Rational Numbers

Rational numbers are a fundamental concept in mathematics and have many uses in our everyday lives. They help us do basic math like addition and subtraction. This chapter will explain what rational numbers are and how they are different from other types of numbers.

Rational Numbers

A rational number is any number that can be written in the form of p/q where both "p" and "q” are integers and "q" is not equal to zero.

Examples, The fraction 7/3 is a rational number. Here, "p" is 7, "q" is 3 and "q" can't be zero.

The natural number 5 is a rational number. 5 can be written as a fraction 5/1. Here, "p" is 5, "q" is 1 and "q" can't be zero.

All natural numbers, whole numbers, integers, fractions and decimals are rational numbers.

Properties of Addition for Rational Numbers

Property 1: Closure Property of Addition

When you add two rational numbers, the result is always another rational number. 

If you have two fractions ab and cd and you add them together (ab + cd), the result will also be a rational number. This is known as the "closure property of addition".

Example:  5/3 and −2/9 are any two rational numbers.

Property 2:  Commutative Property of Addition

When you can add two rational numbers in any order,  the result is always the same.

If you have two fractions like ab and cd, their addition in any order gives the same result.

This is known as the "commutative property of addition".

Example: −7/5 and 4/13 are any two rational numbers.

Property 3: Associative Property of Addition

When we add three or more rational numbers they can be grouped in any order.

If a/b , c/d and e/f are any three rational numbers, then

This is known as the "associative property of addition".

Example: −3/4, 5/2 and 3/7 are any three rational numbers.

Property 4: Distributive Property of Addition

You can either multiply the number by each rational number and then add the results or you can add the rational numbers first and then multiply the sum by the number and you will get the same result.

If a/b, c/d , and ef are any three rational numbers, then

This is known as the "distributive property of addition".

Example: 5/2, 2/7 and −2/3 are any three rational numbers.

Property 5: Additive Identity of Rational Numbers 

When you add any rational number to 0, you get the same rational number. We call 0 the "identity element" for adding rational numbers because it doesn't change the value of the rational number.

If a/b is any rational number, then a/b + 0 = a/b.

This is known as the "additive identity of rational numbers".

Example: If 2/3 is any rational number, then

2/3 + 0 = 2/3

Properties of Subtraction for Rational Numbers

Property 1: Closure Property of Subtraction

The result of subtracting one rational number from another is always another rational number.

If you have two rational numbers like a/b and c/d and you subtract them (a/b − c/d), the result will also be a rational number.
This is known as the "closure property of subtraction".

Example:

Property 2: Non-Commutative Property of  Subtraction

When one rational number is subtracted from another, the result is not the same after subtraction if their order has changed.

If you have two rational numbers like a/b and c/d, then a/b − c/d ≠ c/d − a/b.

This is called the "non-commutative property of subtraction".

Example:

Property 3: Non-Associative Property of  Subtraction

Subtraction of rational numbers is not associative.

If a/b , c/d and e/f are any three rational numbers, then

This is called the "non-associative property of subtraction".

Example:

Property 4: Distributive Property of Subtraction

Multiplying rational numbers and then subtracting is the same as first subtracting the rational numbers and then multiplying. The order in which you do the subtraction and multiplication doesn't change the final result when working with rational numbers.

If ab , cd and ef are any three rational numbers, then

This is called the "distributive property of multiplication over subtraction".

Example:

Negative of a Rational Number

(− a/b) is the "negative" of a rational number a/b. It is also an "additive inverse" of a rational number a/b.

If a rational number ab is added to its additive inverse, then

Properties of Multiplication for Rational Numbers

Property 1: Closure Property under Multiplication

When you multiply two rational numbers, the result is always another rational number.

If you have two fractions like ab and c/d, and you multiply them (a/b × c/d), the result will also be a rational number.

This is known as the "closure property under multiplication".

Example: 

Property 2:  Commutative Property of Multiplication

When you multiply two rational numbers in any order, the result is always the same.

If you have two fractions like a/b and c/d, their multiplication in any order gives the same result.

This rule is called the "commutative property of multiplication".

Example: 

Property 3: Associative Property of Multiplication

When we multiply three or more rational numbers they can be grouped in any order.

If a/b , c/d and e/f are any three rational numbers, then

This is known as the "associative property of multiplication".

Example: 

Property 4: Multiplicative Identity of Rational Numbers

When you multiply any rational number by 1, you get the same rational number. This is because 1 is known as the multiplicative identity.

If a/b is any rational number, then (a/b) × 1 = a/b.

This is known as the "multiplicative identity of rational numbers".

Example: Multiplicative identity of a rational number −5/7 is:

⇒ −5/7 × 1 = −5/7

Property 5: Zero Property of Multiplication

If you multiply any rational number by 0, the result is always 0.

If a/b is any rational number, then a/b × 0 = 0.

This is known as the zero property of multiplication.

Example: Zero property of multiplication of a rational number −57 is:

⇒ ^-5⁄₇ × 0 = 0

Reciprocal of a Rational Number

If you have a rational number ab, the reciprocal of that number is ba.

When you multiply a rational number by its reciprocal, you get the result 1.

The reciprocal of a rational number is also known as its multiplicative inverse. However, it's important to note that zero does not have a reciprocal.

Example:

Properties of Division for Rational Numbers

Property 1: Closure Property of Division

Rational numbers can be divided by other rational numbers, and the result will still be a rational number, except when dividing by zero.

If you have two rational numbers a/b and c/d, then the result of (a/b) ÷ (c/d) is still a rational number.

This is known as the "closure property of division".

Example:

Property 2: If you have a non-zero rational number ab, and you divide it by itself (a/b ÷ a/b) , the result is always 1. 

Example: If a non-zero rational number −5/7 is divided by itself, then

Property 3: When you divide a rational number by 1, you get the same rational number.

If you have a rational number a/b and you divide it by 1, you will simply get a/b.

Example: If a non-zero rational number −5/7 is divided by 1, then

Property 4: When you divide zero by any non-zero rational number, the result is always zero.

If you have a rational number a/b and you divide 0 by it, you will simply get 0.

Example: If divide 0 by a non-zero rational number −5/7, then

 0 ÷ ^-5⁄₇ = 0

Rational Number between two Rational Numbers

If x and y are any two rational numbers, then (x + y)/2 is a rational number lying between x and y.

How to Find Rational Numbers Between Two Rational Numbers?

The rational numbers between two rational numbers are used when

→ Denominators are the same.

→ Denominators are different.

When Denominators are Same

When you have two rational numbers with the same denominators and want to find the rational numbers between them, follow these steps:

Step 1: Look at the numerators of the rational numbers you have.
Step 2: Calculate how much the numerators differ from each other.
Step 3: If the difference between the numerators is large, you can find more rational numbers by arranging them in increasing order based on the numerator.
Step 4: If the difference between the numerators is small, you need more rational numbers, you can multiply both the numerator and denominator of the given rational numbers by multiples of 10 to find additional rational numbers.

Example:  What are 5 rational numbers between 3/7 and 5/7?

Explanation: 3/7 and 5/7 are two rational numbers with the same denominator 7.

Follow these steps:

Step 1: First compare the numerators.

3 < 5 (3 is less than 5)

Step 2: Since there is a very small difference between the numerators 3 and 5, we find the numbers that lie between them which is only 4.

Step 3: If you need more numbers in between, you can multiply both the original rational numbers by 10.

Step 4: Now there is a very large difference between the numerators 30 and 50. There are 20 numbers between the numerators 30 and 50. So, you can pick any 5 rational numbers in between.

The five rational numbers lying between 3/7 and 5/7 with the same denominator are:

Note: There are 20 numbers between the numerators 30 and 50. So, you may also choose the other 5 rational numbers in between.

When Denominators are Different

When you have two rational numbers with different denominators and want to find the rational numbers between them, follow these steps:

Step 1: Start by finding the Least Common Multiple (LCM) of the two rational numbers' denominators.

Step 2: Next, adjust the two rational numbers by multiplying and dividing them, so their denominators become the LCM you found in step 1.

Step 3: Once both numbers have the same denominator (which is now the LCM), you can follow the same process as explained before for rational numbers with the same denominators.

Example:  What are 5 rational numbers between 3/5 and 5/7?

Explanation: 3/5 and 5/7 are two rational numbers with different denominators (5 and 7).

Follow these steps:

Step 1: First, find the Least Common Multiple (LCM) of the denominators (5 and 7).

LCM of denominators (5 and 7) = 5 × 7 = 35

Step 2: Make the denominators the same.

Step 3: The denominators are the same (both are 35). Since, there is a very small difference between the numerators 21 and 25, we find the numbers that lie between them which is 22, 23 and 24.

Step 4: If you need more numbers in between, you can multiply both the original rational numbers by 10.

Now there is a very large difference between the numerators 210 and 250. There are 40 numbers between the numerators 210 and 250. So, you can pick any 5 rational numbers in between.

The five rational numbers lying between 3/5 and 5/7 with the different denominator are:

Note: There are 40 numbers between the numerators 210 and 250. So, you may also choose the other 5 rational numbers in between.

Representation of Rational Numbers on the Number Line

Rational numbers can be shown on a number line. This number line has its centre at zero which is also called the origin (O). Rational numbers that are positive are placed to the right of zero and those that are negative are placed to the left of zero on the number line.

Examples:

→ Representation of a rational number 4/5 on number line.

4/5 is a positive rational number, so it will be to the right of zero on the number line.
To represent it, first divide the part of the number line between 0 and 1 into 5 equal pieces since the denominator of the rational number is 5. Mark the part on the number line with a value equal to the numerator which is 4. This marked point which is denoted as "P" shows the position of the rational number 45 on the number line.

→ Representation of a rational number 17/5 on number line.

→ Representation of a rational number −5/6 on number line.