Question

Prove that
root 5
is irrational​

Answer 1

Given: √5

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√5=p/q

On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number

Hence proved.

Answer 2

Answer:

Step-by-step explanation:

let $\sqrt{5}$ be rational number

so, it can be written as

          $\sqrt{5}$ = $\frac{a}{b}$  {where a and b are co prime }

           5$b^{2}$ = $a^{2}$

  therefore $a^{2}$ is divisible by 5, then it follows a is also divisible by 5

   so, we can write a = 5c for some integer c

substituting value of a=5c in equation

        5$b^{2}$ = (5$c^{2}$)

        5$b^{2}$=25$c^{2}$

          $b^{2}$=5

∴$b^{2}$ is also divisible by 5

∵b is also divisible by 5

  so we can say that a and b are not co prime . so, $\sqrt{5}$ is irrational

   

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