Continuation from Class IX on real numbers and irrational numbers.
Key Concepts
Euclid’s Division Algorithm
Describes how any positive integer ( a ) can be divided by another positive integer ( b ) resulting in a smaller remainder ( r ) (i.e., ( r < b )).
Application: Compute HCF of two positive integers.
Fundamental Theorem of Arithmetic
Every composite number can be factorized uniquely as a product of prime factors, apart from the order of the factors.
Important applications:
Proving the irrationality of numbers (e.g., ( \sqrt{2} ), ( \sqrt{3} ), ( \sqrt{5} )).
Determining when the decimal expansion of a rational number is terminating or non-terminating repeating by analyzing the prime factorization of its denominator.
Fundamental Theorem of Arithmetic
Every composite number ( n ) can be expressed as ( n = p1^{e1} p2^{e2} \cdots pk^{ek} ), where ( pi ) are prime numbers and ( ei ) are their respective powers.
Example: Factorization of 32760 is ( 2^3 \times 3^2 \times 5 \times 7 \times 13 ).
Applications of the Theorem
Finding HCF and LCM using prime factorization:
HCF is the product of the smallest powers of common prime factors.
LCM is the product of the highest powers of all prime factors.
Irrational Numbers
A number is irrational if it cannot be expressed as ( \frac{p}{q} ) where ( p ) and ( q ) are integers and ( q \neq 0 ).
Examples of irrational numbers: ( \sqrt{2}, \sqrt{3}, \pi ).
Proof of Irrationality Using the Fundamental Theorem of Arithmetic
Theorems involved:
If a prime ( p ) divides ( a^2 ), then it divides ( a ).
Classical proofs for proving for ( \sqrt{2} ), ( \sqrt{3} ) using contradiction.
Summary Points
Fundamental Theorem of Arithmetic: unique prime factorization of composites.
Prime divisibility properties related to even powers.
Validated proofs for numbers like ( \sqrt{2} ) and ( \sqrt{3} ) as irrational.