Question

Prove that √5 is a irrational number.

Answer 1

$hii... \\ \\ assume \: that \: \sqrt{5} \: is \: a \: rational \: number \: . \\ then \: \sqrt{5} = \frac{a}{b} where \: a \: and \: b \: are \: integers \: b \: is \: not \: equal \: to \: 0 \: and \: hcf(a \: and \: b \: )is \: 1....(i) \\ = > \sqrt{5} b = a \\ = > 5 {b}^{2} = {a}^{2} (squaring \: both \: side) \\ = > {a}^{2} is \: divisible \: by \: 5 \\ = > a \: is \: divisible \: by \: 5.......(ii) \\ = > a = 5c \: where \: c \: is \: an \: integer. \\ so.. \sqrt{5} b = 5c \\ = > 5 {b}^{2} = 25 {c}^{2} \\ = > {b}^{2} = 5 {c}^{2} \\ = > {b}^{2} is \: divisible \: by \: 5 \\ = > b \: is \: divisible \: by \: 5.....(iii) \\ \\ from \: (ii)and \: (iii)we \: get... \\ \: 5 \: is \: the \: common \: factor \: of \: a \: and \: b \: which \: cntradict \: (i). \\ thus \: our \: assumption \: was \: wrong \: \\ \\ so.. \sqrt{5} is \: an \: irrational \: number. \\ \\ \\ \\ hope \: this \: helps :)$

Answer 2

Heya!!
Here is your answer :-
To prove,
√5 is an irrational number.
We will prove this by the method of contradiction.
In the method of contradiction we assume opposite situation of what is to be proved.

Let √5 be a rational number in the form of p/q where q≠0 and p and q are co-prime integers i.e. they don't have any common factor except 1.

$\sqrt{5} = p \div q$
Squaring both sides of the equation,we get:-
$( { \sqrt{5} })^{2} = (p \div q )^{2}$
$5 = {p}^{2} \div {q}^{2}$
Now using transposition,
$5 {q}^{2} = {p}^{2}( i)$
From (i) , we can see that :-
5 is a factor of p^2
So, 5 is a factor of p (ii)

Now, let p=5x , where x is any integer.
By substituting the value of p, we get:-
$5 {q}^{2} = ({5x})^{2}$
Now, solving it:-
$5 {q}^{2} = 25 {x}^{2}$
Now using transposition,
${q}^{2} = 5 {x}^{2} (iii)$
So, from (iii) we can see that 5 is a factor of q^2.
Therefore, 5 is a factor of q. (iv)

=> So , from the statements (ii) and (iv) we can see that p and q have a common factor 5.
=> This contradicts our supposition that √5 is a rational number.
=> Hence our assumption was wrong.
=> That's why √5 is an irrational number.
=> Hence, proved.


Hope it helps you.