Class 8 Notes - Rational Numbers

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In mathematics, a rational number is any number that can be expressed in the form p/q, where p and q are integers, and q ≠ 0. This definition encompasses a wide range of numbers, including fractions, integers, and finite decimals. Understanding rational numbers is fundamental, as they form the basis for various mathematical concepts and real-world applications.

Properties of Rational Numbers

Rational numbers exhibit several key properties that are essential for performing arithmetic operations and understanding their behavior within the number system:

1. Closure Property

The set of rational numbers is closed under addition, subtraction, and multiplication. This means that performing these operations on any two rational numbers will always yield another rational number.

Example:
  • Addition: (2/3) + (3/4) = 17/12 (a rational number)
  • Subtraction: (5/6) - (1/2) = 1/3 (a rational number)
  • Multiplication: (7/8) × (4/5) = 28/40 = 7/10 (a rational number)

2. Commutative Property

Addition and multiplication of rational numbers are commutative; the order of the numbers does not affect the result.

Example:
  • Addition: (1/2) + (2/3) = (2/3) + (1/2)
  • Multiplication: (3/5) × (4/7) = (4/7) × (3/5)

3. Associative Property

Addition and multiplication of rational numbers are associative; the grouping of numbers does not affect the result.

Example:
  • Addition: [(1/4) + (2/5)] + (3/7) = (1/4) + [(2/5) + (3/7)]
  • Multiplication: [(2/3) × (3/4)] × (4/5) = (2/3) × [(3/4) × (4/5)]

4. Distributive Property

Multiplication is distributive over addition for rational numbers.

Example:

(1/2) × [(2/3) + (3/4)] = [(1/2) × (2/3)] + [(1/2) × (3/4)]

5. Identity Elements

The additive identity for rational numbers is 0, and the multiplicative identity is 1.

Example:
  • Addition: (3/7) + 0 = 3/7
  • Multiplication: (5/9) × 1 = 5/9

6. Inverse Elements

Every rational number has an additive inverse (negative) and a multiplicative inverse (reciprocal), except for zero in the case of multiplication.

Example:
  • Additive Inverse: The additive inverse of (3/5) is -(3/5) or (-3/5), since (3/5) + (-3/5) = 0.
  • Multiplicative Inverse: The multiplicative inverse of (4/7) is (7/4), since (4/7) × (7/4) = 1.

Representation of Rational Numbers on the Number Line

Rational numbers can be represented on the number line. To locate a rational number p/q:

  1. Divide the segment between integers into q equal parts.
  2. Count p parts from zero if p is positive, or from zero in the opposite direction if p is negative.
Example:

To represent (3/4) on the number line:

  • Divide the segment between 0 and 1 into 4 equal parts.
  • Count 3 parts to the right of 0.
  • The point reached represents (3/4).

Rational Numbers Between Two Rational Numbers

Between any two rational numbers, there are infinitely many rational numbers. To find a rational number between two given rational numbers, we can take their average or identify fractions with common denominators.

Example:

Find a rational number between (1/3) and (1/2):

  • Take their average: [(1/3) + (1/2)] ÷ 2
  • Calculate: [(2/6) + (3/6)] ÷ 2 = (5/6) ÷ 2 = (5/12)
  • Answer: (5/12) lies between (1/3) and (1/2).

Operations on Rational Numbers

Addition and Subtraction

To add or subtract rational numbers, first make their denominators the same, and then perform the operation on the numerators.

Example:
  • (1/4) + (1/6) = [(3/12) + (2/12)] = (5/12)
  • (5/8) - (3/4) = [(5/8) - (6/8)] = (-1/8)

Multiplication

To multiply rational numbers, multiply the numerators and the denominators directly.

Example:

(2/5) × (3/4) = (2 × 3) / (5 × 4) = 6/20 = 3/10

Division

To divide one rational number by another, multiply the first number by the reciprocal of the second.

Example:

(3/7) ÷ (2/5) = (3/7) × (5/2) = (3 × 5) / (7 × 2) = 15/14

Practice Questions

  1. Find two rational numbers between (2/5) and (3/5).
  2. Simplify: (7/12) + (5/18).
  3. Multiply: (4/9) × (3/8).
  4. Divide: (5/6) ÷ (2/3).
  5. Represent (7/10) on the number line.

Conclusion

Understanding rational numbers is crucial as they form the foundation for more advanced mathematical concepts. Mastery of their properties, operations, and representations will enhance problem-solving skills and mathematical reasoning. Regular practice and application in real-life contexts will solidify this knowledge for students.