L-03, Ch-1st,

Описание: 📘 Class 10 Maths Chapter 1 – Real Numbers | Full Chapter Summary

In this chapter, “Real Numbers”, we study the classification of numbers, prime factorization, irrational numbers, and decimal expansions. This chapter strengthens the number system foundation and develops logical reasoning used in higher mathematics.

🔹 1. Revisiting Real Numbers

Real numbers include all numbers on the number line:

Natural numbers

Whole numbers

Integers

Rational numbers

Irrational numbers

Together, rational and irrational numbers form the set of real numbers.

🔹 2. Fundamental Theorem of Arithmetic

Statement:

Every composite number can be expressed as a product of prime numbers, and this factorization is unique, apart from the order of factors.

Example:

This theorem is the basis for prime factorization and solving number problems.

🔹 3. Relationship Between HCF and LCM

For any two positive integers a and b:

This relation helps verify calculations and solve application problems.

🔹 4. Irrational Numbers and Proofs

An irrational number cannot be written in the form

Important result:

👉 √2 is irrational

Proof idea (contradiction method):

Assume √2 is rational → √2 = p/q (in simplest form)

After squaring: 2q² = p²

This implies p is even → p = 2k

Substituting again leads to q also being even

This contradicts the assumption that p/q is in lowest form.

Therefore, √2 is irrational.

Similar proofs apply to √3, √5, etc.

🔹 5. Decimal Expansion of Rational Numbers

A rational number � (q ≠ 0) has:

✅ Terminating decimal expansion

if q = �

✅ Non-terminating recurring decimal

if q contains primes other than 2 or 5

Examples:

1/8 → terminating

1/3 → recurring

This explains how decimal behavior depends on prime factors of the denominator.

🎯 Key Learning Outcomes

Understanding rational and irrational numbers

Proving numbers are irrational

Applying the Fundamental Theorem of Arithmetic

Using HCF–LCM relationship

Identifying decimal expansion types

Strengthening number system concepts

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