Rational Numbers: CBSE Class 8 Maths Chapter 1

Welcome to the world of Rational Numbers! You've already met numbers like whole numbers, integers, and fractions in earlier classes. Rational numbers are like a bigger family that includes all of them. Think of it this way: if you can write a number as a fraction, it's a rational number! In this chapter, we'll explore what makes these numbers special. We will learn about their properties, how to add, subtract, multiply, and divide them, and even how to find new rational numbers that sit between two existing ones. Understanding rational numbers is a key step in your maths journey, helping you solve more complex problems in algebra and geometry later on. By the end of this chapter, you will be a pro at handling these useful numbers!

What Are Rational Numbers?

A rational number is any number that can be expressed in the form p/q, where 'p' and 'q' are integers and 'q' is not equal to zero (q ≠ 0).

Let's break this down:

  • 'p' is the numerator. It can be any integer (positive, negative, or zero).
  • 'q' is the denominator. It can be any integer except zero. Why? Because dividing by zero is undefined in mathematics!

Examples of rational numbers include:

  • 1/2: Here, p=1 and q=2.
  • -3/4: Here, p=-3 and q=4.
  • 5: This is a rational number because it can be written as 5/1. So, all integers are rational numbers!
  • 0: This is also a rational number because it can be written as 0/1, 0/2, or 0/any non-zero integer.
  • -2: This can be written as -2/1.

So, the family of rational numbers is very large! It includes all the integers (..., -3, -2, -1, 0, 1, 2, 3, ...) and all the fractions (like 1/2, 5/8, -4/7) you've worked with before.

Key Properties of Rational Numbers

Closure Property
Rational numbers are closed under addition, subtraction, and multiplication. This means if you add, subtract, or multiply any two rational numbers, the result is always another rational number.
Commutative Property
Addition and multiplication are commutative for rational numbers. This means you can swap the order of the numbers without changing the result. For any rational numbers a and b: a + b = b + a and a × b = b × a.
Associative Property
Addition and multiplication are associative for rational numbers. This means you can regroup the numbers without changing the result. For any rational numbers a, b, and c: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
Additive Identity
The number 0 is the additive identity for rational numbers. Adding 0 to any rational number does not change its value (a + 0 = a).
Multiplicative Identity
The number 1 is the multiplicative identity for rational numbers. Multiplying any rational number by 1 does not change its value (a × 1 = a).
Distributive Property
Multiplication distributes over addition. For any rational numbers a, b, and c: a × (b + c) = (a × b) + (a × c).

How to Find Rational Numbers Between Two Numbers

  1. Step 1: Make the Denominators Equal — To find rational numbers between two given numbers, like 1/4 and 1/2, first find a common denominator. The LCM (Least Common Multiple) of 4 and 2 is 4. So, 1/2 can be written as 2/4. Our numbers are now 1/4 and 2/4.
  2. Step 2: Increase the Gap Between Numerators — Between 1/4 and 2/4, it's hard to see any numbers. To create more 'space', multiply the numerator and denominator of both numbers by the same value, like 10. So, 1/4 becomes 10/40 and 2/4 becomes 20/40.
  3. Step 3: List the Numbers in Between — Now you have 10/40 and 20/40. It's easy to list rational numbers between them! For example: 11/40, 12/40, 13/40, 14/40, and 15/40 are all rational numbers between the original 1/4 and 1/2. You can find infinitely many numbers this way.

Exam Tip: Representing on a Number Line

A very common question is to represent a rational number like 5/3 on the number line. Many students make a mistake here. Remember, the denominator '3' tells you how many equal parts to divide EACH unit into. So, you divide the space between 0 and 1 into three parts, the space between 1 and 2 into three parts, and so on. Then, the numerator '5' tells you how many of these small parts to count from 0. You will count 5 steps: 1/3, 2/3, 3/3 (which is 1), 4/3, and finally 5/3. So 5/3 lies between 1 and 2.

Practice Questions with Solutions

  • Q: Evaluate: (5/9) + (-7/12) A: Step 1: Find the LCM of the denominators 9 and 12, which is 36. Step 2: Convert fractions to equivalent fractions with denominator 36. (5/9) = (54)/(94) = 20/36 (-7/12) = (-73)/(123) = -21/36 Step 3: Add the equivalent fractions. 20/36 + (-21/36) = (20 - 21)/36 = -1/36 Final answer: -1/36
  • Q: Multiply: (-3/5) (15/(-9)) A: Step 1: Simplify the second fraction: 15/(-9) = - (15/9) = - (5/3). Step 2: Multiply the numerators and denominators. (-3/5) (-5/3) = ((-3) (-5)) / (5 3) Step 3: Perform the multiplication. = 15 / 15 = 1 Final answer: 1
  • Q: Find two rational numbers between 1/3 and 1/2. A: Step 1: Find a common denominator for 1/3 and 1/2. LCM(3, 2) = 6. 1/3 = 2/6 1/2 = 3/6 Step 2: To find numbers between them, multiply numerator and denominator by a factor, e.g., 3. 2/6 = (23)/(63) = 6/18 3/6 = (33)/(63) = 9/18 Step 3: Two rational numbers between 6/18 and 9/18 are 7/18 and 8/18. Final answer: 7/18, 8/18 (or 4/9)
  • Q: Verify the distributive property of multiplication over addition for rational numbers: (2/3) ((1/2) + (3/4)). A: Step 1: Calculate LHS: (2/3) ((1/2) + (3/4)) (1/2) + (3/4) = (2/4) + (3/4) = 5/4 (2/3) (5/4) = (25) / (34) = 10/12 = 5/6 Step 2: Calculate RHS: (2/3)(1/2) + (2/3)(3/4) (2/3)(1/2) = 2/6 = 1/3 (2/3)*(3/4) = 6/12 = 1/2 Step 3: Add the results of RHS. 1/3 + 1/2 = (2/6) + (3/6) = 5/6 Step 4: Compare LHS and RHS. Both are 5/6. Final answer: LHS = RHS = 5/6. The property is verified.

Frequently Asked Questions

Is zero a rational number?

Yes, zero is a rational number. It can be written in the p/q form, for example, as 0/1, 0/5, or 0/(-10). As long as the denominator is not zero, the value is 0.

What is the difference between a fraction and a rational number?

Fractions usually refer to parts of a whole and are typically positive (like 1/2, 3/4). Rational numbers are a broader set that includes positive fractions, negative fractions (like -3/4), and all integers.

How many rational numbers exist between any two rational numbers?

There are infinitely many rational numbers between any two distinct rational numbers. No matter how close two rational numbers are, you can always find another one in between them.

Is Pi (π) a rational number?

No, Pi (π) is an irrational number. It cannot be written as a simple fraction p/q. Its decimal representation goes on forever without repeating.