REAL NUMBERS
INTRODUCTION
IN THIS WE WILL DISCUSS THE REAL NUMBERS.
IMPORTANT PROPERTIES OF INTEGERS.
- Euclid's division algorithm
- Fundamental theoremof arithmetic
ALGORITHM
An algorithm is a series of well defined steps which give a procedure for solving a problem.
LEMA
Lema is a particular type of theorem .
EUCLID'S DIVISION ALGORITHM
It is defined as
a = bq + r
Given positive integers a and b, there exist unique integers q and r satisfying
a = bq + r, where 0 ≤ r < b, q, r is whole number.
let us discuss the relation of two numbers of a pair (14, 5), (15 ,4),(8, 3),(3, 7),(30, 6).
These pair can be rewritten as
14 = 5 * 2 + 4
15 = 4 * 3 + 3
8 = 3 * 2 + 2
3 = 7 * 0 + 3
30 = 6 * 5 + 0
These relations we obtain by division process. In division Q is called quotient and r called as remainder.
HIGHEST COMMON FACTOR (H.C.F) BY EUCLID DIVISION
Example
We find out H.C.F of the integer 972 and 21.
By euclid algoritm, 972 = 21 * 46 + 6
Here the divisor is 46 and reminder is 6. We applythe division algorithm again on 21 and 6.
21 = 6 * 3 + 3
Now the divisoris 6 and the reminder is 3 . We again apply divisionalgorithm on 6 and 3.
6 = 3 * 2 + 0
Now, the remainder is 0 and H.C.F of 6 and 3 is 3 and we can say that H.C.F of 972 and 21 is 3.
here, H.C.F (972 ,21) = H.C.F(21 , 6)=H.C.F(6 ,3) = 3.
Leta and b be two positive integers if
a = bq + r, 0 ≤ r < b
H.C.F(a ,b) = H.C.F(b ,r).
