Prove √2 + √3 is an irrational number.
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Answered by
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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
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