1 thought on “prove that <br />[tex] \sqrt{2 + \sqrt{3} } [/tex]<br />is irrational”
Answer:
First we prove that
2
is irrational.
for that we assume that
2
is rational. And as rational number can be written as
q
p
form.
where we assume that p and q have no common factor. Squaring in both side. After that
2=
q
2
p
2
which implies that
p
2
=2q
2
thus p
2
is even. The only way this can be true is that p itself is even. But then p
2
is actually divisible by 4. Hence q
2
and therefore q must be even. So p and q are both even which is a contradiction to our assumption that they have no common factors. The square root of 2 cannot be rational.
and we similarly prove for
3
.
And as we know that sum of two irrational no. is always irrational.
Answer:
First we prove that
2
is irrational.
for that we assume that
2
is rational. And as rational number can be written as
q
p
form.
where we assume that p and q have no common factor. Squaring in both side. After that
2=
q
2
p
2
which implies that
p
2
=2q
2
thus p
2
is even. The only way this can be true is that p itself is even. But then p
2
is actually divisible by 4. Hence q
2
and therefore q must be even. So p and q are both even which is a contradiction to our assumption that they have no common factors. The square root of 2 cannot be rational.
and we similarly prove for
3
.
And as we know that sum of two irrational no. is always irrational.