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Mathematics Class 8 - Rational Numbers Notes

Comprehensive study notes for Class 8 - Rational Numbers olympiad preparation

Rational Numbers

Welcome to the chapter on Rational Numbers for Class 8. In this chapter, you will learn what rational numbers are, how to represent them, and how to perform operations on them. By the end of this chapter, you will be able to solve problems involving rational numbers confidently!

Introduction

Rational numbers are numbers that can be written in the form p/q, where p and q are integers and q ≠ 0. All fractions, integers, and some decimals are rational numbers.

Examples of Rational Numbers

  • 2/3, -5/7, 4 (can be written as 4/1), 0 (can be written as 0/1)
  • -8/9, 15/2, 0.5 (can be written as 1/2)

Properties of Rational Numbers

  • Closure: Rational numbers are closed under addition, subtraction, multiplication, and division (except division by zero).
  • Commutative: Addition and multiplication of rational numbers are commutative.
  • Associative: Addition and multiplication of rational numbers are associative.
  • Existence of Identity: 0 is the additive identity, 1 is the multiplicative identity.
  • Existence of Inverse: For every rational number p/q (q ≠ 0), there is an additive inverse (-p/q) and a multiplicative inverse (q/p, p ≠ 0).

Standard Form of Rational Numbers

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.

Comparing Rational Numbers

To compare rational numbers, convert them to have a common denominator and then compare the numerators.

  • Example: Compare 2/5 and 3/7. Find a common denominator (35): 2/5 = 14/35, 3/7 = 15/35. So, 3/7 > 2/5.

Operations on Rational Numbers

  • Addition: Find a common denominator, add numerators.
  • Subtraction: Find a common denominator, subtract numerators.
  • Multiplication: Multiply numerators and denominators.
  • Division: Multiply by the reciprocal of the divisor.

Example:
Addition: 1/4 + 1/6 = (3/12) + (2/12) = 5/12
Multiplication: 2/3 × 3/5 = 6/15 = 2/5

Representation on Number Line

Rational numbers can be shown on a number line. For example, 1/2 is halfway between 0 and 1.

Fun Activity: Rational Number Hunt!

Write down five rational numbers you see in your textbook or daily life. Try to write them in standard form!

Summary

  • Rational numbers are numbers that can be written as p/q, where q ≠ 0.
  • They include integers, fractions, and some decimals.
  • You can add, subtract, multiply, and divide rational numbers.
  • Rational numbers can be compared and shown on a number line.

Practice Questions

  1. Write two rational numbers between 1/3 and 2/3.
  2. Express -12/16 in standard form.
  3. Add: 5/8 + 3/4
  4. Multiply: -2/5 × 3/7
  5. Divide: 4/9 ÷ 2/3

Challenge Yourself

  • Find three rational numbers between 0 and 1.
  • Show -3/4 on a number line.

Did You Know?

  • All integers are rational numbers!
  • The decimal 0.333... is a rational number because it can be written as 1/3.

Glossary

  • Rational Number: A number that can be written as p/q, where p and q are integers and q ≠ 0.
  • Standard Form: When the denominator is positive and numerator and denominator have no common factors except 1.
  • Reciprocal: Flipping the numerator and denominator (e.g., reciprocal of 2/3 is 3/2).

Answers to Practice Questions

  1. 1/2, 3/5 (any two between 1/3 and 2/3)
  2. -3/4
  3. 5/8 + 6/8 = 11/8
  4. -6/35
  5. 4/9 ÷ 2/3 = 4/9 × 3/2 = 12/18 = 2/3

Practice rational numbers every day to become confident in math!

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