Real Number Class 10 ||Maths|| Chapter 1 NCERT Notes

Real Number Class 10 ||Maths|| Chapter 1 NCERT Notes

Real Number Class 10 ||Maths|| Chapter 1 NCERT Notes

1. Euclid’s Division Lemma

Euclid’s division lemma states that for any two positive integers, $a$ and $b$ , there exist unique integers $q$ and $r$ such that:
$a = b q + r where 0 \leq r < b$
a: Dividend
b: Divisor
q: Quotient
r: Remainder
Application: Euclid’s division lemma is useful for finding the HCF (Highest Common Factor) of two numbers using Euclid’s Algorithm.
Steps to find HCF using Euclid’s Algorithm:
Apply the division lemma to the given numbers.
Repeat the process with the divisor and the remainder until the remainder is zero.
The divisor at this stage is the HCF.

2. Fundamental Theorem of Arithmetic

Every composite number can be expressed (or factorized) as a product of primes and this factorization is unique, except for the order of the prime factors.
For example, the number 36 can be expressed as:
$36 = 2^{2} \times 3^{2}$
Applications of the Fundamental Theorem of Arithmetic:
To find the HCF and LCM of two numbers using their prime factorization.
HCF: Take the product of the smallest powers of common prime factors.
LCM: Take the product of the highest powers of all prime factors.

3. Irrational Numbers

A number that cannot be expressed as a ratio of two integers is called an irrational number. It has a non-terminating and non-repeating decimal expansion.
Examples:
$\sqrt{2}, \sqrt{3}, π$ , etc.
Important Results:
The sum or difference of a rational and an irrational number is always irrational.
The product or quotient of a non-zero rational number with an irrational number is always irrational.

4. Decimal Expansions of Rational Numbers

A rational number can either have a terminating or a non-terminating repeating decimal expansion.
Conditions:
A rational number will have a terminating decimal expansion if the denominator (in its simplest form) has prime factors 2 or 5 (or both).
If the denominator has any prime factor other than 2 or 5, the decimal expansion will be non-terminating and repeating.

5. HCF and LCM of Two Numbers

For any two positive integers $a$ and $b$ , the product of their HCF and LCM is equal to the product of the numbers themselves:
$HCF (a, b) \times LCM (a, b) = a \times b$

Key Concepts:

Prime Numbers: Numbers greater than 1 with no divisors other than 1 and itself.
Composite Numbers: Numbers that have divisors other than 1 and itself.
Real Numbers: Include both rational and irrational number