1. Real Numbers
Real Numbers
1.2 Euclid’s Division Lemma
Theorem 1.1 (Euclid’s Division Lemma) :
Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
Euclid’s division algorithm
To obtain the H.C.F of two positive integers, say c and d, with c > d, follow
the steps below:
Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤ r < d.
Step 2 : If r = 0, d is the H.C.F of c and d. If r ≠ 0, apply the division lemma to d and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required H.C.F.
This algorithm works because H.C.F (c, d) = H.C.F (d, r) where the symbol H.C.F (c, d) denotes the H.C.F of c and d, etc.
Remarks :
1. Euclid’s division lemma and algorithm are so closely interlinked that people often call former as the division algorithm also.
2. Although Euclid’s Division Algorithm is stated for only positive integers, it can be extended for all integers except zero, i.e., b ≠ 0.
1.3 The Fundamental Theorem of Arithmetic
Theorem 1.2 (Fundamental Theorem of Arithmetic) :
Every composite number can be expressed ( factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
Note:
HCF (a, b) X LCM (a, b) = a X b.
1.4 Revisiting Irrational Numbers
1.5 Revisiting Rational Numbers and Their Decimal Expansions
