Class 7 Notes - Rational Numbers

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1. Introduction to Rational Numbers

Rational numbers are numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0. All integers, fractions, and terminating or repeating decimals are rational numbers.

Examples:

  • 2/3 is a rational number (p = 2, q = 3)
  • -5/7 is a rational number (p = -5, q = 7)
  • 4 can be written as 4/1, so it is rational
  • 0 can be written as 0/5, so it is rational

2. Properties of Rational Numbers

  • Closure: Rational numbers are closed under addition, subtraction, and multiplication.
  • Commutativity: Addition and multiplication of rational numbers are commutative.
  • Associativity: Addition and multiplication are associative.
  • Existence of Identity: 0 is the additive identity, 1 is the multiplicative identity.
  • Existence of Inverse: Every rational number except 0 has a multiplicative inverse.
  • Distributivity: Multiplication is distributive over addition.

3. Representation on the Number Line

Rational numbers can be represented on the number line. For example, to represent 3/4, divide the segment between 0 and 1 into 4 equal parts and count 3 parts from 0.

4. Standard Form of a Rational Number

A rational number is said to be in standard form if the denominator is positive and the numerator and denominator have no common factors except 1.

  • Example: -6/8 can be written as -3/4 in standard form.

5. Comparison of Rational Numbers

To compare rational numbers, convert them to like denominators and then compare the numerators.

  • Example: Compare 2/5 and 3/7. Find LCM of 5 and 7 (which is 35), convert both: 2/5 = 14/35, 3/7 = 15/35. So, 3/7 > 2/5.

6. Operations on Rational Numbers

  • Addition/Subtraction: Convert to like denominators, then add/subtract numerators.
  • Multiplication: Multiply numerators and denominators directly.
  • Division: Multiply by the reciprocal of the divisor.
Example:
  • 1/2 + 1/3 = (3+2)/6 = 5/6
  • 2/5 × 3/7 = 6/35
  • 4/9 ÷ 2/3 = 4/9 × 3/2 = 12/18 = 2/3

7. Word Problems

  1. What is the sum of 3/4 and -2/5?
  2. Subtract 5/6 from 1/2.
  3. Multiply -3/7 by 2/9.
  4. Divide 7/8 by -1/4.

8. Practice Exercises

  1. Express -12/16 in standard form.
  2. Compare 5/12 and 7/18.
  3. Find the multiplicative inverse of -3/8.
  4. Represent 2/5 on the number line.

9. Summary

  • Rational numbers are numbers that can be written as p/q, where q ≠ 0.
  • They include integers, fractions, and terminating/repeating decimals.
  • They have properties like closure, commutativity, associativity, and distributivity.
  • Operations can be performed using rules for addition, subtraction, multiplication, and division.