Chapter Hub
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
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Division and HCF
Covers Euclid division lemma and Iterative division for HCF.
Open mapPrime factorisation
Covers Fundamental Theorem of Arithmetic, Prime factorisation for HCF, Prime factorisation for LCM and Relation between HCF and LCM.
Open mapDecimal expansion
Covers Terminating decimal expansion and Non-terminating recurring decimal expansion.
Open mapIrrational numbers and proofs
Covers Irrational numbers, Proof of irrationality of sqrt(2), Proof of irrationality of sqrt(3) and Proof of irrationality of sqrt(5).
Open mapKey Concepts
Concepts grouped the way the chapter is taught — open the group that matches what you want to revise.
Division and HCF
Covers Euclid division lemma and Iterative division for HCF.
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.
Prime factorisation
Covers Fundamental Theorem of Arithmetic, Prime factorisation for HCF, Prime factorisation for LCM and Relation between HCF and LCM.
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.
Decimal expansion
Covers Terminating decimal expansion and Non-terminating recurring decimal expansion.
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 and proofs
Covers Irrational numbers, Proof of irrationality of sqrt(2), Proof of irrationality of sqrt(3) and Proof of irrationality of sqrt(5).
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 √2 = p/q in lowest terms. After squaring, 2q2 = p2, 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 √3 = p/q in lowest terms. After squaring, 3q2 = p2, 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 √5 = p/q in lowest terms. After squaring, 5q2 = p2, so p is divisible by 5. That then forces q to be divisible by 5, which contradicts lowest terms.
Exam Intelligence
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
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 √2 proof but forgetting to change the prime in √3 and √5 proofs
How CBSE Asks This Chapter
- Euclid division lemma
Find quotient and remainder for a given division, or verify whether a written division form is valid.
- Iterative division for HCF
Use repeated division to find the HCF of two numbers, or explain why the method works.
- Fundamental Theorem of Arithmetic
State the theorem, use a factor tree, or use prime factorisation while finding HCF and LCM.
- Prime factorisation for HCF
Find the HCF from prime factorisation, or identify which powers should be chosen.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Remember
a = bq + r, where 0 ≤ r < b.Trap to avoid
Students often write a remainder equal to or larger than the divisor, or they forget that the remainder cannot be negative.
Remember
If a = bq + r, then HCF(a,b) = HCF(b,r).Trap to avoid
Students stop after the first remainder or take the quotient instead of the last non-zero remainder.
Fundamental Theorem of Arithmetic
Remember
Every composite number has a unique prime factorisation apart from the order of factors.Trap to avoid
Students stop at a product of composite factors such as 84 = 6 × 14 and call it prime factorisation.
Remember
HCF = product of common prime factors with the least powers.Trap to avoid
Students often take the largest powers of common primes, which gives the LCM idea instead of the HCF.
Remember
LCM = product of all prime factors with the greatest powers.Trap to avoid
Students take the smallest powers and end up finding the HCF instead of the LCM.
Remember
a x b = HCF(a,b) x LCM(a,b).Trap to avoid
Students divide by the wrong value or try to apply the relation to more than two numbers at once.
Remember
In lowest form, a denominator of the form 2m x 5n gives a terminating decimal.Trap to avoid
Students check the original denominator without first reducing the fraction to lowest form.
Non-terminating recurring decimal expansion
Remember
In lowest form, if the denominator has a prime other than 2 or 5, the decimal is recurring.Trap to avoid
Students think every non-terminating decimal is irrational, but repeating decimals are still rational.
Remember
An irrational number cannot be expressed as p/q with integers p and q, where q ≠ 0.Trap to avoid
Students call every non-terminating decimal irrational and forget that repeating decimals are rational.
Proof of irrationality of sqrt(2)
Remember
Assume √2 = p/q in lowest terms, then 2q2 = p2.Trap to avoid
Students stop after proving p is even and forget to show that q is even too.
Proof of irrationality of sqrt(3)
Remember
Assume √3 = p/q in lowest terms, then 3q2 = p2.Trap to avoid
Students repeat the √2 proof without changing the prime from 2 to 3.
Proof of irrationality of sqrt(5)
Remember
Assume √5 = p/q in lowest terms, then 5q2 = p2.Trap to avoid
Students copy the earlier proofs but forget to replace the prime with 5 in every divisibility step.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Euclid division lemma | a = bq + r, where 0 ≤ r < b. | Students often write a remainder equal to or larger than the divisor, or they forget that the remainder cannot be negative. |
| Iterative division for HCF | If a = bq + r, then HCF(a,b) = HCF(b,r). | Students stop after the first remainder or take the quotient instead of the last non-zero remainder. |
| Fundamental Theorem of Arithmetic | Every composite number has a unique prime factorisation apart from the order of factors. | Students stop at a product of composite factors such as 84 = 6 × 14 and call it prime factorisation. |
| Prime factorisation for HCF | HCF = product of common prime factors with the least powers. | Students often take the largest powers of common primes, which gives the LCM idea instead of the HCF. |
| Prime factorisation for LCM | LCM = product of all prime factors with the greatest powers. | Students take the smallest powers and end up finding the HCF instead of the LCM. |
| Relation between HCF and LCM | a x b = HCF(a,b) x LCM(a,b). | Students divide by the wrong value or try to apply the relation to more than two numbers at once. |
| Terminating decimal expansion | In lowest form, a denominator of the form 2m x 5n gives a terminating decimal. | Students check the original denominator without first reducing the fraction to lowest form. |
| Non-terminating recurring decimal expansion | In lowest form, if the denominator has a prime other than 2 or 5, the decimal is recurring. | Students think every non-terminating decimal is irrational, but repeating decimals are still rational. |
| Irrational numbers | An irrational number cannot be expressed as p/q with integers p and q, where q ≠ 0. | Students call every non-terminating decimal irrational and forget that repeating decimals are rational. |
| Proof of irrationality of sqrt(2) | Assume √2 = p/q in lowest terms, then 2q2 = p2. | Students stop after proving p is even and forget to show that q is even too. |
| Proof of irrationality of sqrt(3) | Assume √3 = p/q in lowest terms, then 3q2 = p2. | Students repeat the √2 proof without changing the prime from 2 to 3. |
| Proof of irrationality of sqrt(5) | Assume √5 = p/q in lowest terms, then 5q2 = p2. | Students copy the earlier proofs but forget to replace the prime with 5 in every divisibility step. |
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
Try a 15-question quiz from this chapter. Get instant score and unlock concept-wise analytics.
Help improve this page
Found something confusing, incorrect, or missing?