Rational Numbers

Rational Numbers are the most fundamental subset of the collection of all numbers. A rational number represents the fraction of two integers where the denominator can’t be 0. In various applications, throughout the different streams of mathematics rational numbers and their properties are used such as number theory, algebra, calculus, etc. In this article, we will explore rational numbers in depth with various examples, properties, and operations which we can operate on the collection of rational numbers.

What is a Rational Number?

Rational numbers can be defined as numbers represented in the form of p/q where p and q are both integers and 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, also the collection of all rational numbers is denoted by Q.

• 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.

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

Curious about the nature of π? Read – Is π a rational or irrational number?

How to Identify Rational Numbers?

All the rational numbers follow the following rules, thus using the help of these rules we can identify the rational numbers

• 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 expansion.
• Non-terminating decimals with repeating decimal values are also considered rational numbers as they can be represented in the form of p/q.

Example: Which of the following numbers are rational numbers?

a) -1.75
b) 2/3
c) √5
d) π

Solution:

a) -1.75 is a rational number as it  it has a terminating decimal expansion.

b) 2/3 is also a rational number as it can be expressed in the form of a ratio of two integers.

c) √5 is an irrational number because  it has a decimal expansion with infinitly many digits without any repetation.

d) π is also an irrational number as it has a decimal expansion with infinitly many digits without any repetation.

Thus, only (a) and (b) are the rational numbers out of all the given numbers.

Types of Rational Numbers

Various types of numbers can be represented as rational numbers some of which are discussed below:

Integers: All the integers, i.e. negative integers and positive integers both come under rational numbers. For example, -1, -11, -4, 6. 8. 10. etc all are rational numbers.

As natural numbers, whole numbers, and others come under rational numbers they are all rational numbers.

Fraction Number: A rational number is a ratio of two integers that can be written in the form of p/q where q is not equal to zero. Hence, any fraction with a non-zero denominator is a rational number.

Example: -2 / 5 is a rational number where -2 is an integer being divided by a non-zero integer 5

Decimal Number: A rational number can be also written in the decimal form if the decimal value is definite or has repeating digits after the decimal point.

Example: 0.3 is a rational number. The value 0.3 can be further expressed in the form of a ratio or fraction as p/q

0.3 = 3/10

Also, 1.333333… can be represented as 4/3 hence, 1.33333… is a rational number.

The image added below shows types of Rational Numbers.

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 rational numbers are not fractions.

Operations on Rational Numbers

There are four most common operations for Rational Numbers, which are as follows:

• Addition
• Subtraction
• Multiplication
• Division

Addition

The addition of two rational numbers can be done using the following step-by-step method where the addition of 3/4 and 1/6 is explained as an example.

Step 1: Find the common denominator (LCD) for both the rational number. i.e.,

The common denominator for 4 and 6 is 12.

Step 2: Convert both the rational number to equivalent fractions using the common denominator. i.e.,

3/4 = (3 x 3)/(4 x 3) = 9/12

and 1/6 = (1 x 2)/(6 x 2) = 2/12

Step 3: Add the numerators of the equivalent fractions obtained in step 2. i.e.,

9/12 + 2/12 = (9 + 2)/12 = 11/12

Step 4: Simplify the resulting fraction if possible. i.e.,

11/12 is already in its simplest form.

Thus, Addition of 3/4 and 1/6 is 11/12 .

Subtraction

Subtraction of two Rational Numbers can be done using the following step-by-step method where subtraction of 1/3 and 2/5 is explained.

Step 1: Find the common denominator (LCD) for both the rational number. i.e.,

The common denominator for 3 and 5 is 15.

Step 2: Convert both the rational numbers to equivalent fractions with the common denominator. i.e.,

1/3 = (1 x 5)/(3 x 5) = 5/15

and 2/5 = (2 x 3)/(5 x 3) = 6/15

Step 3: Subtract the numerators of the equivalent fractions obtained in step 2. i.e.,

5/15 – 6/15 = (5 – 6)/15 = -1/15

Step 4: Simplify the resulting fraction if possible. i.e.,

-1/15 is already in its simplest form.

Therefore, 1/3 – 2/5 = -1/15.

Multiplication

Multiplication of two rational numbers can be achieved by simply multiplying the numerator and denominator of the given Rational Numbers. Step by step method with an example of multiplication of -11/3 and 4/5 is as follows:

Step 1: Write both rational number in with multiplication sign(×) in between. i.e.,

-11/3 × 4/5

Step 2: Multiply the numerator and denominator individually. i.e.,

(-11 × 4)/(3 × 5)

Step 3: We get the result of the multiplication. i.e.,

-44/15

Division

Division of two Rational numbers can be achieved in the following steps(where the division of 3/5 and 4/7 is explained):

Step 1:  Write both rational number in with division sign in between. i.e.,

3/5 ÷ 4/7

Step 2: Change “÷” with  “×” and take raciprocal of the second rational number. i.e.,

3/5 × 7/4

Step 3: Multiply the numerator and denominator of the resulting fractions. i.e.,

(3 × 7)/(5 × 4)

Step 4: We get the result of the division. i.e.,

21/20

The image added below shows all the properties of the rational numbers.

Rational Numbers Properties

Rational Numbers show several properties under the different operations (two of such common operations are addition and multiplication), which are as follows:

• Closure Property
• Commutative Property
• Associative Property
• Identity Property
• Inverse Property
• Distributive Property

Closure Property

• Closure Property for Addition: Rational numbers are closed under addition, i.e., for any two rational numbers a and b, the sum a + b is also a rational number.
• Closure Property for Multiplication: Rational numbers are closed under multiplication, i.e., for any two rational numbers a and b, their product ab is also a rational number

Example:

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

Now,  a + b = 3 / 4 + (-1) / 2 

⇒ a + b = (3- 2)/ 4  

⇒ a + b = 1/4, is Rational Number.

 Also, a × b = 3/4 × (-1)/2 = -3/8,  which is also Rational number.

Commutative Property

• Commutative Property for Addition: Rational numbers hold commutative property under addition operation, i.e., for any two rational numbers a and b, a + b = b + a.
• Commutative Property for Multiplication:  Rational numbers hold commutative property under multiplication operation as well, i.e., for any two rational numbers a and b, ab = ba.

Example:

For a = (-7) / 8 and b = 3 / 5
Now, a + b = -7/8 + 3/5 

⇒ a + b = (-7 x 5 + 3 x 8)/40 = (-35 + 24) / 40 

⇒ a + b = (-11) / 40

And, b + a = 3/5 + (-7)/8 

⇒ b + a = (3 x 8 + (-7) x 5)/ 40  = (24 – 35)/40 

⇒ b + a =  -11/40 = a + b 

Now, ab = (-7)/8 x 3/5 = (-7 x 3)/(8 x 5)

⇒ ab = -21/40

And, ba = 3/5 x (-7)/8 = (3 x 7 )/(5 x 8) 

⇒ ba =(-21)/40 = ab

Associative Property

• Associative Property for Addition: Rational Numbers are associative under addition operation, i.e., for any three Rational Numbers a, b, and c, a + (b + c) = (a + b) + c
• Associative Property for Multiplication: Rational Numbers are associative under multiplication operation as well, i.e., for any three Rational numbers a, b, and c, a(bc) = (ab)c

Example:

For three Rational numbers a,b,c where a = -1/2, b = 3/5, c = -7/10
Now,

a + b = -1/2 + 3/5 = (-5 + 6)/10 = 1/10

and (a + b) + c = 1/10 + (-7)/10

⇒ (a + b) + c = (1 – 7)/10 = -6/10 = -3/5

Also, b + c = 3/5 + (-7)/10 

⇒ b + c = (6 – 7)/10 = -1/10

 and, a + (b + c) = -1/2 + (-1)/10

⇒ a + (b + c) = (-5 – 1)/10= -6/10 = -3/5

Thus, (a + b) + c = a + (b + c) is true for Rational Numbers.

Similarly, for multiplication

a × b = -1/2 × 3/5 = -3/10

and, (a × b)× c = -3/10 × -7/10= -3× (-7)/100

⇒ (a × b)× c  = 21/100 

Also, b× c = 3/5 × (-7)/10 = -21/50

and, a × ( b × c ) = -1/2 × (-21)/50

⇒ a × ( b × c ) = 21/100 

Thus, (a× b)× c = a × ( b × c ) is true for Rational Numbers.

Identity Property

• Identity Property for Addition: For any rational number a, there exists a unique rational number 0 such that 0 + a = a = a + 0, where 0 is called the identity of the rational number under the addition operation. 
• Identity Property for Multiplication: For any rational number a, there exists a unique rational number 1 such that a × 1 = a = a × 1, where 1 is called the identity of the rational number under the multiplication operation.

Inverse Property

• Additive Inverse property: For any rational number a, there exists a unique rational number -a such that a + (-a) = (-a) +  a = 0, and -a is called the inverse of element a under the operation of addition. Also, 0 is the additive identity.
• Multiplicative Inverse property: For any rational number b, there exists a unique rational number 1/b such that b × 1 / b = 1 / b × b = 1, and 1/b is called the inverse of the element b under the multiplication operation. Here, 1 is the multiplicative identity.

Example:

For a = -11/23 
a + (-a) = -11/23 – (-11)/23
a + (-a) = -11/23 + 11/23 = (-11 + 11)/23 = 0

Similarly, (-a) + a = 0

Thus, 11/23 is the additive inverse of -11/23.

Now, for b = -17/29 
b × 1/b = -17/29 × -29/17 = 1

Similarly, 1 / b × b = 29/17 × -17/29 = 1

Thus, -29 / 17 is the multiplicative inverse of -17/23.

Distributive Property

Distributive property for any two operations holds if one distributes over the other. For example, multiplicative is distributive over addition for the collection of rational numbers, for any three rational numbers a, b, and c the distributive law of multiplication of addition is 

a × (b +c) = (a× b) + (a × c), and it is true for all the rational numbers. 

Example:

For rational number a, b, c i.e., a = -7 / 9, b = 11 / 18 and c = -14 / 27

Now, b + c = 11/18 + (-14)/27
⇒ b + c = 33/54 + (-28)/54 = (33 – 28)/54 = 5/54

and, a × ( b + c ) = -7/9 × 5/54
⇒ a × ( b + c ) = (-7 × 5)/(9 × 54) = -35/486 . . .(1)

Also, a × b = -7/9 × 11/18
⇒ a × b = (-7 × 11)/9 × 18 = -77/9 × 9 × 2

and a × c = (-7)/ 9 ×(-14)/27
⇒ a × c = (7 × 14)/9 × 9 × 3 = 98/9 × 9 × 3

Now, (a × b) + (a × c) = (-77/9 × 9 × 2 ) + ( 98/9 × 9 × 3)
⇒ (a × b) + (a × c) = (-77 × 3 + 98 × 2)/9 × 9 × 2 × 3
⇒ (a × b) + (a × c) = (-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.

Learn more about, Properties of a Real Number

Read More,

• Natural Numbers
• Whole Numbers
• Prime Numbers

Solved Examples 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 an 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.

Example 4: Simplify (2/3) × (6/8) ÷ (5/3).

Solution:

(2/3) × (6/8) ÷ (5/3) = (2/3) x (6/8) × (3/5)

= (2 × 6 × 3)/(3 × 8 × 5) 

= 36/120 = 3/10

Example 5: Arrange the following rational numbers in ascending order: 

1/3, -1/2, 2/5, and -3/4.

Solution:

The common denominator for 3, 2, 5, and 4 is 60. Thus

1/3 = 20/60,
-1/2 = -30/60,
2/5 = 24/60,
-3/4 = -45/60

With common denominator, rational number with greatest numerator is greatest.

⇒ -30/60 < -45/60 < 20/60 < 24/60
Thus, ascending order of given rational numbers is: -1/2 < -3/4 < 1/3 < 2/5

FAQs on Rational Numbers

Q1: 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…….

Q2: What are rational numbers? 

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.

Q3: 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.

Q4: 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…

Q5: 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.

Q6: 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.

Q7: 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.

Q8: 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.

Q9: How many rational numbers are between 1/2 and 1/3?

Answer:

There are infinitly many rational numbers between any two rational number, thus there are infinitlyl many rational numbers between 1/2 and 1/3, some of those numbers are 11/24, 7/24, 19/48, 13/72, 3/8 etc.

Q10: By which symbol rational number denoted?

Answer:

Rational Numbers are denoted by “Q” in the mathematics.

Q11: How many rational numbers are there?

Answer:

There are infinitly many rational numbers.