prove that 3 + 2 √5 is an irrational number
proof:-
let 3 + 2 √5 is a rational number
then 3+2√5=p/q
(where p and q have not a any common factor and q is not equal to zero)
now, 2√5=p/q - 3
2√5=p/q - 3 /1
2√5=(p- 3q)/q
√5=(p- 3q)/2q
here, (p- 3q)/2q =rational number
√5= irrational number
this relation irrational number =rational number is not possible because rational is not equal to irrational ,so our recognition is failed.
hense, 3 + 2 √5 is an irrational number.
prove that 3√2 is an irrational number.
proof:-
let 3√2 is a rational number
then 3√2 =p/q
(where p and q have not a any common factor and q is not equal to zero)
now, √2 =p/3q
√2=p/3q
here, p/3q =rational number
√2= irrational number
this relation irrational number =rational number is not possible because rational is not equal to irrational ,so our recognition is failed.
hense,3√2 is an irrational number.
prove that 4 - √5 is an irrational number.
proof:-
let 4 - √5 is a rational number
then 4 - √5 =p/q
(where p and q have not a any common factor and q is not equal to zero)
now, 4 -p/q =√5
√5= 4 -p/q
√5= (4q -p)/q
here, (4q -p)/q =rational number
√5= irrational number
this relation irrational number =rational number is not possible because rational is not equal to irrational ,so our recognition is failed.
hense,4 - √5 is an irrational number.
prove that 6+√ 2 is an irrational number.
proof:-
let 6+√ 2 is a rational number
then 6+√ 2=p/q
(where p and q have not a any common factor and q is not equal to zero)
now,√ 2=p/q-6
√ 2=(p-6q)/q
here, (p-6q)/q =rational number
√ 2= irrational number
this relation irrational number =rational number is not possible because rational is not equal to irrational ,so our recognition is failed.
hense,6+√ 2 is an irrational number.
