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CBSE • Class X • Mathematics • Ch 1
Estimated Time: 45 Mins
Study Progress: In Progress

Real Numbers

In Class 10 Mathematics, "Real Numbers" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

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🔢 Have You Ever Wondered?

How do mathematicians prove with absolute logical certainty that $\sqrt{2}$ can never be written as a fraction? The Fundamental Theorem of Arithmetic ...

How do mathematicians prove with absolute logical certainty that $\sqrt{2}$ can never be written as a fraction? The Fundamental Theorem of Arithmetic guarantees that every integer has a unique prime fingerprint.

Why This Chapter Matters

In Class 10 Mathematics, "Real Numbers" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Before You Begin (Prerequisites)

  • Divisibility and prime factorization.
  • Rational and irrational numbers from Class 9.
  • Proof by contradiction.

What You Will Learn (Core Objectives)

  • State and apply the Fundamental Theorem of Arithmetic (Unique Factorization Theorem).
  • Calculate HCF and LCM of integers using prime factorization powers.
  • Prove by contradiction that $\sqrt{2}, \sqrt{3},$ and $\sqrt{5}$ are irrational numbers.
  • Prove that numbers like $3 + 2\sqrt{5}$ are irrational.
  • Explain the connection between prime factorization of denominators and terminating decimals.

Chapter Roadmap & Progression

1 1. The Fundamental Theorem of Arith...
2 2. Proof of Irrationality by Contra...

Complete Concept Guide (100% Curriculum Coverage)

1. The Fundamental Theorem of Arithmetic

Every composite number can be expressed (factorized) uniquely as a product of primes, apart from the order in which the prime factors occur. E.g. $1260 = 2^2 \times 3^2 \times 5 \times 7$.
• HCF: Product of smallest power of each common prime factor.
• LCM: Product of greatest power of each prime factor involved in the numbers.
• $\mathbf{\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b}$.

2. Proof of Irrationality by Contradiction

To prove $\sqrt{2}$ is irrational, assume the opposite: that $\sqrt{2} = \frac{a}{b}$ where $a, b$ are coprime integers ($b \ne 0$). Squaring both sides: $2 = \frac{a^2}{b^2} \implies a^2 = 2b^2$. Thus $2$ divides $a^2$, which implies $2$ divides $a$. Let $a = 2c \implies 4c^2 = 2b^2 \implies b^2 = 2c^2$, so $2$ divides $b$. Hence, 2 divides both $a$ and $b$, contradicting that $a$ and $b$ are coprime! Thus, $\sqrt{2}$ is irrational.

Visual Learning & Conceptual Map

Real Numbers Master Matrix

Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture

1. The Fundamental Theorem of Arithmetic • 2. Proof of Irrationality by Contradiction

Chapter Summary & 10 Key Takeaways

Takeaway 1
Fundamental Theorem: Every composite number has a unique prime factorization.
Takeaway 2
Relationship: $\text{HCF} \times \text{LCM} = a \times b$ holds for two positive integers.
Takeaway 3
Irrationality: Proved using proof by contradiction.
Takeaway 4
Coprime: Integers sharing no common factor other than 1.
Takeaway 5
High Board Priority: Proving $\sqrt{2}$ or $3 + \sqrt{5}$ irrational is a guaranteed 3-mark CBSE question.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
Prove that $\sqrt{5}$ is an irrational number.
Reveal Answer & Explanation
Answer: Assume $\sqrt{5} = a/b$ (coprime). $5b^2 = a^2 \implies 5$ divides $a$. Let $a = 5c \implies 5b^2 = 25c^2 \implies b^2 = 5c^2 \implies 5$ divides $b$. This contradicts that $a$ and $b$ are coprime. Hence $\sqrt{5}$ is irrational.
Proof by contradiction using divisibility by 5.
2
Find the HCF and LCM of 96 and 404 using prime factorization.
Reveal Answer & Explanation
Answer: $96 = 2^5 \times 3$; $404 = 2^2 \times 101$. $\text{HCF} = 2^2 = 4$. $\text{LCM} = \frac{96 \times 404}{4} = 9,696$.
HCF = 4, LCM = 9696.
3
Show that $5 \times 11 \times 13 + 13$ is a composite number.
Reveal Answer & Explanation
Answer: $13(5 \times 11 + 1) = 13(55 + 1) = 13 \times 56$. Since it has factors other than 1 and itself, it is composite.
Factor out 13.
4
Explain why $6^n$ can never end with the digit 0 for any natural number $n$.
Reveal Answer & Explanation
Answer: For a number to end in 0, its prime factorization must contain both 2 and 5. The prime factorization of $6^n = (2 \times 3)^n = 2^n \times 3^n$. Since 5 is not a factor, $6^n$ can never end in 0.
Requires prime factor 5.
5
If $\text{HCF}(306, 657) = 9$, find $\text{LCM}(306, 657)$.
Reveal Answer & Explanation
Answer: $\text{LCM} = \frac{306 \times 657}{9} = 34 \times 657 = 22,338$.
LCM = (a * b) / HCF.
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