Class 10 Mathematics Chapter 1 Real Numbers — Formulas & Key Points

Three pillars support every problem in Real Numbers: Euclid's Division Lemma, the Fundamental Theorem of Arithmetic, and the decimal expansion theorem. The 2024 CBSE board paper carried a 3-mark proof question directly invoking the Fundamental Theorem. Understanding not just the statement but the logical flow of each theorem saves marks in long-answer proofs. Euclid's Division Lemma underpins the HCF algorithm; the Fundamental Theorem guarantees unique prime factorisation, which in turn powers HCF-LCM calculations and irrationality proofs. Commit the exact NCERT wording to memory — examiners reward verbatim theorem statements in Step 1 of proofs.

Euclid's Division Lemma states: Given integers a and b with b > 0, there exist unique integers q (quotient) and r (remainder) such that a = bq + r and 0 ≤ r < b. This lemma is the engine of Euclid's Division Algorithm for finding HCF. To find HCF(a, b), apply the lemma repeatedly: divide a by b to get remainder r₁, then divide b by r₁ to get r₂, continue until remainder is zero; the last non-zero remainder is HCF(a, b). The 2023 CBSE paper asked students to find HCF of 867 and 255 using this algorithm, awarding 2 marks for correct intermediate steps. Always write each step in a = bq + r form to earn full method marks even if arithmetic slips occur.

The Fundamental Theorem of Arithmetic asserts that every integer greater than 1 is either a prime number or can be expressed uniquely as a product of prime numbers, disregarding the order of factors. For example, 60 = 2² × 3 × 5 is the only prime factorisation of 60. This uniqueness is critical: it allows us to compare factorisations to find HCF and LCM, and it underpins proofs of irrationality. In the 2025 sample paper, a 4-mark question asked students to prove √5 is irrational using contradiction and the Fundamental Theorem. The theorem guarantees that if p² is divisible by 5, then p must be divisible by 5 — a step you must justify explicitly. When writing proofs, state 'By the Fundamental Theorem of Arithmetic' to signal rigour. Examiners deduct marks for vague phrases like 'obviously' or 'clearly' without citing the theorem.

Once you have prime factorisations, HCF and LCM follow mechanical rules. Write each number as a product of prime powers. For HCF, take each common prime and raise it to the smallest exponent appearing in any factorisation. For LCM, take every prime (common or not) and raise it to the largest exponent. The product relationship HCF(a,b) × LCM(a,b) = a × b holds for any two positive integers and is a favourite shortcut in competitive exams and board word problems. The 2024 board paper featured a 2-mark question: 'The HCF of two numbers is 12 and their product is 3456; find their LCM.' Answer: LCM = 3456 ÷ 12 = 288. Always verify your HCF and LCM by checking divisibility and ensuring their product equals a×b. This cross-check catches arithmetic slips and secures method marks even when final answers differ slightly due to calculation errors.

A fraction p/q in its lowest terms (HCF(p,q)=1) has a terminating decimal expansion if and only if the denominator q can be expressed as 2^m × 5^n where m and n are non-negative integers. If q has any prime factor other than 2 or 5, the decimal is non-terminating repeating. This theorem appears in 2-mark 'without actual division' questions: for example, 'Does 23/200 terminate?' Write 200=2³×5², so yes, it terminates. The 2023 board paper asked students to determine the nature of 17/8, 64/455, 15/1600 without performing long division. You score full marks by factorising the denominator and citing the theorem. Remember to reduce the fraction first: 64/455 is already in lowest terms (HCF(64,455)=1), and 455=5×7×13 contains 7 and 13, hence non-terminating repeating. Practice factorising denominators up to 1000 for speed; keep a mental list of cubes and products of small primes.

To prove a number is irrational, assume the contrary (that it is rational), express it as p/q in lowest terms, derive a contradiction using the Fundamental Theorem of Arithmetic, and conclude the assumption was false. The template: Assume √n = p/q, HCF(p,q)=1. Square both sides: n = p²/q², so p² = nq². Hence n divides p², implying n divides p (by Fundamental Theorem, if a prime divides a square it divides the base). Write p = nk. Substitute: n = (nk)²/q² → nk² = q², so n divides q² and thus q. Now both p and q are divisible by n, contradicting HCF(p,q)=1. Therefore √n is irrational. This proof structure earned 4 marks in the 2024 board paper for √3. Examiners award 1 mark for correct assumption, 1 for squaring and rearrangement, 1 for invoking the Fundamental Theorem, and 1 for stating the contradiction clearly. Missing any step costs marks. Practise writing the proof in under three minutes to leave time for numerical questions. At CBSETUTOR.ai, students upload photo doubts of irrationality proofs and receive step-by-step LaTeX breakdowns within seconds — a flat ₹999/month subscription for Classes 6–12 with a 3-day free trial ensures no concept is left unclear before the boards.

Precision in terminology distinguishes top-scoring answers from average ones. Real numbers encompass all rational and irrational numbers. Rational numbers are those expressible as p/q where p and q are integers and q ≠ 0; they include integers, fractions, terminating decimals and non-terminating repeating decimals. Irrational numbers cannot be written as fractions; their decimal expansions are non-terminating and non-repeating. Prime numbers have exactly two distinct divisors (1 and themselves); composite numbers have more than two divisors; 1 is neither prime nor composite. Co-prime (or relatively prime) integers have HCF equal to 1 — for instance, 8 and 15 are co-prime despite both being composite. The NCERT textbook uses 'Fundamental Theorem of Arithmetic' and not 'Unique Factorisation Theorem'; stick to NCERT terms in board exams. The 2025 marking scheme specifically looks for 'HCF' and 'LCM' in capital letters, not 'GCD' or 'GCF'. When writing proofs, spell out 'highest common factor' and 'least common multiple' at first mention, then abbreviate. These small consistencies signal exam readiness and can tip borderline marks in your favour, especially in 1-mark definitional questions that appeared in Section A of recent papers.

Remembering when a decimal terminates: 'Only Twos and Fives, Otherwise Repeats' — if the denominator (in lowest terms) has only factors 2 and 5, the decimal terminates; any other prime factor means it repeats. For Euclid's algorithm steps, the mnemonic 'Divide, Replace, Repeat, Done' helps: divide the larger by the smaller, replace the larger with the smaller and the smaller with the remainder, repeat until remainder is zero, the last non-zero remainder is done (the HCF). To recall that HCF × LCM = product of two numbers, think 'Highest × Lowest = Product' as a symmetry rule. When proving irrationality, 'Assume, Square, Factor, Contradict' maps the four proof steps. For the Fundamental Theorem of Arithmetic, remember 'Unique Prime Product' — every number breaks into primes in exactly one way. These shortcuts are not a substitute for understanding, but they prevent blanking under exam pressure. The 2024 topper from Delhi used flashcards with one mnemonic per card and cycled through them daily for a month before boards. Additionally, writing formulas on a single A4 sheet and sticking it above your study desk keeps them in peripheral vision during revision sessions. If you find yourself forgetting steps during practice tests, photograph your working and upload it to CBSETUTOR.ai; the AI tutor highlights exactly where the method deviates from the NCERT proof template, reinforcing correct habits faster than passive re-reading.

Mistake 1: Writing HCF and LCM without reducing the fraction first. Always confirm HCF(p,q)=1 before applying the decimal expansion theorem; failing to simplify 64/128 to 1/2 cost many students the 2023 board marks. Mistake 2: Confusing 'p divides a²' with 'p divides a'. The Fundamental Theorem states that if a prime p divides a², then p must divide a; this implication is one-way and must be cited explicitly in proofs — omitting it loses the critical step mark. Mistake 3: In Euclid's algorithm, writing a = bq + r but forgetting to state 0 ≤ r < b; the uniqueness of q and r hinges on this inequality. Mistake 4: Mixing up HCF (product of minimum powers) with LCM (product of maximum powers); a simple swap here flips the entire answer. Mistake 5: Declaring a number irrational by decimal approximation instead of proof; saying '√2 ≈ 1.414… looks non-repeating' earns zero marks — only the contradiction method is acceptable. Mistake 6: Using = instead of ≈ when writing non-terminating decimals; π = 3.14 is wrong, π ≈ 3.14 or π = 3.141… is correct. Mistake 7: Assuming co-prime means both numbers are prime; 8 and 9 are co-prime but both composite. The 2024 marking scheme deducted half a mark for each notational slip, so clean presentation matters. Practice writing solutions in the NCERT format: numbered steps, algebraic expressions on separate lines, and boxed final answers. At CBSETUTOR.ai, uploading a worked solution triggers an instant audit against the official CBSE marking rubric, catching these errors before they become habits. A subscription costs just ₹999 per month for unlimited doubt-solving across all subjects in Classes 6 to 12, with a 3-day free trial to experience the precision feedback loop firsthand.

Example 1 (HCF by Euclid's algorithm): Find HCF(1260, 504). Step 1: 1260 = 504×2 + 252. Step 2: 504 = 252×2 + 0. HCF is 252. Example 2 (Terminating or not): Without division, determine if 27/400 is terminating. First reduce: HCF(27,400)=1 (27=3³, 400=2⁴×5², no common factor). Denominator 400=2⁴×5² has only 2 and 5, so terminating. Example 3 (Prove irrationality): Prove 3+√2 is irrational. Assume 3+√2 is rational, say 3+√2 = r (r rational). Then √2 = r−3. Since r and 3 are rational, r−3 is rational, implying √2 is rational — contradiction (we know √2 is irrational by the standard proof). Hence 3+√2 is irrational. These examples mirror the exact style and mark distribution of CBSE board questions. Practice reproducing them without referring to notes; aim for each solution in under 90 seconds. The 2024 topper from Chennai credited her speed to writing 50 such mini-examples on index cards and shuffling them for random drill every evening. When doubt arises mid-solution, snap a photo and upload it to CBSETUTOR.ai's doubt engine; you will receive annotated steps highlighting where your reasoning diverges from the NCERT method, turning confusion into clarity within minutes. The platform's ₹999/month flat fee covers unlimited queries for every CBSE class from 6 to 12, and the 3-day trial lets you test the AI tutor's responsiveness with zero financial commitment.

Keep this box open on exam morning for a final sweep of Chapter 1 Real Numbers. Euclid's Division Lemma: a = bq + r, 0 ≤ r < b. Fundamental Theorem of Arithmetic: every integer > 1 has unique prime factorisation. HCF = product of common primes to minimum power; LCM = product of all primes to maximum power. HCF×LCM = product of the two numbers. Decimal terminates ↔ denominator (in lowest terms) is 2^m × 5^n only. Irrationality proof steps: Assume rational p/q, square, factor, show contradiction with HCF(p,q)=1. Co-prime: HCF=1. Prime: exactly two divisors. Composite: more than two divisors; 1 is neither. √2, √3, √5, π, e are irrational. 2 is the only even prime. Numbers ending in 0,2,4,6,8 are divisible by 2; ending in 0,5 by 5. Sum of digits divisible by 3 → number divisible by 3; by 9 → divisible by 9. Two consecutive positive integers are always co-prime. This box condenses the chapter into 120 seconds of reading time. Print it, laminate it, and slide it into your pencil box; many students report that a quick glance before entering the exam hall calms pre-paper anxiety and brings formulas to the forefront of working memory. Pair this sheet with the three solved examples above and you have a complete revision kit. If any formula feels shaky, use CBSETUTOR.ai's 24×7 photo-upload feature to clarify doubts on the spot — whether it is 6 a.m. before your exam or 11 p.m. the night before, the AI tutor delivers instant, syllabus-aligned explanations at ₹999/month for all classes, with a risk-free 3-day trial to start your journey.