NCERT Class 10 Mathematics Mathematics: Chapter 1 — 1 Introduction
This chapter, "Real Numbers," for Class 10 Mathematics, revisits the concept of real numbers, including irrational numbers, introduced in Class IX. It focuses on two key properties of positive integers: Euclid's division algorithm and the Fundamental Theorem of Arithmetic. Euclid's algorithm, related to divisibility, is used to compute the Highest Common Factor (HCF). The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of primes. This theorem is applied to prove the irrationality of numbers like √2, √3, and √5, and to determine when the decimal expansion of a rational number is terminating or non-terminating repeating, based on the prime factorization of its denominator. The chapter begins by exploring the concept of prime factorization.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 10 |
| Subject | Mathematics |
| Book | Mathematics |
| Chapter | Chapter 1 — 1 Introduction |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 4 minutes |
| Word count | 666 |
Learning outcomes
- Understand the properties of real numbers, including irrational numbers.
- Apply Euclid's division algorithm to compute HCF.
- Understand and apply the Fundamental Theorem of Arithmetic.
- Prove the irrationality of certain numbers.
- Determine the nature of decimal expansions of rational numbers.
Vocabulary
| Word | Meaning |
|---|---|
| Real Numbers | Numbers that can be represented on a number line, including rational and irrational numbers. |
| Irrational Numbers | Numbers that cannot be expressed as a simple fraction p/q, where p and q are integers and q is not zero. |
| Euclid's division algorithm | A method to find the HCF of two positive integers based on the division lemma. |
| Fundamental Theorem of Arithmetic | States that every composite number can be expressed as a product of primes in a unique way. |
| HCF | Highest Common Factor, the largest positive integer that divides two or more integers. |
| Composite Number | A positive integer that has at least one divisor other than 1 and itself. |
| Prime Factorisation | Expressing a composite number as a product of its prime factors. |
| Terminating Decimal Expansion | A decimal expansion that ends after a finite number of digits. |
| Non-terminating Repeating Decimal Expansion | A decimal expansion that continues indefinitely with a repeating pattern of digits. |
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Practice questions
- What is the Fundamental Theorem of Arithmetic? Answer: It states that every composite number can be expressed as a product of primes in a unique way, apart from the order of the factors.
- How is Euclid's division algorithm useful? Answer: It is useful for computing the HCF of two positive integers.
- Give an example of an irrational number. Answer: √2, √3, √5, π are examples of irrational numbers.
- What does the prime factorization of the denominator of a rational number tell us? Answer: It tells us whether the decimal expansion of the rational number is terminating or non-terminating repeating.
Frequently asked questions
What are real numbers in Class 10 Mathematics?
Real numbers include all rational and irrational numbers, which can be represented on a number line.
What is Euclid's division algorithm used for?
It is primarily used to find the Highest Common Factor (HCF) of two positive integers.
What is the Fundamental Theorem of Arithmetic?
It states that every composite number can be uniquely expressed as a product of prime numbers.
Can the Fundamental Theorem of Arithmetic be used to prove irrationality?
Yes, it is used to prove the irrationality of numbers like √2, √3, and √5.
How does the prime factorization of a denominator help with decimal expansions?
It helps determine if a rational number's decimal expansion is terminating or non-terminating repeating.
What is the difference between terminating and non-terminating repeating decimals?
A terminating decimal ends after a finite number of digits, while a non-terminating repeating decimal continues indefinitely with a repeating pattern.
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Topics covered
NCERT Class 10 Mathematics — Mathematics — Chapter 1 — 1 Introduction. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.