Real Numbers Notes Class 10 Maths
Real Numbers
Board: CBSE · Session: 2026-27 · Class: 10 · Subject: Mathematics (041 / 241)
Last academically reviewed: 23 September 2026 · Reviewed by: Pandey Sir
Primary source: CBSE Class 10 Mathematics syllabus 2026-27. Euclid’s division lemma and decimal expansions are outside the current board scope and are marked as enrichment at the end.
Why this chapter matters
Every whole number is built from primes, and the way it is built is unique.
You will prove why √2, √3 and √5 can never be written as a fraction, then use primes to find HCF and LCM at speed.
Chapter Learning Map
Follow the spine top to bottom: the theorem, the tools it gives you, then the proofs.
1 · Fundamental Theorem of Arithmetic
Every integer greater than 1 is either a prime or can be written as a product of primes, and that factorisation is unique apart from the order of the primes.
In one line: primes are the building blocks, and every number has exactly one set of them.
96 = 25 · 3 (2×2×2×2×2×3). 404 = 22 · 101. However you split them, you always land on the same primes.
“One number, one set of primes.” The order is the only thing that can change.
Stopping at any factor pair that is not all primes. Keep splitting until every factor is prime.
2 · HCF and LCM by Prime Factorisation
Write each number as a product of primes. HCF = product of the common primes at their lowest power. LCM = product of all primes at their highest power.
96 = 25 · 3 and 404 = 22 · 101. Common prime is only 2, lowest power 22 → HCF = 4. Highest powers of everything → LCM = 25 · 3 · 101 = 9696. Check: 4 × 9696 = 38784 = 96 × 404 ✓
For two numbers, use HCF × LCM = a × b to find the one you were not asked for. It saves a whole step.
| Numbers | Prime forms | HCF | LCM |
|---|---|---|---|
| 6, 20 | 6 = 2·3, 20 = 22·5 | 2 | 22·3·5 = 60 |
| 96, 404 | 25·3, 22·101 | 4 | 25·3·101 = 9696 |
Applying HCF × LCM = a × b to three numbers. It is true only for two numbers; for three, use prime factorisation.
3 · Irrational Numbers
A number is irrational if it cannot be written as pq where p and q are integers and q ≠ 0.
The key lemma: if a prime p divides a2, then p divides a. Every irrationality proof in this chapter rests on it.
Proof idea: assume the opposite. Write the number as pq in lowest terms, then show p and q must share a factor — which contradicts “lowest terms”.
Assume √2 = pq in lowest terms. Then 2 = p2q2, so p2 = 2q2. Hence p2 is even, so p is even; write p = 2k. Then 4k2 = 2q2, so q2 = 2k2, making q even too. Both p and q even contradicts “lowest terms”. Therefore √2 is irrational.
√2, √3, √5, √7 … are irrational. A rational plus an irrational is always irrational.
Do not call √4 irrational. √4 = 2, which is rational. Check for a perfect square first.
These two topics were removed from the Class 10 Mathematics syllabus. They are useful background for olympiads and higher classes, but they will not be assessed in the 2026-27 board paper — so learn them only after the core above is secure.
Euclid’s division lemma. For positive integers a and b, there exist unique q and r with a = bq + r, 0 ≤ r < b. Repeating the division gives Euclid’s algorithm for the HCF.
Decimal expansions. A rational number pq in lowest terms is terminating only when q has no prime factor other than 2 and 5.
4 · Solved NCERT Highlights
Given: assume 3√2 = pq in lowest terms. Then √2 = p3q. The right side is a fraction of two integers, so it is rational — but √2 is irrational. Contradiction, so 3√2 is irrational.
Why it works: a non-zero rational times an irrational is always irrational. Use the irrationality of √2 as your stepping stone.
For “prove 5 − 2√3 is irrational”, isolate √3 first: if 5 − 2√3 were rational, then √3 = 5 − r2 would be rational too. Contradiction.
5 · Topper Secrets & Extra Examples
For the HCF of three numbers, use the prime method only — HCF × LCM = product does not hold for three numbers.
In any irrationality proof, p and q must be in lowest terms. That single line is what makes the contradiction land.
Three bells ring every 6, 8 and 12 minutes. When do they ring together next? LCM(6, 8, 12) = 24 minutes. Small numbers, full marks.
Find the smallest number by which 396 must be multiplied to make it a perfect square. 396 = 22 · 32 · 11. The lone 11 needs a partner, so multiply by 11.
Writing “√2 is irrational” without proof in a 3-mark question. State the assumption, the contradiction and the conclusion — the marks are for the reasoning.
Visual Board — Factor Trees
Board problems reward clean trees. Use circles and small rectangles only.
Draw factor trees with circles and rectangles, not freehand curves. Keep primes in the small boxes so the examiner can spot them instantly.
⚡ Formula Vault
Revise these before every test. Each carries marks.
Real Numbers is part of Unit I, Number Systems, which carries 6 of the 80 board marks. Expect short questions — mostly 1, 2 and 4 marks — on prime factorisation, HCF and LCM, and one irrationality proof.
Important Question Bank
Tap “Show answer”.
A · Multiple Choice (1 mark each)
(a) 2 (b) 4 (c) 8 (d) 12
Show answer
Answer: (b) 4. 96 = 25·3, 404 = 22·101 → common prime 2 at lowest power 22.
(a) 24 · 6 (b) 25 · 3 (c) 2 · 48 (d) 3 · 32
Show answer
Answer: (b) 25·3. Only (b) has every factor prime: 96 = 2×2×2×2×2×3.
(a) rational (b) irrational (c) an integer (d) a whole number
Show answer
Answer: (b) irrational. The square root of a non-perfect-square integer is irrational.
(a) 30 (b) 60 (c) 120 (d) 12
Show answer
Answer: (b) 60. 6 = 2·3, 20 = 22·5 → LCM = 22·3·5 = 60.
(a) √9 (b) √16 (c) √7 (d) 34
Show answer
Answer: (c) √7. 7 is not a perfect square, so √7 is irrational. √9 = 3 and √16 = 4 are integers; 3/4 is rational.
B · Assertion–Reason (1 mark each)
(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.
Assertion: The HCF of 96 and 404 is 4.
Reason: The HCF is the product of the common prime factors at their lowest powers.
Show answer
Answer: (a). Both are true and the reason correctly explains the assertion. 96 = 25·3, 404 = 22·101, common lowest power 22 = 4.
Assertion: 7 × 11 × 13 + 13 is a composite number.
Reason: A number with more than two factors is composite.
Show answer
Answer: (a). 13(7×11 + 1) = 13 × 78, which has more than two factors; the reason is true and explains it.
Assertion: √4 is irrational.
Reason: The square root of a prime number is irrational.
Show answer
Answer: (d). The assertion is false (√4 = 2, which is rational); the reason is true but does not apply here.
C · Case-Based (4 marks)
A sweet shop packs laddoos in boxes of 18 and barfis in trays of 24. The owner wants whole boxes and whole trays only.
(i) Write the prime factorisation of 18 and 24. (1) (ii) Find the LCM of 18 and 24. (1) (iii) What is the smallest number of laddoos and barfis that can be packed so the two counts are equal? (1) (iv) Find the HCF and state what it represents. (1)
Show answer
(i) 18 = 2 · 32 and 24 = 23 · 3.
(ii) LCM = 23 · 32 = 72.
(iii) 72 of each — 4 boxes of laddoos and 3 trays of barfis.
(iv) HCF = 2 · 3 = 6. It is the largest group size into which both counts divide exactly.
D · Short & Long Answer
Show answer
Given: 96 = 25·3, 404 = 22·101
Step 1: HCF = 22 = 4
Step 2: LCM = 25·3·101 = 9696
Final: 4 × 9696 = 38784 and 96 × 404 = 38784 → verified.
Show answer
Assume √5 = pq in lowest terms, q ≠ 0.
Step 1: 5 = p2q2 → p2 = 5q2. So p2 is divisible by 5, hence p is divisible by 5. Let p = 5k.
Step 2: 25k2 = 5q2 → q2 = 5k2, so q is divisible by 5 too.
Step 3: Both p and q are multiples of 5, contradicting “lowest terms”.
Final: Therefore √5 is irrational.
E · Board-Pattern Practice
Show answer
Take 13 common: 13(7 × 11 + 1) = 13 × 78 = 2 × 3 × 13 × 13.
It has more than two factors, so it is composite.
Common Mistakes Table
| Mistake students make | Correct idea |
|---|---|
| Using HCF × LCM = a·b for three numbers | It holds only for two numbers; for three, use prime factorisation. |
| Claiming √4 is irrational | √4 = 2, which is rational. Non-perfect squares are the ones that stay irrational. |
| Preparing Euclid’s division lemma for the board paper | It was removed from the Class 10 syllabus. Use the Fundamental Theorem of Arithmetic instead. |
| Quoting a result without proof in a 3-mark question | Show the assumption, the contradiction and the conclusion — that is where the marks are. |
Active Recall & Revision
Cover the answers. Say them aloud. Then check.
Q: State the Fundamental Theorem of Arithmetic.
Every integer greater than 1 is a unique product of primes, apart from the order of the factors.
Q: Why is √2 irrational?
Assuming √2 = p/q in lowest terms forces both p and q to be even, which contradicts “lowest terms”.
Q: When can you use HCF × LCM = a × b?
Only for two numbers, never for three.
Revise after 2 days, then 3 days, then 5 days, then 7 days. This locks memory.
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