If you are preparing for your board exam, JKBOSE Class 10th Maths Real Numbers Notes is the first thing you must master. This is Chapter 1 of JKBOSE 10th Maths and questions from this chapter appear every year. Most are short answer, direct and scoring.
When I prepared for my Class 10 board exam, I made sure Real Numbers was 100% clear before moving ahead. That gave me quick marks in the paper.
Quick Summary JKBOSE 10th Maths Chapter 1
| Topic | Details |
|---|---|
| Chapter | Real Numbers (Chapter 1) |
| Subject | Mathematics |
| Class | 10th JKBOSE |
| Important Topics | Euclid Division Lemma, HCF, LCM, Irrational Proof |
| Common Marks Area | 1 mark, 2 mark and sometimes 3 – 4 mark |
| Level | Easy to Medium |
What are Real Numbers
Real numbers are all numbers that can be placed on the number line.
This includes whole numbers, integers, fractions, decimals, rational numbers and irrational numbers.
Simply remember this if you can show a number on the number line, it is a real number.
JKBOSE Class 10th Maths Real Numbers Notes
Euclid's Division Lemma
For any two positive integers a and b, you can always write:
a = bq + r
Where:
- a = dividend
- b = divisor
- q = quotient
- r = remainder
Condition: 0 ≤ r < b
Example: If a = 17 and b = 5
17 = 5 × 3 + 2
Here q = 3 and r = 2
This formula is the base of Euclid’s Algorithm used to find HCF.
How to Find HCF Using Euclid's Algorithm
Step 1 – Divide the larger number by the smaller number.
Step 2 – If remainder becomes 0, divisor is HCF.
Step 3 – If remainder is not 0, divide previous divisor by remainder.
Step 4 – Repeat until remainder becomes 0.
Example: Find HCF of 135 and 225
225 = 135 × 1 + 90
135 = 90 × 1 + 45
90 = 45 × 2 + 0
HCF = 45
I practiced this algorithm on many number pairs before my exam. After 5–6 sums, it becomes automatic.
Fundamental Theorem of Arithmetic
Every composite number can be written as a product of prime numbers in one unique way, except for order.
Example:
12 = 2 × 2 × 3 = 2² × 3
This theorem is used in HCF and LCM by prime factorization.
- HCF = smallest powers of common prime factors
- LCM = greatest powers of all prime factors
- HCF × LCM = Product of two numbers
HCF and LCM by Prime Factorization
Example: Find HCF and LCM of 12 and 18
Prime factorization:
12 = 2² × 3
18 = 2 × 3²
HCF = 2¹ × 3¹ = 6
LCM = 2² × 3² = 36
Check: 6 × 36 = 216 = 12 × 18 ✓
This type of question appears regularly in JKBOSE 10th Maths Chapter 1.
Rational and Irrational Numbers
| Type | Definition | Examples |
|---|---|---|
| Rational | Can be written as p/q where q ≠ 0 | 1/2, 3, 0.5, -4 |
| Irrational | Cannot be written as p/q | √2, √3, π |
Important: Decimal of irrational number is non-terminating and non repeating.
The proof of √2 being irrational came directly in my board paper. Practice writing it fully step by step.
Decimal Expansions
| Type | Meaning | Example |
|---|---|---|
| Terminating | Decimal ends | 1/4 = 0.25 |
| NonmTerminating Repeating | Decimal repeats | 1/3 = 0.333... |
| Non Terminating Non-Repeating | Decimal never repeats | √2 = 1.41421... |
Rule: Write fraction in lowest form p/q. If q has only 2 and/or 5 as prime factors, decimal is terminating.
Example: 7/20 → 20 = 2² × 5 → Terminating ✓
Key Formulas Real Numbers
| Formula | Use |
|---|---|
| a = bq + r | Euclid's Division Lemma |
| HCF × LCM = a × b | Relation between HCF and LCM |
| HCF = smallest common prime powers | Prime factorization |
| LCM = greatest prime powers | Prime factorization |
JKBOSE Class 10th Real Numbers Handwritten Notes PDF
I have written these notes by hand exactly how I prepared for my board exam and scored 466/500.
- Clear step-by-step solutions
- Important formulas highlighted
- Irrational proof written in exam format
- Practice questions included
Important Questions for Practice
1. Use Euclid's Division Algorithm to find HCF of 135 and 225.
2. Find HCF and LCM of 12, 15 and 21.
3. Prove that √2 is irrational.
4. Can 6ⁿ end with digit 0 for any natural number?
5. Without division, state whether 17/8 is terminating.
Exam Tips from Dar Zaid
Priority: Euclid Algorithm and prime factorization.
Writing method: Show every division step clearly. Do not skip lines.
Common mistake: Students mix smallest and greatest powers in HCF and LCM. Write this rule on rough column before solving.
Time tip: Attempt this chapter early in the paper. It builds confidence.
For all chapters, visit JKBOSE Class 10th Maths Notes – Chapter Wise.
Frequently Asked Questions
What is Euclid Division Lemma JKBOSE Class 10?
It states that for any two positive integers a and b, there exist integers q and r such that a = bq + r where 0 ≤ r < b. It is used to find HCF step by step.
How many marks does Real Numbers carry in JKBOSE?
Usually short answer questions of 1 or 2 marks appear. Sometimes a 3–4 mark HCF question is asked from this chapter.
Is Real Numbers important for JKBOSE exam?
Yes. It is Chapter 1 and one of the most scoring chapters in JKBOSE Class 10 Maths.
What is the Fundamental Theorem of Arithmetic?
It states that every composite number can be written as a product of prime numbers uniquely, except for the order of factors.
What is the difference between rational and irrational numbers?
Rational numbers can be written in the form p/q where q ≠ 0. Irrational numbers cannot be written in this form. Their decimal expansion is non-terminating and non-repeating.
