Chapter 1: Real Numbers

Overview

This page provides comprehensive Chapter 1: Real Numbers - Board Exam Notes. Class 10 Maths Chapter 1 Real Numbers Detailed Notes, HCF LCM Calculator, Irrationality Proofs, Decimal Expansions, and Interactive Study Material for Board Exams.

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.

Concept Check: Numbers ending with digit Zero
For any number like $a^n$ (e.g., $6^n$ or $4^n$) to end with the digit $0$, its prime factorisation must contain the primes 2 and 5. For example, $6^n = (2 \times 3)^n$ does not contain 5 as a prime factor, so $6^n$ can never end with the digit 0 for any natural number $n$.

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.

Summary (Theorems 1.5, 1.6 & 1.7): A rational number $x = \frac{p}{q}$ has a terminating decimal expansion if and only if the prime factorisation of $q$ is of the form $2^n 5^m$. To convert terminating fractions to decimals without long division, multiply the numerator and denominator by suitable powers of 2 or 5 to make the denominator a power of 10.

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 v3, v5 follows the same steps as v2. Only replace 2 by the respective prime.

PYQ Trend:
CBSE alternates between v2 and v3 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 = 2n5? ? terminating decimal
? Otherwise ? non-terminating recurring
? vp (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: