Real Numbers
Study Points
1. A
lemma is a proven statement used for proving another statement.
2. An
Algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
3.
Muḥammad ibn Mūsā al-Khwārizmī, a persian mathematician coined the term 'algorithm'.
4.
Euclid's division lemma: Given positive integers
a and
b there exist whole numbers
q and
r satisfying a = bq +r, 0 ≤ r < b.
5.
Euclid's division algorithm: In order to compute the HCF of two positive integers say
a and
b, with a > b by using Euclid's algorithm we follow the given below steps:
| STEP-❶: | Apply Euclid's division lemma to a and b and obtain whole numbers q1 and r1
such that a = bq1 + r1, 0 ≤ r < b |
| STEP-❷: | If r1 = 0, b is the HCF of a and b. |
| STEP-❸: | If r1
≠ 0, apply Euclid's division lemma to b and r1 and obtain two whole numbers q1 and r2 such that b = q1r1 + r2 |
| STEP-❹: | If r2 = 0, r1 is the HCF of a and b |
| STEP-❺: | If r2
≠ 0, apply Euclid's division lemma to r1 and r2 and continue the above process till the remainder rn is zero. The divisor at this stage i.e. rn-1or the non-zero remainder at the previous stage is the HCF of a and b. |
6. Euclid’s Division Algorithm is stated for only positive integers but it can be extended for all integers except zero, i.e,
b ≠ 0.
7.
The Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorized) as product of primes, and this factorization is unique except for the order in which the prime factors occur.
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