Question

Under root 5 is a irrational number

Answer 1

√5 Is a irrational number.

Because it is not rational number.

Rational Numbers :

Those numbers which are in the form of p/q where both p and q are integers and q≠0 .

Example - 4,⅔ etc .

Both Rational Numbers and irrational numbers with each other forms the set of real numbers .

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Answer 2

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√5 is an irrational number.

$\bold{explanation}$

Let we assume that √5 is an rational number .

Hence, √5 can be written in the form of a/b

where a and b (b $\cancel{=}$ 0 } are co - prime numbers ( no common factor other than 1 )

So,
√5 = a/b
√5b = a

Squaring on both sides

(√5b)² = a²

5b² = a²
b² = a²/5

Here 5 divides a²

[Theorem: If p is a prime number , and p divided a² , then p divides a , where $\bold{a}$ is a positive number.]

So,
5 divides a also .......(1.)

Therefore, we can say
a/5 = c (where c is some integer )

So,
a = 5c

Now, we know that
5b² = a²
(putting a = 5c here )

5b²= (5c)²

5b² = 25 c²

b² = 1/5 * 25c²

b² = 5c²

c² = b² / 5

Hence, 5 divides b² also

[Theorem: If p is a prime number , and p divided a² , then p divides a , where $\bold{a}$ is a positive number.]

So,
b divides 5 also .......(2.)

By eq.(1.) and (2.)

5 divides both a and b .
Hence, 5 is a factor of a and b.

So, a and b have a factor 5

Hence, our assumption is wrong .

°•° By contradiction,

$\underline{root5\: is \:an \:irrational \:number }$

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