CBSE Class 10 Maths Real Numbers: Chapter Guide

If your child's school has just finished or is currently teaching Chapter 1, this is the moment to get the syllabus boundaries right, not just the maths itself. Real Numbers is usually one of the first two or three chapters covered once the CBSE academic year begins in April, and because it is short and largely procedural, it is also one of the easiest chapters to either master completely or quietly get wrong by studying the wrong exercises.

What the Chapter Actually Covers

CBSE's current syllabus keeps Real Numbers to two exercises, and both lean on the same core idea — the Fundamental Theorem of Arithmetic. It's a short chapter, typically worth around six marks in the 80-mark paper, but it sets up prime factorisation and proof-writing skills that show up again later in the year.

• The Fundamental Theorem of Arithmetic. Every composite number can be written as a product of prime numbers, and that factorisation is unique except for the order in which the primes are written — 140, for instance, is always 2 × 2 × 5 × 7, however you arrive at it. This exercise uses that fact to find the HCF and LCM of two numbers: write both numbers as products of primes, then the HCF is the product of the smallest power of every prime common to both, and the LCM is the product of the greatest power of every prime that appears in either number.
• Proving numbers irrational. The second exercise asks students to prove that numbers such as √2, √3, and √5, and expressions built from them such as 5 - 2√3, are irrational. Every proof follows the same shape — proof by contradiction: assume the number is rational, express it as a fraction in lowest terms, and use the Fundamental Theorem of Arithmetic to show that assumption breaks down.

What's No Longer in the Syllabus

The older version of this chapter had four exercises, and CBSE's rationalisation has dropped two of them entirely from what gets taught and examined:

• Euclid's Division Lemma and Algorithm. This used to be the opening exercise — a step-by-step method for finding the HCF of two numbers through repeated division rather than prime factorisation. It is no longer part of the CBSE-assessed syllabus.
• Decimal expansions of rational numbers. The exercise on whether a fraction's decimal expansion terminates or repeats, based on the prime factors of its denominator, has also been removed.

This matters more than it sounds like it should. Because Real Numbers used to be a longer chapter, a large share of the solution PDFs, YouTube walkthroughs, and even printed guidebooks still in circulation cover all four exercises, not two. If a resource is walking through Euclid's division algorithm to find an HCF, or asking whether a fraction like 17/40 terminates, it was built for an older syllabus, not the one currently being examined. There's nothing wrong with the maths in it, but time spent there this week is time not spent on what CBSE will actually ask.

Where Students Actually Lose Marks

The chapter is short, but three habits quietly cost marks even among students who understand the concepts:

• Applying the HCF × LCM shortcut to more than two numbers. HCF(a, b) × LCM(a, b) = a × b is true — but only for exactly two numbers. Students who reach for the same shortcut with three numbers get a confidently wrong answer.
• Losing track of powers while factorising. Missing one factor of 2 or double-counting a 3 while breaking a larger number into primes throws off the HCF and LCM that follow, even when the method is correct. Writing the factorisation out in full — 2 × 2 × 2 × 2 × 2 × 3, not a shortcut done in your head — catches this before it becomes a wrong answer.
• Skipping the middle step in irrationality proofs. A common shortcut is to write "assume √2 is rational, so √2 = p/q" and jump straight to the contradiction. CBSE's marking scheme awards a specific step in between — showing that if p² is divisible by 2, then p itself must be divisible by 2 — and skipping it costs a mark even when the final conclusion is correct.

A Practical Way to Study This Chapter

Because the chapter is now genuinely short, the goal this week is precision, not coverage:

• Re-derive HCF and LCM for five or six number pairs. Factorise each pair fully, find the HCF and LCM by hand, and check the answer against the product rule — remembering it only holds for pairs, not three numbers at once.
• Write out three or four irrationality proofs in full. Don't just recall the shape of the proof; write every line, including the "must be divisible by" step, until it's automatic rather than memorised.
• Leave Euclid's division algorithm and decimal expansion alone. If an old guidebook or a sibling's notebook pulls you toward either, that's time better spent on more prime-factorisation and irrationality practice — the two things actually being examined this year.
• Take a short, chapter-only test before moving to Polynomials. A quick check now costs a few minutes; discovering the gap during first-term revision in September costs a lot more.

Where IndiaSchool.ai Fits

IndiaSchool.ai has a free AI lesson and unlimited board-pattern practice for every chapter of CBSE Class 10 Maths, including Real Numbers, built around the current rationalised syllabus rather than an older edition — so practice questions don't waste time on exercises that have already been dropped. A chapter-only paper, limited to just what's currently examined, takes under two minutes to generate from the custom question paper tool, and the full Class 10 Maths chapter list is on the syllabus explorer. One subscription, at ₹500 a month or ₹5,000 a year, covers every board and subject on the platform, with a 14-day free trial that needs no card to start.

The Bottom Line

Real Numbers rewards precision over memorising: knowing exactly which two exercises are still being examined saves as much time as knowing how to solve them. A student who can factorise confidently, find the HCF and LCM of two numbers without reaching for the wrong shortcut, and write a complete irrationality proof — middle step included — has banked an easy, low-drama set of marks before Polynomials even starts.