Root 5 irrational
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Sia ? 6 years, 5 months ago
Let {tex}\sqrt 5{/tex} is a rational number.
{tex}\sqrt { 5 } = \frac { a } { b }{/tex}(a, b are co-primes and b{tex}\neq{/tex}0)
or, {tex}a = b \sqrt { 5 }{/tex}
On squaring both the sides, we get
a2=5b2 ---------------------------------(1)
Hence 5 is a factor of a2
so 5 is a factor of a
Let a = 5c, (c is some integer)
{tex}\therefore{/tex} a2 = 25c2
From equation(1) putting the value of a2
or, 5b2 = 25c2
or b2=5c2
so 5 is a factor of b2
or 5 is a factor of b
Hence 5 is a common factor of a and b
But this contradicts the fact that a and b are co-primes.
This is because we assumed that {tex}\sqrt 5{/tex} is rational
{tex}\therefore{/tex} {tex}\sqrt 5{/tex} is irrational.
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