Question

Prove that following numbers are irrational
a) 7√5
b)6+√ 2
c)1/√2
d)√2+√3

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Answer 1

1. Let us assume that 7√5 is a rational number.
we know that the rational numbers are in the form of p/q form where p,q are intezers.
so, 7√5 = p/q
     p = 7√5q
we know that 'p' is a rational number. so 7√5 q must be rational since it equals to p
but it doesnt occurs with 7√5 since √5  its not an intezer
therefore, p =/= 7√5q
this contradicts the fact that 7√5 is an irrational number
hence our assumption is wrong and 7√5 is an irrational number.

2. Let us assume that 6+√2 is a rational number.
we know that the rational numbers are in the form of p/q form where p,q are intezers.
so, 6+√2 = p/q
     p = (6+√2)q
we know that 'p' is a rational number. so (6+√2)q must be rational since it equals to p
but it doesnt occurs with it because sum or rational n irrational number results irrational. so, (6+√2) since its not an intezer
therefore, p =/= (6+√2)q
this contradicts the fact that (6+√2)q is an irrational number
hence our assumption is wrong and 6+√2 is an irrational number.

3. Let us assume that 1/√2 is a rational number.
we know that the rational numbers are in the form of p/q form where p,q are intezers.
so, 1/√2 = p/q
     p = q/√2we know that 'q' is a rational number. so q/√2 must be rational since it equals to p 

but it doesnt occurs with q/√2 since its not an intezer. (since √2 isnt rational)therefore, p =/= q√2

this contradicts the fact that 1/√2 is an irrational number
hence our assumption is wrong and 1/√2 is an irrational number.

4. Let us assume that √2+√3 is a rational number.
we know that the rational numbers are in the form of p/q form where p,q are intezers.
so, √2+√3 = p/q
     p = (√2+√3)q
we know that 'p' is a rational number. so (√2+√3)q must be rational since it equals to p
but it doesnt occurs with it because √2, √3 since are not an intezers
therefore, p =/= (√2+√3)q
this contradicts the fact that (√2+√3)q is an irrational number
hence our assumption is wrong and √2+√3 is an irrational number.

Answer 2

1. Let us assume that 7√5 is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are intezers.

so, 7√5 = p/q

p = 7√5q

we know that 'p' is a rational number. so 7√5 q must be rational since it equals to p

but it doesnt occurs with 7√5 since √5 its not an intezer

therefore, p =/= 7√5q

this contradicts the fact that 7√5 is an irrational number

hence our assumption is wrong and 7√5 is an irrational number.

2. Let us assume that 6+√2 is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are intezers.

so, 6+√2 = p/q

p = (6+√2)q

we know that 'p' is a rational number. so (6+√2)q must be rational since it equals to p

but it doesnt occurs with it because sum or rational n irrational number results irrational. so, (6+√2) since its not an intezer

therefore, p =/= (6+√2)q

this contradicts the fact that (6+√2)q is an irrational number

hence our assumption is wrong and 6+√2 is an irrational number.

3. Let us assume that 1/√2 is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are intezers.

so, 1/√2 = p/q

p = q/√2we know that 'q' is a rational number. so q/√2 must be rational since it equals to p

but it doesnt occurs with q/√2 since its not an intezer. (since √2 isnt rational)therefore, p =/= q√2

this contradicts the fact that 1/√2 is an irrational number

hence our assumption is wrong and 1/√2 is an irrational number.

4. Let us assume that √2+√3 is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are intezers.

so, √2+√3 = p/q

p = (√2+√3)q

we know that 'p' is a rational number. so (√2+√3)q must be rational since it equals to p

but it doesnt occurs with it because √2, √3 since are not an intezers

therefore, p =/= (√2+√3)q

this contradicts the fact that (√2+√3)q is an irrational number

hence our assumption is wrong and √2+√3 is an irrational number.