Real Numbers Notes: Class 10 Maths Chapter 1
Class 10 Maths Chapter 1 Real Numbers covers Euclid's Division Lemma, the Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, and proofs of irrationality of certain numbers.
Overview of Topics & Subtopics (Click to Jump)
1. Euclid's Division Lemma
Short Note:
• For any two positive integers a and b, there exist unique whole numbers q and r such that a = bq + r, where 0 ≤ r < b.
• Euclid's Division Algorithm is a technique based on this lemma to compute the HCF of two positive integers by repeated division.
2. The Fundamental Theorem of Arithmetic
Short Note:
• Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
• This theorem is used to find the HCF and LCM of two or more positive integers using prime factorisation: HCF = product of the smallest power of each common factor; LCM = product of the greatest power of each factor involved.
• For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b.
3. Revisiting Irrational Numbers
Short Note:
• A number is irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
• Using the Fundamental Theorem of Arithmetic, it can be proved that √2, √3, √5, etc. are irrational, typically using proof by contradiction.