Class 10 Maths Ex 1.1

Class 10 Maths Ex 1.1: A Deep Dive into Real Numbers

This article provides a comprehensive guide to Class 10 Maths Exercise 1.1, focusing on the fundamental concepts of real numbers. We'll explore the key definitions, theorems, and problem-solving strategies, ensuring a thorough understanding of this crucial chapter. Understanding real numbers forms the bedrock for many advanced mathematical concepts, making this exercise particularly important. We will cover everything from Euclid's division lemma to the application of these concepts in solving real-world problems. Let's begin!

Introduction to Real Numbers

Real numbers encompass all numbers that can be represented on a number line, including rational and irrational numbers. This exercise focuses on solidifying your understanding of these number systems and their properties. We'll delve into how to represent these numbers, identify their types, and apply various theorems to solve problems effectively. Mastering this section lays a strong foundation for your future mathematical studies.

Euclid's Division Lemma and Algorithm

One of the core concepts in this exercise is Euclid's Division Lemma. It states that for any two positive integers, a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Here, a is the dividend, b is the divisor, q is the quotient, and r is the remainder.

This seemingly simple statement forms the basis of the Euclid's Division Algorithm, which is a method used to find the Highest Common Factor (HCF) of two or more numbers. The algorithm repeatedly applies the division lemma until the remainder becomes zero. The last non-zero remainder is the HCF.

Example: Let's find the HCF of 45 and 105 using Euclid's division algorithm.

1. Divide 105 by 45: 105 = 45 × 2 + 15
2. Now, divide 45 by the remainder 15: 45 = 15 × 3 + 0

Since the remainder is 0, the HCF is the last non-zero remainder, which is 15.

Understanding and applying Euclid's division algorithm is crucial for solving many problems in this exercise.

Fundamental Theorem of Arithmetic

Another cornerstone of this chapter is the Fundamental Theorem of Arithmetic, also known as the Unique Factorization Theorem. It states that every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order of the factors. This theorem is fundamental to many number theory concepts and provides a powerful tool for solving problems related to prime factorization.

For example, the prime factorization of 72 is 2³ × 3². This means that 72 can only be expressed as the product of 2 and 3, and no other prime numbers. This uniqueness is a critical aspect of the theorem.

Irrational Numbers and Their Properties

This exercise also introduces a deeper exploration of irrational numbers. Irrational numbers are numbers that cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0. They are non-repeating, non-terminating decimals. Famous examples include π (pi) and √2. The exercise might involve proving the irrationality of certain numbers, often through the method of contradiction.

Proving the Irrationality of √2: This is a classic proof that demonstrates the power of the method of contradiction. We assume √2 is rational, meaning it can be expressed as p/q, where p and q are co-prime integers (their HCF is 1). Squaring both sides, we get 2 = p²/q², which implies 2q² = p². This shows that p² is an even number, and therefore p must also be even. We can then write p = 2k*, where k is an integer. Substituting this into the equation, we get 2q² = (2k)² = 4k², which simplifies to q² = 2k². This shows that q² is also even, and therefore q must be even. However, this contradicts our initial assumption that p and q are co-prime, as they both are divisible by 2. Therefore, our assumption that √2 is rational must be false, proving that √2 is irrational.

Similar methods can be used to prove the irrationality of other numbers.

Decimal Representation of Rational Numbers

The exercise also explores the decimal representation of rational numbers. Rational numbers always have either terminating or non-terminating but recurring decimal expansions. Understanding this connection between the fractional form and the decimal representation of a rational number is important. A terminating decimal can be easily converted to a fraction, and a non-terminating, recurring decimal can also be converted to a fraction using a specific method.

Example: Convert 0.333... (recurring) to a fraction.

Let x = 0.333... Then 10x = 3.333... Subtracting the first equation from the second: 9x = 3 Therefore, x = 3/9 = 1/3

This demonstrates how recurring decimals can be converted into fractions.

Solved Examples and Practice Problems

Exercise 1.1 typically includes a variety of problems that test your understanding of the concepts discussed above. These problems might involve:

• Finding the HCF of two or more numbers using Euclid's algorithm.
• Expressing numbers as a product of their prime factors.
• Proving the irrationality of numbers.
• Converting decimals to fractions and vice versa.
• Applying the concepts of real numbers to solve word problems.

Let's work through a few example problems:

Problem 1: Find the HCF of 1260 and 7344 using Euclid’s division algorithm.

Solution:

1. 7344 = 1260 × 5 + 1044
2. 1260 = 1044 × 1 + 216
3. 1044 = 216 × 4 + 180
4. 216 = 180 × 1 + 36
5. 180 = 36 × 5 + 0

The HCF is 36.

Problem 2: Prove that 3 + √5 is irrational.

Solution: We will use the method of contradiction. Assume 3 + √5 is rational. Then 3 + √5 = p/q, where p and q are co-prime integers. This implies √5 = p/q - 3 = (p - 3q) / q. Since p and q are integers, (p - 3q) / q is rational. But we know that √5 is irrational. This contradiction proves that our assumption is incorrect, and therefore 3 + √5 is irrational.

Frequently Asked Questions (FAQ)

Q1: What is the difference between rational and irrational numbers?

A1: Rational numbers can be expressed as a fraction p/q, where p and q are integers, and q ≠ 0. Irrational numbers cannot be expressed in this form; they are non-repeating, non-terminating decimals.

Q2: Why is Euclid's division lemma important?

A2: Euclid's division lemma is fundamental because it forms the basis for Euclid's algorithm, a method for finding the HCF of two numbers efficiently. This algorithm is widely used in various mathematical applications.

Q3: How can I effectively prepare for problems in Exercise 1.1?

A3: Thoroughly understand the definitions of rational and irrational numbers, Euclid's division lemma and algorithm, and the Fundamental Theorem of Arithmetic. Practice solving a wide range of problems, paying close attention to the different methods used for proving irrationality and converting between fractions and decimals.

Conclusion

Mastering Class 10 Maths Exercise 1.1 is crucial for building a solid foundation in mathematics. This exercise reinforces fundamental concepts of real numbers, including rational and irrational numbers, prime factorization, and Euclid's algorithms. By understanding these concepts and practicing problem-solving, you'll not only excel in this exercise but also build the necessary skills for more advanced mathematical topics in the future. Remember, consistent practice and a deep understanding of the underlying principles are key to success. Don't hesitate to review the concepts and examples repeatedly until you feel confident in your understanding. Good luck!