Rational Numbers Class 7 Notes Maths Chapter 1

Introduction

In your study of numbers you began by counting objects around you. These counting numbers and their extensions form different sets that are useful for different purposes.

• Natural numbers: The numbers you use for counting are called counting numbers or natural numbers. They are 1, 2, 3, 4, and so on.
• Whole numbers: Natural numbers together with 0, that is 0, 1, 2, 3, ...
• Integers: Whole numbers together with their negatives, for example ..., -3, -2, -1, 0, 1, 2,...
• Fractions: Numbers written in the form numerator/denominator, where the numerator is an integer and the denominator is a non-zero positive integer.
• Need for rational numbers: Some situations require fractional values (for example, -3/4 km to indicate below sea level). This need leads us to enlarge the number system to include rational numbers.
• Focus of the chapter: We introduce rational numbers and study their standard form, properties and how to add, subtract, multiply, divide and compare them. We also learn to represent them on the number line and to find rational numbers between two given rational numbers.

What are Rational Numbers?

A number is called a rational number if it can be written in the form p/q, where p and q are integers and q ≠ 0.

• The denominator q cannot be zero.
• Whole numbers, integers, fractions and terminating or repeating decimals can be expressed in the form p/q, so they are all rational.
• Positive rational number: When p and q have the same sign. Positive rational numbers lie to the right of 0 on the number line.
• Negative rational number: When p and q have opposite signs. Negative rational numbers lie to the left of 0 on the number line.
• Equivalent rational numbers: If you multiply or divide both numerator and denominator of a rational number by the same non-zero integer, you obtain an equivalent rational number. For example 2/3 and 4/6 are equivalent.
• Standard form of a rational number: A rational number is in standard form when the numerator and denominator are coprime (their greatest common divisor is 1) and the denominator is positive. To convert to standard form, divide numerator and denominator by their highest common factor (HCF). If the denominator is negative, factor out the negative sign so that the denominator becomes positive.

Properties of Rational Numbers

(a) Closure Property

Definition: A set is said to be closed under an operation if applying that operation to members of the set always yields a member of the same set. Rational numbers are closed under addition, subtraction, multiplication and division (except division by zero).

(b) Commutativity Property

Definition: An operation is commutative if changing the order of the operands does not change the result. Addition and multiplication of rational numbers are commutative:

• For any rationals a and b, a + b = b + a.
• For any rationals a and b, a × b = b × a.

(c) Associativity Property 

Definition: An operation is associative if the way in which operands are grouped does not affect the result. Addition and multiplication of rational numbers are associative:

• For any rationals a, b and c, (a + b) + c = a + (b + c).
• For any rationals a, b and c, (a × b) × c = a × (b × c).

The Role of Zero (0):

Zero is the additive identity. Adding zero to any number does not change that number.

1. Addition with whole numbers: 2 + 0 = 0 + 2 = 2. In general, for any whole number a, a + 0 = 0 + a = a.
2. Addition with integers: -5 + 0 = 0 + (-5) = -5. In general, for any integer b, b + 0 = 0 + b = b.
3. 
4. Addition with rational numbers: Adding zero to a rational number leaves it unchanged. In general, for any rational number c, c + 0 = 0 + c = c.
5. Identity for addition: Zero is called the additive identity for whole numbers, integers and rational numbers.

The Role of One (1):

One is the multiplicative identity. Multiplying any number by 1 leaves it unchanged.

1. Multiplication with whole numbers: 5 × 1 = 1 × 5 = 5. In general, for any whole number a, a × 1 = 1 × a = a.
2. Multiplication with rational numbers: 27 × 1 = 1 × 27 = 27. In general, for any rational number r, r × 1 = 1 × r = r.
3. Multiplicative identity: 1 is the multiplicative identity for whole numbers, integers and rational numbers.

Negative of a Number (Additive Inverse):

For any integer or rational number x, there exists a number -x (called the additive inverse) such that x + (-x) = 0. Examples:

• 1 + (-1) = 0, so the additive inverse of 1 is -1.
• 2 + (-2) = 0, so the additive inverse of 2 is -2 and vice versa.
• In general, for any integer or rational a, a + (-a) = 0.

Reciprocal of a Number (Multiplicative Inverse):

The reciprocal or multiplicative inverse of a rational number  is another rational number  

For example,
Zero has no reciprocal because there is no rational number that, when multiplied by 0, results in 1.
Zero has no reciprocal because there is no rational number that, when multiplied by 0, results in 1.
In general, ifis the reciprocal of  , then

Operations on Rational Numbers

Addition of Rational Numbers

• Same Denominator: If two rational numbers have the same denominator, simply add their numerators. The denominator remains unchanged.
  Example: $\frac{3}{4}+\frac{1}{4}=\frac{4}{4}$ =1

• Different Denominators: Convert the rational numbers to have a common denominator before adding. Find equivalent fractions with the same denominator, then add the numerators. Example: $\frac{1}{2}+\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{5}{6}$

• Additive Inverses: Two rational numbers whose sum is zero are additive inverses of each other. Example: $\frac{2}{3}$ & $-\frac{2}{3}$

Subtraction of Rational Numbers

• Same Denominator: If the rational numbers have the same denominator, subtract their numerators. The denominator remains the same. Example: $\frac{5}{8}-\frac{2}{8}=\frac{3}{8}$

• Different Denominators: Convert to equivalent rational numbers with a common denominator before subtracting. Subtract the numerators and keep the common denominator. Example: $\frac{3}{4}-\frac{1}{6}=\frac{9}{12}-\frac{2}{12}=\frac{7}{12}$

Multiplication of Rational Numbers

• Multiply the numerators together and the denominators together. Maintain the signs correctly. Example: $\frac{-}{1}\times \frac{2}{5}=\frac{-}{2}$

• Reciprocals: Two rational numbers whose product is 1 are reciprocals of each other. Example: $\frac{-}{1}$ and $\frac{-}{\mathrm{3/}}$;  $\frac{-}{1}\times \frac{-}{3}=1$

• A rational number and its reciprocal will always have the same sign.

Division of Rational Numbers

• To divide by a rational number, multiply by its reciprocal. Example: $\frac{2}{5}÷\frac{3}{4}=\frac{2}{5}\times \frac{4}{3}=\frac{8}{15}$

Some Solved Examples on Above Concepts

Example 1: Simplify the rational number -16/(-24).

Ans:

Step 1: Identify the common factors of both numerator and denominator. In this case, the common factors are 2, 4, and 8.
Step 2: Divide both numerator and denominator by the greatest common factor, which is 8.
(-16) ÷ 8 = - 2
(-24) ÷ 8 = - 3
So, -16/(-24) simplifies to -2/-3 =x 2/3

Example 2: Add the rational numbers 2/5 and 3/10.

Ans:

Step 1: Find the least common denominator (LCD) of both denominators. In this case, the LCD is 10.
Step 2: Convert both fractions to equivalent fractions with the LCD as the new denominator.
2/5 = 4/10 (multiply both numerator and denominator by 2)
3/10 = 3/10 (no change needed)
Step 3: Add the equivalent fractions.
4/10 + 3/10 = (4 + 3) / 10 = 7/10
So, 2/5 + 3/10 = 7/10.

Example 3: Using appropriate properties, find: -2/3 × 3/5 + 5/2 - 3/5 × 1/6.

Solution:

Rearrange terms to group common factors by commutativity:

-2/3 × 3/5 - 3/5 × 1/6 + 5/2

Factor 3/5 from the first two products:

= 3/5 (-2/3 - 1/6) + 5/2

Compute the expression in parentheses. Find common denominator 6:

-2/3 = -4/6, so -4/6 - 1/6 = -5/6

Thus 3/5 × (-5/6) + 5/2

= (3 × -5) / (5 × 6) + 5/2

= -15/30 + 5/2

Reduce -15/30 = -1/2

= -1/2 + 5/2

= (-1 + 5)/2 = 4/2 = 2

Example 4: Use appropriate properties to find: 2/5 × (-3/7) - 1/6 × 3/2 + 1/14 × 2/5.

Solution:

Rearrange terms by commutativity:

2/5 × (-3/7) + 1/14 × 2/5 - 1/6 × 3/2

Combine the first two terms which share factor 2/5:

= 2/5 × (-3/7 + 1/14) - 3/12

Compute inside parentheses with denominator 14: -3/7 = -6/14, so -6/14 + 1/14 = -5/14

Thus = 2/5 × (-5/14) - 1/4

= (2 × -5) / (5 × 14) - 1/4

= -10/70 - 1/4

Reduce -10/70 = -1/7

= -1/7 - 1/4

Find common denominator 28: (-4)/28 - 7/28 = (-11)/28

Therefore, the value is -11/28.

Example 5: Verify that -(-x) = x for:

(i) x = 11/15

(ii) x = -13/17

Solution:

(i) x = 11/15
We have, x = 11/15
The additive inverse of x is – x (as x + (-x) = 0).
Then, the additive inverse of 11/15 is – 11/15 (as 11/15 + (-11/15) = 0).
The same equality, 11/15 + (-11/15) = 0, shows that the additive inverse of -11/15 is 11/15.
Or, – (-11/15) = 11/15
i.e., -(-x) = x

(ii) -13/17
We have, x = -13/17
The additive inverse of x is – x (as x + (-x) = 0).
Then, the additive inverse of -13/17 is 13/17 (as 13/17 + (-13/17) = 0).
The same equality (-13/17 + 13/17) = 0, shows that the additive inverse of 13/17 is -13/17.
Or, – (13/17) = -13/17,
i.e., -(-x) = x

Comparing Two Rational Numbers

To compare rational numbers:

• If two positive rational numbers have the same denominator, the number with the larger numerator is larger. For example, between 3/7 and 5/7, 5/7 is larger.
• A positive rational number is always greater than a negative rational number.
• For negative rational numbers with the same denominator, compare numerators ignoring the negative sign: the one with the larger numerator (in absolute terms) is actually the smaller number. For example, -5/7 is less than -3/7.
• To compare rational numbers with different denominators, convert them to equivalent fractions with the same denominator (usually the LCM) and then compare numerators.
• Between any two distinct rational numbers there are infinitely many rational numbers.

Representation of Rational Numbers on the Number Line

Whole numbers, natural numbers and integers are represented on a number line by marking equal intervals and placing numbers accordingly. Rational numbers can also be represented on the number line: positive rationals are to the right of 0 and negative rationals are to the left of 0.

To represent a rational number p/q on the number line, divide each unit interval into q equal parts and count p such parts to the right of 0 if p is positive, or to the left if p is negative.

Between any two rational numbers there are infinitely many rational numbers; they can be found by taking averages or by finding finer equivalent fractions between them.

Rational Numbers between Two Rational Numbers

To find rational numbers between two given rational numbers, you can either make their denominators equal and list intermediate numerators, or repeatedly take the mean (average) of two numbers to get more numbers between them.

Example: Find three rational numbers between 1/4 and 1/2.

Ans:

Find the mean of 1/4 and 1/2.

Mean = (1/4 + 1/2) / 2

= (1/4 + 2/4) / 2

= (3/4) / 2 = 3/8

Now find the mean of 1/4 and 3/8 to get one rational number between them:

Mean = (1/4 + 3/8) / 2

= (2/8 + 3/8) / 2 = (5/8) / 2 = 5/16

Find the mean of 3/8 and 1/2 to get another rational number between them:

Mean = (3/8 + 1/2) / 2

= (3/8 + 4/8) / 2 = (7/8) / 2 = 7/16

Hence three rational numbers between 1/4 and 1/2 are 5/16, 3/8 and 7/16.

Example: Find rational numbers between 3/5 and 3/7.

Make the denominators the same. LCM of 5 and 7 is 35.

3/5 = 21/35 and 3/7 = 15/35.

Now any fractions with denominator 35 and numerators between 15 and 21 are rational numbers between them, for example 16/35, 17/35, 18/35, 19/35, 20/35.

If desired, multiply numerator and denominator of 3/5 and 3/7 by the same integer to get equivalent fractions with larger denominators and thus obtain more rational numbers between them.

Remark: Between any two distinct rational numbers there are infinitely many rational numbers. They need not be integers.

For further practice and visual explanation you may watch the following video: