Real Numbers — الرياضيات – الصف 10 (CBSE) بشرح مبسّط

ما ستتعلّمه

1. 1

   الدرس 1: Prime Factors and the Fundamental Theorem of Arithmetic

   5 دقائق
   • From Positive Integers to Irrational Numbers
   • Read Numbers as Products of Prime Factors
   • Construct Larger Numbers from Primes
   • Follow the Complete Factor Tree of 32760
   • State the Fundamental Theorem of Arithmetic
   • The Textbook's Historical Noteغير مُقيَّم
   • Changing Order Does Not Change Prime Factorisation
2. 2

   الدرس 2: Applications: HCF, LCM and Prime Factorisation

   5 دقائق
   • Example 1: Can a Power of Four End in Zero?
   • Example 2: Find the HCF and LCM of 6 and 20
   • The Product Relationship for Two Positive Integers
   • Example 3: HCF First, Then LCM
   • Example 4: HCF and LCM of Three Integers
   • The Two-Number Identity Does Not Extend Unchanged
   • Exercise 1.1: Choose the Appropriate Method
3. 3

   الدرس 3: Why Root Two and Root Three Are Irrational

   5 دقائق
   • Recall What an Irrational Number Is
   • Theorem 1.2: From a Square to Its Base
   • Optional Enrichment: Proof of Theorem 1.2غير مُقيَّم
   • Theorem 1.3: Assume Root Two Is Rational
   • Root Two: Reach the Contradiction
   • Example 5: Begin the Proof for Root Three
   • Example 5: Complete the Proof for Root Three
4. 4

   الدرس 4: Irrational Expressions, Three-Number Relationships and Practice

   6 دقائق
   • Combine Rational and Irrational Numbers Carefully
   • Example 6: Five Minus Root Three
   • Example 7: Three Times Root Two
   • Exercise 1.2: Plan Your Irrationality Proofs
   • Section 1.4: Retrieve the Chapter's Three Summary Points
   • A Note to the Reader: Relationships for Three Integers
   • Chapter Recap: Factorise, Justify and Practise

المفاهيم الأساسية

Prime products and factorisation

Instead of starting with a number and finding its factors, start with prime numbers and multiply them.

Constructing integers from prime products

A prime may also be repeated, with a power recording those repeated copies.

A complete prime factor tree

The highlighted leaves now give the complete prime factorisation.

The meaning of uniqueness

Once this order is fixed, the written factorisation is unique.

Revisiting Irrational Numbers

Rational and irrational numbers together make up the real numbers.

Properties of rational–irrational combinations

The sum or difference of a rational number and an irrational number is irrational.

تركيز الامتحان

Board exam

CBSE assesses this chapter in the 2026-27 year-end board examination.

Syllabus

REAL NUMBERS: 1. Fundamental Theorem of Arithmetic - statements after reviewing work done earlier and after illustrating and motivating through examples. 2. Proofs of irrationality of root 2, root 3, root 5. Learning outcomes: describes Fundamental Theorem of Arithmetic with examples; proves algebraically the irrationality of numbers like root 2, root 3, root 5, 3 + 2 root 5 etc.; applies the Fundamental Theorem of Arithmetic to solve problems related to real-life contexts.

Not assessed

These parts go beyond the 2026-27 syllabus and are taught for understanding only: The Textbook's Historical Note and Optional Enrichment: Proof of Theorem 1.2.

Exam note

The NCERT proof of Theorem 1.2 is marked 'not from the examination point of view' in the textbook.

Paper pattern

Mathematics Standard: 80 marks, 3 hours, 38 questions. Section A: 18 MCQs + 2 assertion-reason (1 mark each); B: 5 VSA (2 marks); C: 6 SA (3 marks); D: 4 LA (5 marks); E: 3 case-study questions (4 marks). No calculators.

Question style

About 50% of board questions are competency-focused.

Practice in the app

84 board-pattern questions with marking schemes: 29 multiple choice (1 mark), 13 assertion–reason (1 mark), 16 very short answer (2 marks), 14 short answer (3 marks), 3 long answer (5 marks) and 9 case-based (4 marks).

أسئلة نموذجية مع نقاط التصحيح

أسئلة بنمط امتحان المجلس من مجموعة تدريبات AI Lessons لهذا الفصل. توضّح نقاط التصحيح ما يبحث عنه المصحّح؛ والإجابات النموذجية الكاملة في التطبيق.

1. Very short answerدرجتان

   A student accepts 253 = 11 × 23 but rejects 4 = 2 × 2, saying that a prime may occur only once. Correct the student's reasoning.

   نقاط التصحيح

   • Identifies both representations as valid prime factorisations.1
   • Explains that prime factors may repeat, as 2 does in the factorisation of 4.1

2. Short answer3 درجات

   A calculation record correctly lists HCF(96, 404) = 4 but gives LCM(96, 404) = 96 × 404. Identify the missing operation and correct the LCM, showing the identity used.

   نقاط التصحيح

   • States the two-integer HCF–LCM product identity.1
   • Identifies division by the HCF and substitutes (96 × 404)/4.1
   • Obtains the correct LCM, 9696.1

3. Long answer5 درجات

   Prove that √5 is irrational. Hence explain why 7 √5 is irrational.

   نقاط التصحيح

   • Assumes √5 = a/b with coprime positive integers a and b.0.5
   • Obtains b √5 = a and 5b² = a².0.5
   • Uses the prime-divisibility theorem to infer that 5 divides a.0.5
   • Writes a = 5c and substitutes to obtain 5b² = 25c².0.5
   • Simplifies to b² = 5c².0.5
   • Uses the theorem again to infer that 5 divides b.0.5
   • Identifies the common factor 5 as contradicting coprimality.0.5
   • Rejects the assumption and concludes that √5 is irrational.0.5
   • Uses the non-zero rational multiplier 7 to justify that 7 √5 is irrational.1

الأسئلة الشائعة

فصول أخرى في الرياضيات – الصف 10

• الفصل 2Polynomialsالدروس: 4 · الدقائق: 21→
• الفصل 3Pair of Linear Equations in Two Variablesالدروس: 4 · الدقائق: 24→
• الفصل 4Quadratic Equationsالدروس: 4 · الدقائق: 22→
• الفصل 7Coordinate Geometryالدروس: 3 · الدقائق: 21→

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