  = -35 / 486………………………………………………………………….(1) 

Also,
a × b = -7 / 9 × 11 / 18 

         = (-7) × 11 / 9 × 18 

         = -77 / 9 × 9 × 2

a ∗ c = (-7) /  9 ×(-14) / 27 

         = 7 × 14 / 9 × 9 × 3

         = 98 / 9 × 9 × 3

(a × b) + (a × c) = (-77 / 9 × 9 × 2 ) + ( 98 / 9 × 9 × 3)

                         = (-77) × 3 + 98 × 2 / 9 × 9 × 2 × 3

                         = -231 + 196 / 486 

                         = (-35) / 486…………………………………………………………….(2)

( 1 ) and( 2 ) shows that  a × ( b + c ) = ( a × b ) + ( a × c ). Hence,multiplication is distributive over addition for the collection Q of rational numbers.

Solved Example on Rational Numbers

Example 1: Check which of the following is irrational or rational: 1/2, 13, -4, √3, and π.

Solution:

Rational numbers are numbers that can be expressed in the form of p/q, where q is not equal to 0.

1/2, 13, and -4 are rational numbers as they can be expressed as p/q.

√3, and π are irrational numbers as they can not be expressed as p/q.

Example 2: Check if a mixed fraction, 3(5/6) is a rational number or irrational number.

Solution:

The simplest form of 3(5/6) is 23/6

Numerator = 23, which is an integer

Denominator = 6, is an integer and not equal to zero.

So, 23/6 is a rational number.

Example 3: Determine whether the given numbers are rational or irrational.

(a) 1.33  (b) 0.1  (c) 0  (d) √5

Solution:

a) 1.33 is a rational number as it can be represented as 133/100.

b) 0.1 is a rational number as it can be represented as 1/10.

c) 0 is a rational number as it can be represented as 0/1.

d) √5 is an irrational number as it can not be represented as p/q.

FAQs on Rational Numbers

Question 1: What is the difference between rational and irrational numbers?

Answer:

A rational number is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero, whereas an irrational number cannot be expressed in the form of fractions. Rational numbers are terminating decimals but irrational numbers are non-terminating and non-recurring. An example of a rational number is 10/2, and an irrational number is a famous mathematical value Pi(π) which is equal to 3.141592653589…….

Question 2: What are rational numbers? Give Examples.

Answer:

A rational number is a number that is in the form of p/q, where p and q are integers, and q is not equal to 0. Some examples of rational numbers include 1/3, 2/4, 1/5, 9/3, and so on.

Question 3: Is 0 a rational number?

Answer:

Yes, 0 is a rational number because it is an integer that can be written in any form such as 0/1, 0/2, where b is a non-zero integer. It can be written in the form: p/q = 0/1. Hence, we conclude that 0 is a rational number.

Question 4: Is Pi(π) a rational number?

Answer:

No, Pi (π) is not a rational number. It is an irrational number and its value equals 3.142857…

Question  5: Are fractions rational numbers?

Answer:

Fractions are numbers that are represented in the form of (numerator/denominator) which is equivalent to p/q form so fractions are considered rational numbers. Example 3/4 is a fraction but is also a rational number.

Question 6: Are all rational numbers integers?

Answer:

No, all rational numbers are not integers but the opposite is true. i.e. “all integers are rational numbers.” For example, 1/2 is a rational number but not an integer whereas -7 is an integer and is also a rational number.

Question 7: Can rational numbers be negative?

Answer:

Yes, a rational number can be negative i.e. all negative number comes under rational numbers. Example -1.25 is a rational number.

Question 8: Are all whole numbers rational numbers?

Answer:

Yes, all whole number are considered as rational numbers. For example 1 is a whole number and is also a rational number.