Question

root5 is irrational​

Answer 1

yes as

let root 5 be rational

then it must in the form of p/q [q is not equal to 0][p and q are co-prime]

root 5=p/q

=> root 5 * q = p

squaring on both sides

=> 5*q*q = p*p ------> 1

p*p is divisible by 5

p is divisible by 5

p = 5c [c is a positive integer] [squaring on both sides ]

p*p = 25c*c --------- > 2

sub p*p in 1

5*q*q = 25*c*c

q*q = 5*c*c

=> q is divisble by 5

thus q and p have a common factor 5

there is a contradiction

as our assumsion p &q are co prime but it has a common factor

so √5 is an irrational

mark me as brainlist

Answer 2

$\huge\underline{\bold{\red{Answer\::}}}$

Let us assume that √5 is a rational number

√5 = $\dfrac{a}{b}$

[Here a and b are co-prime numbers]

b√5 = a

Squaring on both sides we get;

5b² = a² ___(1)

b² = $\dfrac{{a}{^2}}{5}$

Here 5 divides a²

Now....

a = 5c

Here c is integer

Squaring on both sides we get;

a² = 25c²

5b² = 25c²

[From (1)]

b² = 5c²

c² = $\dfrac{{b}^{2}}{5}$

Here 5 divides b²

Both a and b are co-prime numbers. And 5 divides both of them.

So, our assumption is wrong.

√5 is irrational number.

Hence proof.