Question

prove that root 5 is irrational

Answer 1

by the contradiction method we can find it

Answer 2

WE HAVE TO PROVE THAT ROOT 5 IS IRRATIONAL 
LET US ASSUME THE OPPOSITE,
i.e ROOT 5 IS RATIONAL
HENCE ROOT 5 CAN BE WRITTEN IN THE FORM OF a|b
WHERE A AND B(B IS NOT EQUALS TO 0)ARE CO PRIME (NO COMMON FACTOR OTHER THAN 1
HENCE ROOT 5=A/B 
ROOT 5B=A
SQUARING OB BOTH THE SIDES 
(ROOT5B SQUARE)= A SQUARE
 A SQUARE/5= B SQUARE
HENCE 5 DIVIDES A SQUARE 
BY THE THEOREM :IF P IS A PRIME NUMBER AND DIVIDES A SQUARE,THEN P DIVIDES A, WHERE A IS A POSITIVE NUMBER.
SO, 5 SHALL DIVIDE A ALSO                                                    ....(1)
HENCE,WE CAN SAY
A/5=C WHERE C IS SOME INTEGER
SO, A =5C
NOW WE KNOW THAT 5B SQUARE =A SQUARE
PUTTING A=5C 
5B SQUARE = (5C)SQUARE

5B SQUARE=25C SQUARE
B SQUARE=1/5*25C SQUARE
B SQUARE=5C SQUARE
BSQUARE/5 =C SQUARE
HENCE 5 DIVIDES B SQUARE  
SO 5 DIVIDES B ALSO                                                         ....(2)
BY (1) AND (2)
5 DIVIDES BOTH A AND B 
HENCE 5 IS A FACTOR OF A AND B
SO A AND B HAVE A FACTOR 5 
THEREFORE, A AND B ARE NOT CO PRIME.
HENCE OUR ASSUMPTION IS WRONG
THIS CONTRADICTS THAT, ROOT 5 IS IRRATIONAL NO.