NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers

The first chapter of Class 10 Mathematics, Real Numbers, is one of the most important foundations for board exams and competitive tests. It introduces students to concepts like Euclid’s Division Lemma, prime factorization, irrational numbers, and decimal expansions. At Convex Classes Jaipur, we provide step‑by‑step NCERT solutions, detailed notes, and exam‑oriented practice so students can master this chapter with confidence.

Euclid’s Division Lemma Explained

Euclid’s Division Lemma states that for any two positive integers $a$ and $b$, there exist integers $q$ and $r$ such that:

$$a=bq+r,0\le r<b$$

This lemma is used to find the HCF of numbers. For example, to find the HCF of 135 and 225, we repeatedly divide until the remainder becomes zero. The last divisor, 45, is the HCF. Our solutions explain this process step by step, making it easy for students to apply in exams.

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Fundamental Theorem of Arithmetic

Every composite number can be expressed uniquely as a product of prime numbers. This theorem is essential for solving HCF and LCM problems. For instance, 980 can be factorized as $980=2\times 2\times 5\times 7\times 7$. Understanding prime factorization not only helps in board exams but also builds a strong base for IIT‑JEE and NDA mathematics.

Proofs of Irrational Numbers

Students also learn how to prove that numbers like $\sqrt{2},\sqrt{3},\sqrt{5}$ are irrational. These proofs strengthen logical reasoning and are frequently asked in exams. For example, assuming $\sqrt{5}$ is rational leads to a contradiction, proving it is irrational. At Convex Classes Jaipur, our faculty simplifies these proofs and provides additional practice problems to ensure clarity.

Decimal Expansions of Rational Numbers

The chapter concludes with decimal expansions, showing that rational numbers either terminate or repeat. For example, $\frac{1}{7}$ expands to 0.142857…, which repeats endlessly. This concept connects number theory with practical applications and is useful for both board exams and competitive exams.

NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers

Convex Classes Jaipur Advantage

At Convex Classes Jaipur, we don’t stop at NCERT solutions. We provide:

• Detailed notes with shortcuts and tips.
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• Video lectures and doubt‑clearing sessions.
• Rank‑focused mentorship to help students aim for top scores.

Conclusion

Chapter 1 of Class 10 Maths is more than just an introduction to real numbers; it is the foundation for advanced mathematics. By mastering Euclid’s Lemma, prime factorization, irrational proofs, and decimal expansions, students build confidence for board exams and competitive tests. With Convex Classes Jaipur, students not only get complete NCERT solutions but also the guidance and support needed to excel in their academic journey.