Class 7 Maths Rational Numbers Notes

Introduction:

The chapter “Rational Numbers” in Class 7 Maths introduces students to the concept of rational numbers and their properties. Rational numbers are numbers that can be expressed as a ratio of two integers. This chapter focuses on understanding rational numbers, performing operations on them, and recognizing their properties. Students will also learn how to represent rational numbers on a number line and how to compare and order them.

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Key Concepts Covered:

1. What is a Rational Number?

• A rational number is any number that can be expressed as the ratio of two integers. It is written in the form $\frac{p}{q}$qp​, where:
  • $p$p and $q$q are integers.
  • $q \neq 0$q=0 (the denominator cannot be zero).
• Example: $\frac{3}{4}$43​, $-\frac{5}{2}$−25​, and $0$0 are all rational numbers because they can be written as ratios of integers.

2. Types of Rational Numbers:

Rational numbers can be:

• Positive Rational Numbers: When both the numerator and denominator are positive.
• Negative Rational Numbers: When one of the numerator or denominator is negative.
• Zero: Zero is a rational number because it can be written as $\frac{0}{1}$10​.

3. Representation of Rational Numbers on the Number Line:

• Rational numbers can be represented on the number line just like integers and fractions.
• To plot a rational number, you can divide the number line into equal parts corresponding to the denominator and count the appropriate number of units from zero.

Example:
To represent $\frac{3}{4}$43​ on the number line, divide the segment between 0 and 1 into 4 equal parts and mark 3 of these parts.

4. Operations on Rational Numbers:

Rational numbers can be added, subtracted, multiplied, and divided just like integers. Here are the rules for each operation:

Addition and Subtraction:

• Same Denominator: To add or subtract rational numbers with the same denominator, add or subtract the numerators and keep the denominator the same.
• Example: $$\frac{3}{5} + \frac{2}{5} = \frac{3+2}{5} = \frac{5}{5} = 1$$53​+52​=53+2​=55​=1
• Different Denominators: First find the LCM of the denominators, convert the fractions to have the same denominator, and then add or subtract the numerators.

Multiplication:

• To multiply two rational numbers, multiply the numerators and multiply the denominators.
• Example: $$\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$$32​×43​=3×42×3​=126​=21​

Division:

• To divide one rational number by another, multiply the first number by the reciprocal of the second number.
• Example: $$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4} = \frac{10}{12} = \frac{5}{6}$$32​÷54​=32​×45​=3×42×5​=1210​=65​

5. Properties of Rational Numbers:

Rational numbers have similar properties to integers. The key properties include:

Closure Property:

• Addition and Multiplication of rational numbers are closed, meaning the result of adding or multiplying two rational numbers will always be a rational number.

Commutative Property:

• Addition and Multiplication of rational numbers are commutative, meaning: $$a + b = b + a \quad \text{and} \quad a \times b = b \times a$$a+b=b+aanda×b=b×a

Associative Property:

• Addition and Multiplication of rational numbers are associative, meaning: $$(a + b) + c = a + (b + c) \quad \text{and} \quad (a \times b) \times c = a \times (b \times c)$$(a+b)+c=a+(b+c)and(a×b)×c=a×(b×c)

Additive Inverse:

• Every rational number has an additive inverse, meaning for any rational number $\frac{p}{q}$qp​, its additive inverse is $-\frac{p}{q}$−qp​, and: $$\frac{p}{q} + \left(-\frac{p}{q}\right) = 0$$qp​+(−qp​)=0

Multiplicative Inverse (Reciprocal):

• Every non-zero rational number has a multiplicative inverse, meaning for any rational number $\frac{p}{q}$qp​, its reciprocal is $\frac{q}{p}$pq​, and: $$\frac{p}{q} \times \frac{q}{p} = 1$$qp​×pq​=1

6. Comparison of Rational Numbers:

• Rational numbers can be compared by converting them to a common denominator or by converting them into decimal form.
• Example:
  To compare $\frac{3}{5}$53​ and $\frac{4}{7}$74​, find a common denominator and compare the numerators, or convert both fractions to decimals (0.6 and 0.5714, respectively).

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Important Questions with Answers:

1. What is a rational number?
   • Answer: A rational number is a number that can be expressed as the ratio of two integers, i.e., $\frac{p}{q}$qp​, where $p$p and $q$q are integers and $q \neq 0$q=0.
2. How do you represent rational numbers on a number line?
   • Answer: Rational numbers can be represented by dividing the number line into equal parts according to the denominator and locating the appropriate point.
3. How do you add rational numbers with the same denominator?
   • Answer: To add rational numbers with the same denominator, add the numerators and keep the denominator the same.
4. What is the Pythagorean Theorem?
   • Answer: The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
5. How do you multiply rational numbers?
   • Answer: To multiply rational numbers, multiply the numerators and multiply the denominators.
6. What is the additive inverse of 34\frac{3}{4}43​?
   • Answer: The additive inverse of $\frac{3}{4}$43​ is $-\frac{3}{4}$−43​.
7. What is the reciprocal of 56\frac{5}{6}65​?
   • Answer: The reciprocal of $\frac{5}{6}$65​ is $\frac{6}{5}$56​.