Ch 1 revisits real numbers with formal proofs. Students learn Euclid's division lemma for computing HCF, the Fundamental Theorem of Arithmetic, and prove the irrationality of numbers like √2 and √3.
For positive integers a and b, there exist unique integers q and r such that a = bq + r (0 ≤ r < b). This lemma is used to find the HCF of two numbers through repeated division (Euclid's algorithm).
Every composite number can be expressed as a product of primes, and this factorisation is unique (apart from the order). This is used to find HCF and LCM efficiently and to prove irrationality of numbers.
To prove √2 is irrational: assume √2 = p/q (reduced), then p² = 2q², so p is even → p = 2k → 4k² = 2q² → q² = 2k² → q is even. Both p, q even contradicts reduced form. Hence √2 is irrational.
Download: https://ncert.nic.in/textbook/pdf/jemh101.pdf | Complete book: https://ncert.nic.in/textbook/pdf/jemh1ps.zip
Given positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b. It is the basis of Euclid's algorithm for finding HCF.
It guarantees unique prime factorisation of every natural number > 1. This uniqueness is used to find HCF/LCM, prove irrationality, and forms the basis of number theory in higher mathematics.
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