CBSE Class 8 Maths: Rational Numbers Ex 1.1 (NCERT)

Welcome, students! Let's dive into the first chapter of your Class 8 Maths journey: Rational Numbers. You've already learned about natural numbers, whole numbers, and integers. Rational numbers are the next step, representing any number that can be written as a fraction p/q, where p and q are integers and q is not zero.

NCERT Exercise 1.1 is all about understanding the 'rules' or 'properties' that rational numbers follow. Think of these properties as superpowers! They allow you to rearrange and regroup numbers to make calculations much simpler. In this lesson, we will master the Commutative, Associative, and Distributive properties. We'll also become experts at finding the additive and multiplicative inverses. Understanding these concepts is the key to solving the problems in Ex 1.1 with confidence.

Key Properties of Rational Numbers for Ex 1.1

To solve the questions in Exercise 1.1, you need to be very comfortable with three main properties. These properties help you simplify complex-looking expressions.

  1. Commutative Property: This property says that you can change the order of numbers in addition and multiplication without changing the result.
  • For Addition: a + b = b + a. For example, 1/2 + 1/4 = 3/4 and 1/4 + 1/2 = 3/4.
  • For Multiplication: a × b = b × a. For example, (1/2) × (3/5) = 3/10 and (3/5) × (1/2) = 3/10.
  • Note: This does NOT work for subtraction or division!
  1. Associative Property: This property is about grouping. When you add or multiply three or more numbers, you can group them in any way.
  • For Addition: (a + b) + c = a + (b + c). Example: (1/2 + 1/3) + 1/4 = 5/6 + 1/4 = 13/12. And 1/2 + (1/3 + 1/4) = 1/2 + 7/12 = 13/12.
  1. Distributive Property: This is the most important property for Ex 1.1! It links multiplication and addition.
  • a × (b + c) = (a × b) + (a × c). This lets you 'distribute' the number outside the bracket to the numbers inside. It's super useful for factoring out common terms to simplify problems.

Essential Terms: Identity and Inverse

Additive Identity
The number '0' is the additive identity for rational numbers. Adding 0 to any rational number doesn't change its value. For any rational number a, a + 0 = a.
Multiplicative Identity
The number '1' is the multiplicative identity. Multiplying any rational number by 1 doesn't change its value. For any rational number a, a × 1 = a.
Additive Inverse (Negative)
The additive inverse of a rational number is what you add to it to get 0. For a rational number a/b, its additive inverse is -a/b. Example: The additive inverse of 2/3 is -2/3 because 2/3 + (-2/3) = 0.
Multiplicative Inverse (Reciprocal)
The multiplicative inverse of a non-zero rational number is what you multiply it by to get 1. For a/b, its multiplicative inverse is b/a. Example: The multiplicative inverse of 5/8 is 8/5 because (5/8) × (8/5) = 1.

Step-by-Step: Solving Problems Using Properties

  1. Question: Simplify using appropriate properties: 2/5 × (-3/7) - 1/6 × 3/2 + 1/14 × 2/5 — This is a typical question from Exercise 1.1. Let's solve it together.
  2. Step 1: Identify and Rearrange Common Terms — Look for terms that share a common rational number. Here, the terms 2/5 × (-3/7) and 1/14 × 2/5 both have 2/5. Let's bring them together using the Commutative Property. Don't forget to move the sign with the term! 2/5 × (-3/7) + 1/14 × 2/5 - 1/6 × 3/2
  3. Step 2: Apply the Distributive Property — Now, factor out the common term 2/5 from the first two parts. This is the Distributive Property in reverse. 2/5 × [(-3/7) + 1/14] - 1/6 × 3/2
  4. Step 3: Solve the Bracket — First, solve the expression inside the square brackets. The LCM of 7 and 14 is 14. [(-3 × 2) / 14 + 1/14] = (-6 + 1)/14 = -5/14. Our expression now is: 2/5 × (-5/14) - 1/6 × 3/2
  5. Step 4: Simplify the Remaining Terms — Now, perform the multiplications. We can cancel out common factors. For the first part: (2/5) × (-5/14) becomes (1/1) × (-1/7) = -1/7. For the second part: 1/6 × 3/2 becomes 1/2 × 1/2 = 1/4. The expression is now: -1/7 - 1/4
  6. Step 5: Final Calculation — Find the LCM of 7 and 4, which is 28, and subtract the fractions. (-1 × 4)/28 - (1 × 7)/28 = (-4 - 7)/28 = -11/28. The final answer is -11/28.

Exam Tip: Common Mistakes to Avoid

1. Confusing Additive and Multiplicative Inverse: This is a very common error. Always ask yourself: 'Do I need to get 0 (additive inverse) or 1 (multiplicative inverse)?' The additive inverse of a/b is -a/b. The multiplicative inverse is b/a.

2. Forgetting the Sign: When you rearrange terms using the commutative property, the sign to the left of the number is part of that number! For example, in 5 - 3, the terms are +5 and -3. If you rearrange it, it becomes -3 + 5.

3. Incorrectly Applying Distributive Property: Remember, a × (b + c) = ab + ac. You cannot use this for other operations. For instance, a ÷ (b + c) is NOT equal to (a ÷ b) + (a ÷ c).

Practice Questions with Solutions

  • Q: Find the additive inverse of -7/19. A: Step 1: The additive inverse of a rational number 'a' is '-a'. Step 2: So, the additive inverse of -7/19 is -(-7/19). Step 3: Simplifying -(-7/19) gives 7/19. Final answer: 7/19
  • Q: Find the multiplicative inverse (reciprocal) of -13/19. A: Step 1: The multiplicative inverse of a non-zero rational number 'a/b' is 'b/a'. Step 2: So, the multiplicative inverse of -13/19 is 19/(-13). Step 3: This can be written as -19/13. Final answer: -19/13
  • Q: Using appropriate properties, evaluate: (3/7) (-2/5) + (3/7) (-3/5). A: Step 1: Identify the common factor (3/7) and apply the distributive property a(b+c) = ab + ac. Step 2: (3/7) [(-2/5) + (-3/5)]. Step 3: Simplify inside the brackets: (3/7) [(-2-3)/5] = (3/7) (-5/5). Step 4: Multiply: (3/7) * (-1) = -3/7. Final answer: -3/7
  • Q: Verify that -(-x) = x for x = -13/17. A: Step 1: Given x = -13/17. Step 2: First, find -x. The negative of -13/17 is -(-13/17) = 13/17. Step 3: Next, find -(-x). This means the negative of (13/17), which is -(13/17) = -13/17. Step 4: Compare the result with the original value of x. We have -(-x) = -13/17 and x = -13/17. Step 5: Since -(-x) = x, the property is verified. Final answer: Verified

Frequently Asked Questions

What is the difference between the associative and commutative properties?

The commutative property is about changing the *order* of two numbers (a+b = b+a). The associative property is about changing the *grouping* of three or more numbers (a+(b+c) = (a+b)+c) without changing their order.

Why does a rational number p/q require q to be non-zero?

Division by zero is undefined in mathematics. The fraction bar represents division, so if q were 0, you would be dividing p by 0, which is not possible. That's why the definition of a rational number specifically excludes a zero denominator.

Is 0 a rational number?

Yes, 0 is a rational number. It can be written in the form p/q, for example, as 0/1, 0/2, or 0/5. In all these cases, the denominator is not zero, so it fits the definition perfectly.

What is the multiplicative inverse of -1?

The multiplicative inverse of a number is its reciprocal. The reciprocal of -1 is 1/(-1), which simplifies to -1. So, -1 is its own multiplicative inverse, just like 1.