# Real Numbers Class 10 Solutions PDF Free Download

**Size**

357 KB

## Description

Real numbers are a fundamental concept in mathematics, and a strong understanding of this topic is essential for Class 10 students.

Real number solutions form the basis for many advanced mathematical concepts and are applicable to various real-world scenarios.

**Solution 1: Understanding Real Numbers**

**Q: What are real numbers?**

A: Real numbers are a collection of all rational and irrational numbers. They include all counting numbers, whole numbers, integers, fractions, terminating and non-terminating decimals, and irrational numbers like √2 and π.

**Solution 2: Types of Numbers**

**Q: Classify the following numbers as rational or irrational: √3, 0.25, -7, 3/4.**

A: √3 is irrational, 0.25 and 3/4 are rational (fractions), -7 is a rational (integer).

**Solution 3: Properties of Real Numbers**

**Q: State the commutative property of addition and multiplication.**

A: The commutative property of addition states that for any real numbers a and b, a + b = b + a. The commutative property of multiplication states that for any real numbers a and b, a * b = b * a.

**Solution 4: Euclid’s Division Lemma**

**Q: Use Euclid’s division lemma to find the HCF (Highest Common Factor) of 56 and 84.**

A: Using Euclid’s division lemma, 84 can be expressed as 56 * 1 + 28. Applying the lemma again, 56 can be expressed as 28 * 2 + 0. Therefore, the HCF of 56 and 84 is 28.

**Solution 5: Fundamental Theorem of Arithmetic**

**Q: Explain the Fundamental Theorem of Arithmetic.**

A: The Fundamental Theorem of Arithmetic states that every composite number can be expressed as the product of primes, and this factorization is unique, except for the order in which the prime factors are written.

**Solution 6: Irrational Numbers**

**Q: Show that √5 + √2 is an irrational number.**

A: Assume, √5 + √2 is a rational number. That means √5 + √2 = p/q, where p and q are coprime integers. Squaring both sides gives 5 + 2√10 + 2 = p^2/q^2. Rearranging, we obtain √10 = (p^2 – 7q^2)/2q^2. This implies √10 is rational, which contradicts the fact that √10 is irrational. Hence, our initial assumption was incorrect, and √5 + √2 is an irrational number.

**PDF Name:** Real Numbers Class 10 Solutions PDF**PDF Size:** 357 KB**Category:** Class 10**Quality:** Excellent**Language:** English