Prove that (√2 + √3) is irrational
Answers
Answer
Let us assume that
2
+
3
is a rational number
Then. there exist coprime integers p, q,q
=0 such that
2
+
3
=
q
p
=>
q
p
−
3
=
2
Squaring on both sides, we get
=>(
q
p
−
3
)
2
=(
2
)
2
=>
q
2
p
2
−2
q
p
3
+(
3
)
2
=2
=>
q
2
p
2
−2
q
p
3
+3=2
=>
q
2
p
2
+1=2
q
p
3
=>
q
2
p
2
+q
2
×
2p
q
=
3
=>
2pq
p
2
+q
2
=
3
Since, p,q are integers,
2pq
p
2
+q
2
is a rational number.
=>
3
is a rational number.
This contradicts the fact that
3
is irrational.
Thus, our assumption is incorrect.
Therefore,
2
+
3
is a irrational.
Since, p,q are integers,
2pq
p ^2+q ^2 is a rational number.
=>
√3 is a rational number.
This contradicts the fact that
√3 is irrational.
Thus, our assumption is incorrect.
Therefore,
√2+√3 is a irrational.
