RS Aggarwal Class 9: Exercise 1B Solutions Explained

Aug 26, 2025 by Olex Johnson 53 views

RS Aggarwal Class 9 Chapter 1 Number System Exercise 1B Solutions

Introduction to Number Systems

Number systems are fundamental to mathematics, providing a framework for representing and manipulating numerical values. In Class 9, RS Aggarwal's textbook introduces students to various types of numbers, including rational and irrational numbers, and how they are represented on the number line. Exercise 1B specifically focuses on understanding and working with these concepts. Mastering number systems is crucial as it lays the groundwork for more advanced mathematical topics. To successfully tackle this chapter, you'll need to familiarize yourself with key concepts such as rationalizing denominators, identifying rational and irrational numbers, and performing operations on these numbers. By understanding these principles, you will be well-equipped to solve the problems in Exercise 1B and strengthen your overall understanding of number systems.

Understanding number systems involves grasping the properties that define different types of numbers. Rational numbers, for instance, can be expressed as a fraction ${ \frac{p}{q} }$ , where ${ p }$ and ${ q }$ are integers and ${ q \neq 0 }$ . Examples include 2, -3, ${ \frac{1}{2} }$ , and 0.5. On the other hand, irrational numbers cannot be expressed in this form; they have non-repeating, non-terminating decimal expansions. Common examples of irrational numbers are ${ \sqrt{2} }$ , ${ \pi }$ , and ${ e }$ . Recognizing the difference between rational and irrational numbers is essential for Exercise 1B, as many problems require you to classify numbers and perform operations with them.

Rationalizing the denominator is another critical technique covered in this chapter. This process involves eliminating any square roots (or other radicals) from the denominator of a fraction. For example, to rationalize the denominator of ${ \frac{1}{\sqrt{2}} }$ , you would multiply both the numerator and denominator by ${ \sqrt{2} }$ , resulting in ${ \frac{\sqrt{2}}{2} }$ . This technique is particularly useful when simplifying expressions and comparing numbers. In Exercise 1B, you'll encounter several problems that require you to rationalize denominators to simplify expressions or determine the nature of numbers.

Moreover, being able to represent numbers on the number line is an essential skill. The number line provides a visual representation of numbers, allowing you to compare their values and understand their relative positions. Rational numbers can be precisely located on the number line, while irrational numbers can be approximated to a certain degree of accuracy. Understanding how to plot numbers on the number line enhances your understanding of number systems and is a valuable tool for solving problems in Exercise 1B. By mastering these fundamental concepts, you will be well-prepared to tackle the challenges presented in RS Aggarwal Class 9 Chapter 1 Exercise 1B.

Solving Problems from Exercise 1B

In this section, we will delve into solving specific problems from RS Aggarwal Class 9 Exercise 1B. Each problem will be broken down step-by-step to provide a clear understanding of the solution process. By examining these solutions, you will gain insights into how to apply the concepts discussed earlier. Let's proceed with a detailed look at some representative problems.

Problem 1: Show that ${ 5 + \sqrt{2} }$ is an irrational number.

To prove that ${ 5 + \sqrt{2} }$ is irrational, we use the method of contradiction. Assume that ${ 5 + \sqrt{2} }$ is rational. This means that ${ 5 + \sqrt{2} = \frac{p}{q} }$ , where ${ p }$ and ${ q }$ are integers and ${ q \neq 0 }$ .

Rearrange the equation to isolate ${ \sqrt{2} }$ : ${ \sqrt{2} = \frac{p}{q} - 5 }$ .
Simplify the right side: ${ \sqrt{2} = \frac{p - 5q}{q} }$ .
Since ${ p }$ and ${ q }$ are integers, ${ \frac{p - 5q}{q} }$ is a rational number.
However, we know that ${ \sqrt{2} }$ is an irrational number. This leads to a contradiction, as we have an irrational number equal to a rational number.
Therefore, our initial assumption that ${ 5 + \sqrt{2} }$ is rational must be false. Hence, ${ 5 + \sqrt{2} }$ is an irrational number.

Problem 2: Rationalize the denominator of ${ \frac{1}{\sqrt{5} + \sqrt{2}} }$ .

To rationalize the denominator of ${ \frac{1}{\sqrt{5} + \sqrt{2}} }$ , we multiply both the numerator and the denominator by the conjugate of the denominator, which is ${ \sqrt{5} - \sqrt{2} }$ .

Multiply the numerator and denominator by the conjugate: ${ \frac{1}{\sqrt{5} + \sqrt{2}} \times \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}} }$ .
Expand the denominator using the difference of squares formula, ${ (a + b)(a - b) = a^2 - b^2 }$ : ${ \frac{\sqrt{5} - \sqrt{2}}{(\sqrt{5})^2 - (\sqrt{2})^2} }$ .
Simplify the denominator: ${ \frac{\sqrt{5} - \sqrt{2}}{5 - 2} }$ .
Further simplification gives: ${ \frac{\sqrt{5} - \sqrt{2}}{3} }$ .
Thus, the rationalized form of ${ \frac{1}{\sqrt{5} + \sqrt{2}} }$ is ${ \frac{\sqrt{5} - \sqrt{2}}{3} }$ .

Problem 3: Simplify ${ \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} }$ .

To simplify this expression, we rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is ${ \sqrt{7} - \sqrt{5} }$ .

Multiply the numerator and denominator by the conjugate: ${ \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} \times \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}} }$ .
Expand the numerator and denominator: ${ \frac{(\sqrt{7} - \sqrt{5})^2}{(\sqrt{7})^2 - (\sqrt{5})^2} }$ .
Expand the numerator using the formula ${ (a - b)^2 = a^2 - 2ab + b^2 }$ : ${ \frac{(\sqrt{7})^2 - 2(\sqrt{7})(\sqrt{5}) + (\sqrt{5})^2}{7 - 5} }$ .
Simplify the numerator and denominator: ${ \frac{7 - 2\sqrt{35} + 5}{2} }$ .
Combine like terms in the numerator: ${ \frac{12 - 2\sqrt{35}}{2} }$ .
Factor out a 2 from the numerator: ${ \frac{2(6 - \sqrt{35})}{2} }$ .
Cancel the common factor of 2: ${ 6 - \sqrt{35} }$ .
Therefore, the simplified form of ${ \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} }$ is ${ 6 - \sqrt{35} }$ .

Tips and Tricks for Mastering Exercise 1B

To effectively master Exercise 1B and similar problems, here are some valuable tips and tricks:

Understand the Basics: Ensure you have a solid understanding of rational and irrational numbers. Know their definitions, properties, and how to identify them.
Practice Rationalizing Denominators: This is a key technique in many problems. Practice various examples to become proficient.
Use Conjugates: When rationalizing denominators involving square roots, remember to use the conjugate of the denominator.
Simplify Expressions: Always simplify expressions as much as possible. Look for opportunities to combine like terms and cancel common factors.
Review Algebraic Identities: Familiarize yourself with common algebraic identities like ${ (a + b)^2 }$ , ${ (a - b)^2 }$ , and ${ (a + b)(a - b) }$ . These identities can simplify complex expressions.
Work Through Examples: Work through as many examples as possible. The more you practice, the better you will become at recognizing patterns and applying the correct techniques.
Check Your Answers: Whenever possible, check your answers to ensure they are correct. This can help you identify and correct any mistakes.
Seek Help When Needed: Don't hesitate to ask for help from your teacher, classmates, or online resources if you are struggling with a particular problem.

Common Mistakes to Avoid

While solving problems in Exercise 1B, it's easy to make common mistakes. Here are some pitfalls to watch out for:

Incorrectly Identifying Rational and Irrational Numbers: Make sure you accurately identify whether a number is rational or irrational. Remember that irrational numbers have non-repeating, non-terminating decimal expansions.
Forgetting to Use Conjugates: When rationalizing denominators, forgetting to multiply by the conjugate can lead to incorrect solutions.
Making Algebraic Errors: Be careful when expanding and simplifying expressions. Double-check your work to avoid algebraic errors.
Not Simplifying Completely: Ensure you simplify your answers as much as possible. Leaving expressions unsimplified can result in loss of marks.
Ignoring Negative Signs: Pay close attention to negative signs, especially when dealing with algebraic identities.

Summary of Key Concepts

To recap, here are the key concepts covered in RS Aggarwal Class 9 Exercise 1B:

Rational Numbers: Numbers that can be expressed as a fraction ${ \frac{p}{q} }$ , where ${ p }$ and ${ q }$ are integers and ${ q \neq 0 }$ .
Irrational Numbers: Numbers that cannot be expressed as a fraction and have non-repeating, non-terminating decimal expansions.
Rationalizing the Denominator: The process of eliminating square roots (or other radicals) from the denominator of a fraction.
Conjugates: The conjugate of ${ a + b }$ is ${ a - b }$ , and vice versa. Multiplying by the conjugate is a key step in rationalizing denominators.
Algebraic Identities: Common algebraic identities like ${ (a + b)^2 }$ , ${ (a - b)^2 }$ , and ${ (a + b)(a - b) }$ are useful for simplifying expressions.

FAQ Section

Here are some frequently asked questions related to RS Aggarwal Class 9 Exercise 1B:

Q: What is a rational number? A: A rational number is a number that can be expressed in the form ${ \frac{p}{q} }$ , where ${ p }$ and ${ q }$ are integers and ${ q \neq 0 }$ .

Q: What is an irrational number? A: An irrational number is a number that cannot be expressed as a fraction and has a non-repeating, non-terminating decimal expansion. Examples include ${ \sqrt{2} }$ and ${ \pi }$ .

Q: How do I rationalize the denominator? A: To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. This eliminates the square root from the denominator.

Q: What is the conjugate of ${ \sqrt{3} + 2 }$ ? A: The conjugate of ${ \sqrt{3} + 2 }$ is ${ \sqrt{3} - 2 }$ .

Q: Why do we rationalize the denominator? A: Rationalizing the denominator simplifies expressions and makes it easier to compare numbers.

Conclusion

Mastering RS Aggarwal Class 9 Exercise 1B requires a solid understanding of number systems, particularly rational and irrational numbers. By practicing rationalizing denominators, simplifying expressions, and avoiding common mistakes, you can improve your problem-solving skills. Remember to review the key concepts and seek help when needed. Keep practicing, and you'll ace this exercise in no time! If you found this guide helpful, share it with your friends and classmates, and continue practicing to strengthen your understanding of number systems. Happy studying!