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Real Numbers
Real Numbers in Class 10 builds the base for divisibility, factorisation, decimal forms, and irrational numbers. A student should first understand division with remainder, then use prime factorisation for HCF and LCM, and finally connect denominators to decimal expansion. The chapter also trains proof thinking. Once a student can handle Euclid's division lemma, the fundamental theorem of arithmetic, and contradiction proofs for square roots, many exam questions become direct and scoring.
Difficulty
Medium
Study time
96-120 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 96 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
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Key Concepts
Concepts grouped the way the chapter is taught — open the bucket that matches what you want to revise.
Core Concepts
high priorityOpen the chapter concepts in a clean revision order.
Euclid division lemma
If one integer a is divided by another positive integer b, we can write a = bq + r, where q is the quotient and r is the remainder, with 0 <= r < b.
Iterative division for HCF
Iterative division for HCF is a step-by-step method where the larger number is divided by the smaller number, then the divisor and remainder are used again until the remainder becomes 0. The last non-zero remainder is the HCF.
Fundamental Theorem of Arithmetic
Every composite number can be written as a product of prime numbers, and this prime factorisation is unique apart from the order of the primes.
Prime factorisation for HCF
To find the HCF by prime factorisation, write both numbers as products of primes and take only the common prime factors with the smallest powers.
Prime factorisation for LCM
To find the LCM by prime factorisation, write both numbers as primes and take every prime that appears in either number, using the highest power needed.
Relation between HCF and LCM
For two positive integers a and b, the product of the numbers equals the product of their HCF and LCM.
Terminating decimal expansion
A fraction in lowest form has a terminating decimal expansion only when its denominator has prime factors 2 and/or 5 and no other prime factors.
Non-terminating recurring decimal expansion
If a fraction in lowest form has any prime factor in its denominator other than 2 or 5, its decimal expansion is non-terminating recurring.
Irrational numbers
An irrational number cannot be written in the form p/q, where p and q are integers and q != 0.
Proof of irrationality of sqrt(2)
Assume sqrt(2) = p/q in lowest terms. After squaring, 2q^2 = p^2, so p is even. That leads to q also being even, which contradicts the fraction being in lowest terms.
Proof of irrationality of sqrt(3)
Assume sqrt(3) = p/q in lowest terms. After squaring, 3q^2 = p^2, so p is divisible by 3. That then forces q to be divisible by 3, which contradicts lowest terms.
Proof of irrationality of sqrt(5)
Assume sqrt(5) = p/q in lowest terms. After squaring, 5q^2 = p^2, so p is divisible by 5. That then forces q to be divisible by 5, which contradicts lowest terms.
Exam Intelligence
Use this section to decide what deserves the most revision time.
High Probability Topics
- Euclid division lemma
- Iterative division for HCF
- Fundamental Theorem of Arithmetic
- Prime factorisation for HCF
- Prime factorisation for LCM
- Relation between HCF and LCM
- Terminating decimal expansion
- Non-terminating recurring decimal expansion
Common Traps
- Treating any remainder as valid even when it is larger than the divisor
- Using the quotient instead of the last non-zero remainder for HCF
- Stopping prime factorisation before all factors are prime
- Taking largest powers for HCF or smallest powers for LCM
- Forgetting to reduce a fraction to lowest form before checking decimal type
- Calling every repeating decimal irrational
- Copying the sqrt(2) proof but forgetting to change the prime in sqrt(3) and sqrt(5) proofs
Likely Question Types
- MCQ: concept checks, applications, and common mistakes
- Very short answer: definitions, formulas, or conditions
- Short answer: worked method, example, or reason-based explanation
- Case-based: chapter scenario with concept-linked subparts
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
- Euclid division lemma gives the basic division form used everywhere in the chapter
- HCF can be found by repeated division or by prime factorisation
- LCM is found by taking highest prime powers, while HCF uses lowest common prime powers
- The product of two numbers equals HCF x LCM
- Terminating decimals come from denominators with only 2s and 5s after simplification
- Recurring decimals arise when another prime remains in the denominator
- Irrational numbers cannot be written as p/q and are proved by contradiction in square-root cases
- Euclid division lemma: If one integer a is divided by another positive integer b, we can write a = bq + r, where q is the quotient and r is the remainder, with 0…
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
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