Prove that √13is an irrational number by contradiction method. Is – 5 a rational or an irrational number? Give reasons using contradiction method.
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Suppose for the sake of contradicton that
3
is rational.
We know that rational
numbers are those numbers which can be expressed in the form
q
p
, where p and q are integers and q
=0
Hence,
3
=
q
p
where p and q are integers with no factor in common.
Squaring both sides,
3=
q
2
p
2
=p
2
=3q
2
-- (1)
That is, since p
2
=3q
2
, which is multiple of 3, mean p itself must be a multiple of 3 such as p=3n.
Now we have that p
2
=(3n)
2
=9n
2
---- (2)
From (1) and (2),
9n
2
=3q
2
=>3n
2
=q
2
This means, q is also a multiple of 3, contradicting the fact that p and q had no common
factors.
Hence,
3
is an irrational number.
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