Prove that 3-2√7 is irrational, given that √7 is irrationa
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ATQ, We have to prove that 3 + 2√7 is an irrational number. ... Step 2: With the help of Step 1, prove 3 + 2√7 is an irrational number. Let us assume that √7 is a rational number. This implies that √7 can be expressed in the form p/q where q ≠ 0 and p & q are co-primes.
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We have to prove that 3+
7
is irrational.
Let us assume the opposite, that 3+
7
is rational.
Hence 3+
7
can be written in the form
b
a
where a and b are co-prime and b
=0
Hence 3+
7
=
b
a
⇒
7
=
b
a
−3
⇒
7
=
b
a−3b
where
7
is irrational and
b
a−3b
is rational.
Since,rational
= irrational.
This is a contradiction.
∴ Our assumption is incorrect.
Hence 3+
7
is irrational.
Hence proved.
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