Question

prove that root 5 is irrational

Answer 1

Hey!!
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 Let √5 is a rational number.
 ∴ It can be expressed in the form a/b where a,b are co-prime integers.

=)√5=a/b
=)5=a²/b²  {Squaring both the sides}
=)5b²=a²     .....  (1)
=)a² is a multiple of 5. {Euclid's Division Lemma}
=)a is also a multiple of 5. {Fundamental Theorm of arithmetic}
=)a=5m, for some integer m
⇒a²=25m²  ..... (2)
From equations (1) and (2), we get,
5b²=25m²
=)b²=5m²
=)b² is a multiple of 5. {Euclid's Division Lemma}
=)b is a multiple of 5.{Fundamental Theorm of Arithmetic}

Hence, a,b have a common factor 5. this contradicts that they are co-primes. Therefore, p/q is not a rational number. This proves that √5 is an irrational number.

Hope it helps! ^_^

Answer 2

Heya

Your answer is given below :
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Solution :

Let resume on the contrary that root 5 is irrational number then, there exist coprime positive integers a and b such that,

$\sqrt{5} =a |b \: \\ 5 {b}^{2} = {a}^{2} \\ 5 |a \\ a = 5c \: \: \: \: \: \: (by \: theorem) \: \\ {a = 25 {c}^{2} } \\ 5 {b}^{2} = 25 {c}^{2} \\ {b}^{2} = 5 {c}^{2}$
$5 | {b}^{2} | \: \: \: \: \: (by \: theorem) \\ \frac{5}{b }$
Hence we observed that a and b have at least 5 as a common factor this content it's the fact that A and B are coprime.

Hence,√5 is an irrational number.

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I hope it will help you

thank you