Chapter 1: Real Numbers

Fundamental Concepts, Prime Factorization, Irrationality & Decimal Expansions

Exam Weightage & Blueprint

Total: ~6 Marks

Real Numbers is the foundational chapter. The board focus is strictly on prime factorization, proving irrationality, and decimal nature of rationals.

Question Type	Marks	Frequency	Focus Topic
MCQ	1	High	HCF/LCM, Terminating Decimals
Short Answer	2	Medium	Fundamental Theorem, Word Problems
Long Answer	3	Very High	Proof of Irrationality ($\sqrt{2}, \sqrt{3}$)

Fundamental Theorem of Arithmetic

Theorem 1.1: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Prime Factorization Engine

Enter a number to see its unique prime factors (Tree Structure).

HCF and LCM

For any two positive integers $a$ and $b$:

HCF (Highest Common Factor)

Product of the smallest power of each common prime factor.

LCM (Least Common Multiple)

Product of the greatest power of each prime factor involved.

$$ HCF(a, b) \times LCM(a, b) = a \times b $$

HCF-LCM Verifier

Enter two numbers to calculate and verify the formula.

⚠️ Common Mistake: The formula $ HCF \times LCM = a \times b \times c $ is NOT TRUE for three numbers.

Standard Board Question Types:

Find HCF using prime factorisation
Find LCM using prime factorisation
Find smallest number divisible by given numbers
Find greatest number dividing given numbers

Golden Rule:
Smallest number → use LCM
Greatest number → use HCF

Rational Numbers & Decimal Expansions

Let $x = p/q$ be a rational number (where $p, q$ are co-prime).

Terminating Decimal
If prime factorization of $q$ is of the form $2^n 5^m$ (where $n, m$ are non-negative integers).

Non-Terminating Recurring
If prime factorization of $q$ contains factors other than 2 or 5.

Decimal Detective

Enter a fraction $p/q$. Will it terminate?

Revisiting Irrational Numbers

Theorem 1.3: Let $ p $ be a prime number. If $ p $ divides $ a^2 $, then $ p $ divides $ a $, where $a$ is a positive integer.

Proof Builder: $\sqrt{2}$ is Irrational

Click the steps to reveal the logic flow used in Board Exams.

Step 1: Assumption (Contradiction Method) 0.5 Mark

Step 2: Squaring Both Sides 1 Mark

Step 3: Substitution 1 Mark

Step 4: Conclusion 0.5 Mark

Board Pattern Alert:
Proof of irrationality of √3, √5 follows the same steps as √2. Only replace 2 by the respective prime.

PYQ Trend:
CBSE alternates between √2 and √3 every few years.

Competency Based Question (Case Study)

Scenario: A seminar is being conducted by an Educational Organisation. The number of participants in Hindi, English, and Mathematics are 60, 84, and 108 respectively.

Q1: Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.

🧠 One-Page Board Revision Checklist

✔ A composite number has more than two factors
✔ Every composite number has a unique prime factorisation
✔ HCF = smallest power of common primes
✔ LCM = greatest power of all primes
✔ HCF × LCM = product of two numbers (only for two)
✔ If denominator = 2ⁿ5ᵐ → terminating decimal
✔ Otherwise → non-terminating recurring
✔ √p (p prime) is irrational

Exam Tip:
Writing correct definitions fetches full marks even if calculation goes wrong.

Concept Mastery Quiz 🎯

Test your readiness for the board exam.

1. The HCF of two consecutive even numbers is always:

2. The decimal expansion of $\frac{23}{2^3 5^2}$ will terminate after:

3. The product of a non-zero rational and an irrational number is: