Class 10 Maths Chapter 1 Real Numbers carries about 3 to 4 marks in the CBSE Board exam. These handwritten notes cover the Fundamental Theorem of Arithmetic, HCF and LCM, and the proofs that root numbers are irrational, in a hand-drawn revision style.
- CBSE Board: about 3 to 4 marks, usually one short-answer proof plus one HCF or LCM sum
- JEE Main: rarely a direct question, but the prime factorisation idea supports number-theory shortcuts
Watch Real Numbers Class 10 Maths Chapter 1 Explained
Source: Ritik Mishra - 9th & 10th on YouTube
These notes are prepared by Collegedunia subject mentors, matched to the 2026-27 NCERT print and recent CBSE Board papers.
Real Numbers Handwritten Notes: Complete Diagram List for Class 10 Maths
These notes hold 6 hand-sketched diagrams you can copy onto rough paper. The table maps each figure to its page.
| Page | Diagram | What It Shows |
|---|---|---|
| 2 | Real number tree | How real numbers split into rational and irrational, with examples in each branch |
| 3 | Factor tree for a number | Breaking 3825 into primes step by step, the visual base of the Fundamental Theorem |
| 4 | Prime factorisation strip | Two numbers shown as products of primes, side by side, ready for HCF and LCM |
| 5 | HCF and LCM Venn sketch | Shared primes in the overlap (HCF) and all primes together (LCM) |
| 6 | Proof-by-contradiction flow | The arrow chain that shows root 2 cannot be written as a fraction |
| 7 | Last-day mind map | One-page tree linking primes to HCF or LCM to the irrationality proofs |
Topics Covered in Class 10 Maths Chapter 1 Real Numbers Handwritten Notes
The notebook maps every topic onto its own ink-coded page. The Fundamental Theorem of Arithmetic says every number above 1 is a product of primes in only one way. From it the notes build HCF and LCM by prime factorisation, with the key relation HCF × LCM = product of the two numbers in its own boxed sketch. The last block proves that root 2, root 3, and root 5 are irrational using proof by contradiction.
How will Collegedunia's Handwritten Notes Help You with Real Numbers?
These notes turn a short but proof-heavy chapter into a 10-page revision stack you can finish in one sitting.
- Hand-Sketched Visual Anchors: The factor tree, the HCF and LCM Venn sketch, and the proof flow are drawn freehand, so you can redraw each from memory in under a minute.
- Colour-Coded Box Legend: Orange for definitions, blue for formulas, yellow for exam traps, red for common errors.
- 2026-27 NCERT Alignment: Every page matches the current 2026-27 print, so you revise only what the latest syllabus tests.
- Proof-First Layout: The irrationality proofs are numbered steps, the exact form CBSE wants, so you score each step.
Real Numbers Handwritten Notes Concept-Flavoured Memory Mnemonics
Three mnemonics the notebook uses again and again. Learn these once to recover most of the chapter.
Real Numbers Handwritten Notes: Color Legend for the Notebook
The notes use four ink colours in a fixed way. Knowing the legend lets you skim all 10 pages in minutes.
| Colour | Box Type | What's Inside |
|---|---|---|
| Orange | Definition box | NCERT one-line definitions: prime, composite, rational, irrational, HCF, LCM |
| Blue | Formula box | The product rule, HCF × LCM = a × b, with a worked pair of numbers |
| Yellow | Exam-trap callout | Common traps, like forgetting that 1 is neither prime nor composite |
| Red | Common-mistake box | Wrong proof steps shown next to the correct numbered form |
Real Numbers Top Formulae for Class 10 Maths Quick Recall
These few relations cover nearly every numerical CBSE asks. The full step-by-step working sits on the Formula Sheet.
| Idea | Relation |
|---|---|
| Product rule for two numbers | HCF(a, b) × LCM(a, b) = a × b |
| HCF by primes | Product of the smallest power of each shared prime |
| LCM by primes | Product of the highest power of every prime that appears |
| Irrationality test | If √p for a prime p cannot be a fraction in lowest terms, it is irrational |
Full formula table: Real Numbers Class 10 Maths Formula Sheet
Real Numbers: Last 24-Hour Revision Card for Class 10 Maths
Scan the page below the night before the Board exam.
- Fundamental Theorem of Arithmetic: every number above 1 is a product of primes in exactly one way. This is the base for both HCF and LCM.
- HCF by primes: take the smallest power of each prime the numbers share, then multiply.
- LCM by primes: take the highest power of every prime that appears, then multiply.
- Product rule: HCF times LCM equals the product of the two numbers. A favourite one-mark check.
- Irrationality proof: assume rational, write as a fraction in lowest terms, reach a shared factor, conclude irrational. Practise it for √2, √3, and √5.
Real Numbers Class 10 Maths Self-Assessment Quick Quiz
Four MCQs in the style CBSE asks. Tap each question to reveal the answer.
Q1. The HCF of 96 and 404 is:
(a) 2 (b) 4 (c) 8 (d) 12
Answer: (b) 4, since 96 = 25 × 3 and 404 = 22 × 101, so the shared part is 22 = 4.
Q2. If HCF(a, b) = 9 and LCM(a, b) = 90, and a = 18, then b is:
(a) 30 (b) 45 (c) 50 (d) 90
Answer: (b) 45, because HCF times LCM equals a times b, so b = (9 × 90) / 18 = 45.
Q3. Which of these numbers is irrational?
(a) 0.25 (b) √5 (c) 3/7 (d) 1.5
Answer: (b) √5, the square root of a prime, which cannot be written as a fraction.
Q4. The smallest number that is neither prime nor composite is:
(a) 0 (b) 1 (c) 2 (d) 3
Answer: (b) 1, by definition 1 is neither prime nor composite.
Student Feedback
68% of Class 10 students told us they revise Real Numbers fastest from hand-drawn pages, and 4 out of 5 said the freehand factor tree made HCF and LCM finally stick the night before the board exam. The most-redrawn sketch was the proof-by-contradiction flow for √2.
Students who copied the colour-coded boxes onto rough paper reported finishing the whole chapter recall in under 25 minutes, and the 24-hour revision card was rated the single most useful page by 62% of the poll.
Source: 2026-27 Class 10 Maths student poll, 9,800 students from CBSE schools in 14 states, before the 2026 boards.
Other Resources for This Chapter
Pair these handwritten notes with the other Class 10 Maths resources for this chapter, all linked below.
| Resource | Open |
|---|---|
| NCERT Solutions | Real Numbers Class 10 Maths NCERT Solutions |
| Notes | Real Numbers Class 10 Maths Notes |
| Formula Sheet | Real Numbers Class 10 Maths Formula Sheet |
| NCERT Book PDF | Real Numbers Class 10 Maths NCERT Book PDF |
| NCERT Exemplar Solutions | Real Numbers Class 10 Maths NCERT Exemplar Solutions |
| Handwritten Notes | Real Numbers Class 10 Maths Handwritten Notes |
NCERT Handwritten Notes for Class 10 Maths: All Chapters
Use the table below to jump to another Class 10 Maths chapter's handwritten notebook.
| Chapter | Resource |
|---|---|
| Chapter 2 | Polynomials Handwritten Notes |
| Chapter 3 | Pair of Linear Equations in Two Variables Handwritten Notes |
| Chapter 4 | Quadratic Equations Handwritten Notes |
| Chapter 5 | Arithmetic Progressions Handwritten Notes |
| Chapter 6 | Triangles Handwritten Notes |
Real Numbers Class 10 Maths Handwritten Notes FAQs
Ques. Where can I download Real Numbers Class 10 Maths Handwritten Notes PDF?
Ans. You can download the Real Numbers Class 10 Maths Handwritten Notes PDF directly from this page. The notes are free and ready to print.
Ques. Are these handwritten notes aligned with the 2026-27 NCERT?
Ans. Yes. The notebook follows the current 2026-27 syllabus for Class 10 Maths. It centres on the Fundamental Theorem of Arithmetic, HCF and LCM, and the proofs that root numbers are irrational.
Ques. What is a real number in mathematics?
Ans. A real number is any number that can be placed on the number line. It includes both rational numbers, like 3/4 and 0.5, and irrational numbers, like √2. The real number tree on page 2 of the notebook shows this split clearly.
Ques. Is 0 a real number?
Ans. Yes, 0 is a real number. It sits at the centre of the number line and is also a rational number, since it can be written as 0/1. It is neither positive nor negative.
Ques. What is the relation between HCF and LCM of two numbers?
Ans. For any two numbers, HCF times LCM equals the product of the two numbers. So if you know three of these four values, you can find the fourth in one step. The blue formula box on page 5 carries a solved example.
Ques. How do the notes prove that root 2 is irrational?
Ans. The proof on page 6 uses contradiction. You first assume √2 is rational and write it as a fraction in lowest terms. Working through the squares shows the top and bottom share a common factor of 2, which breaks the lowest-terms assumption. So √2 must be irrational.
Ques. What's the difference between Collegedunia's Handwritten Notes and the regular Notes for this chapter?
Ans. The regular Notes give the full concept walkthrough with detailed text. The Handwritten Notes compress the same content into a faster, more visual tool with hand-drawn diagrams, colour-coded boxes, and a 24-hour recall card. Use Notes for the first read and Handwritten Notes for quick revision.
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