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title: "Real Numbers"
board: "CBSE"
curriculum: "CBSE"
class: "Class 10"
subject: "Mathematics"
book: "Mathematics"
description: "In this chapter, we explore real numbers, focusing on Euclid's division algorithm and the Fundamental Theorem of Arithmetic, including properties of irrational numbers and their implications."
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---

# Real Numbers
In this chapter, we continue our exploration of real numbers, focusing specifically on the properties and applications of positive integers, particularly through Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.

---

## Knowledge Snapshot

| Field | Details |
| :--- | :--- |
| Class | Class 10 |
| Subject | Mathematics |
| Book | Mathematics |
| Chapter | Real Numbers |
| Pages | 1-9 |

---

## Chapter Summary

### Short Summary
This chapter discusses real numbers, examining properties of integers, the Fundamental Theorem of Arithmetic, and irrational numbers.

### Detailed Summary
The chapter elaborates on Euclid’s division algorithm, which states that any positive integer 'a' can be divided by another positive integer 'b' leaving a remainder 'r' smaller than 'b'. The Fundamental Theorem of Arithmetic asserts that every composite number can be uniquely expressed as a product of prime numbers. This theorem aids in proving the irrationality of numbers such as √2, √3, and √5, as well as in determining the nature of decimal expansions of rational numbers by analyzing the prime factorization of the denominator.

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## Topic-Wise Explanation

### Introduction
The chapter begins by building upon the concept of real numbers introduced in Class IX, focusing on properties of integers and the exploration of irrational numbers.

### The Fundamental Theorem of Arithmetic
This theorem posits that every composite number can be factored uniquely into prime numbers. It is illustrated through examples and the importance of prime factorization in determining the properties of integers.

### Euclid's Division Algorithm
This section discusses how to compute the highest common factor (HCF) using Euclid's algorithm and its applications in integer divisibility.

### Revisiting Irrational Numbers
Irrational numbers are defined and examples such as √2 and √3 are examined, including proofs of their irrationality based on the Fundamental Theorem of Arithmetic.

### Applications of the Fundamental Theorem
The chapter highlights the applications of the Fundamental Theorem of Arithmetic in various mathematical contexts, including finding the HCF and LCM of numbers using prime factorization.

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## Core Ideas

| Idea | Explanation |
| :--- | :--- |
| Fundamental Theorem of Arithmetic | Every composite number can be expressed as a product of primes in a unique way, barring the order of factors. |
| Euclid's Division Algorithm | A method to determine the HCF of two integers based on their divisibility properties. |
| Irrational Numbers | Numbers that cannot be expressed as a fraction of two integers, such as √2 and √3. |

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## Key Concepts

| Concept | Meaning |
| :--- | :--- |
| Composite Number | A natural number greater than 1 that is not prime, i.e., it has divisors other than 1 and itself. |
| Prime Factorization | Expressing a composite number as a product of its prime factors in unique combinations. |

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## Important Points for Revision
* Euclid’s division algorithm finds the HCF of integers.
* The Fundamental Theorem of Arithmetic ensures unique prime factorization.
* Irrational numbers such as √2 and √3 cannot be expressed as p/q.
* The HCF and LCM relationship is fundamental: HCF(a, b) × LCM(a, b) = a × b.
* Decimal expansions depend on the prime factorization of the denominator in a fraction.

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## Vocabulary and Glossary

| Word / Phrase | Meaning |
| :--- | :--- |
| Irrational Number | A number that cannot be expressed as a ratio of two integers. |
| Prime Number | A natural number greater than 1 that has no positive divisors other than 1 and itself. |

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## Practice Questions

### Short Answer Questions
1. State Euclid’s division algorithm.
2. What does the Fundamental Theorem of Arithmetic state?
3. Give an example of an irrational number.
4. How can you find the HCF of two numbers using prime factorization?
5. Prove that √2 is irrational.

### Long Answer Questions
1. Explain the significance of the Fundamental Theorem of Arithmetic with examples.
2. Prove that √3 is irrational using the proof by contradiction technique.
3. Discuss the applications of Euclid's division algorithm in real-life scenarios.

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## Related Concepts
* Highest Common Factor (HCF)
* Lowest Common Multiple (LCM)
* Decimal Expansions

---

## Source Attribution

| Field | Value |
| :--- | :--- |
| Source | Edzy |
| Reference Type | examSubjectBookChapter |
| Reference ID | 69f86799293c3123114d2e14 |
| Canonical URL | https://www.edzy.ai/cbse-class-10-mathematics-real-numbers |
| Markdown URL | https://www.edzy.ai/okf/chapter/cbse-class-10-mathematics-real-numbers.md |