Question

prove that
$\sqrt{5}$
is irrational​

Answer 1

Let us assume that _/5 is a rational number.

$= > \sqrt{5} = \frac{x}{y}$

=>Suppose x/y have common factor,then we divide by the common factor to get-----

$= > \sqrt{5} = \frac{x}{y} (co - prime)$

$= > x = \sqrt{5}y$

Squaring both sides,we get----

=>x²=5y² -----(1)

=>x² is divisible by 5.

=>x=5c

Now Substituting value of eq.1.

=>(5c)²=5y²

=>25c²=5y²

=>5c²=y²

=>y²=5c²

y² is divisible by 5.This contradicts to the fact that x and y have no common factor.This happens because of our wrong assumption.

Thus,

_/5 is irrational.

Answer 2

Step-by-step explanation:

Let us assume that _/5 is a rational number.

= > \sqrt{5} = \frac{x}{y}=>

5

=

y

x

=>Suppose x/y have common factor,then we divide by the common factor to get-----

= > \sqrt{5} = \frac{x}{y} (co - prime)=>

5

=

y

x

(co−prime)

= > x = \sqrt{5}y=>x=

5

y

Squaring both sides,we get----

=>x²=5y² -----(1)

=>x² is divisible by 5.

=>x=5c

Now Substituting value of eq.1.

=>(5c)²=5y²

=>25c²=5y²

=>5c²=y²

=>y²=5c²

y² is divisible by 5.This contradicts to the fact that x and y have no common factor.This happens because of our wrong assumption.

Thus,

_/5 is irrational.