Class 10 Mathematics – Real Numbers Notes (Conceptual)

1. Introduction

• Real numbers are all the numbers that can be represented on the number line.

• They include:

  • Rational numbers (fractions, decimals that terminate or repeat)

  • Irrational numbers (decimals that never terminate and never repeat)

• Important idea: Every number on the number line is a real number.

2. Fundamental Concepts

a) Prime Numbers

• Numbers greater than 1 with exactly two factors: 1 and itself.

b) Composite Numbers

• Numbers greater than 1 with more than two factors.

c) Co-prime Numbers

• Two numbers are co-prime if their only common factor is 1.

3. Properties of Real Numbers

a) Closure Property

• Addition or multiplication of any two real numbers always gives a real number.

b) Commutative Property

• Order does not matter in addition or multiplication.

c) Associative Property

• Grouping does not matter in addition or multiplication.

d) Distributive Property

• Multiplication distributes over addition.

e) Identity Elements

• Additive identity: Number that does not change another number when added (0).

• Multiplicative identity: Number that does not change another number when multiplied (1).

4. Division Algorithm Concept

• Any number can be divided by another (except zero) to give:

  • Quotient: How many times the divisor fits.

  • Remainder: What is left.

• Key rule: The remainder is always smaller than the divisor.

5. HCF and LCM (Highest Common Factor & Lowest Common Multiple)

• HCF: Largest number that divides two or more numbers exactly.

• LCM: Smallest number that is a multiple of two or more numbers.

Important idea:

• HCF and LCM are closely related to the product of the numbers.

• Prime factorization helps to find HCF and LCM easily.

6. Rational and Irrational Numbers

• Rational numbers: Can be written as fractions. Their decimals terminate or repeat.

• Irrational numbers: Cannot be written as fractions. Their decimals never terminate and never repeat.

• All real numbers are either rational or irrational.

7. Euclid’s Division Lemma (Conceptual Idea)

• Every number can be expressed as a combination of another number, giving a quotient and remainder.

• This is useful for:

  • Finding HCF

  • Proving properties of numbers

8. Important Points to Remember

• 1 is neither prime nor composite.

• Prime factorization is key for HCF and LCM.

• Irrational numbers cannot be expressed as fractions.

• Real numbers are closed under addition, subtraction, multiplication, and division (except by zero).

• Division algorithm helps in understanding remainders and quotients clearly.