Question

$\sqrt{5}$
prove this is irrational no................❤️❤️❤️​

Answer 1

Let's take $\sqrt{5}$ as rational number .

We can write $\sqrt{5} = \frac{a}{b}$ such that ,

a & b are integers , $b ≠ 0$ & there are no factors common to a & b .

Multiplying b on both sides , we get ;

$b \sqrt{5} = a$

To remove root , squaring on both sides , we get , $5b² = a²$ _____ ( i )

That means , 5 is a factor of a² .

For any prime no. p which is a factor a² then it will be the factor of a also .

So , 5 is a factor of a . _____ ( ii )

Hence , we can write a = 5c for some integer c .

Putting the value of a in equation ( i ) we get ,

$5b² = (5c)²$

$5b² = 25c²$

Divide by 5 ; we get ,

$b² = 5c²$

It means 5 is a factor of b²

Thus , 5 is a factor of b _____ ( iii )

From ( ii ) & ( iii ) , we can say that 5 is the factor of both a & b .

This contradicts our theory , as we stated that a & b have no factors in common

Thus , our assumption that $\sqrt{5}$ is rational is wrong .

Hence , $\sqrt{5}$ is irrational number .

Answer 2

Answer:

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