Question

$prove \: that \: \sqrt{5 \: } is \: irrational \: number$
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Answer 1

Question -

Prove that √5 is irrational number.

Answer ==>

Let √5 be Rational number.

Then,

√5 = p/q (where p and q are integers and q is not equal to zero)

Squaring Both sides

(√5)² = (p/q)²

5 = p²/q²

5q² = p²

Here,

we see that

5 divide p

and

5 also divide p².......(¡)

Let p be 5

then,

putting p = 5 on 5q² = p²

we get

5q² = (5)²

5q² = 25

q² = 5

Here,

we see that

5 divide q

And

5 also divide q² ......(¡¡)

from (¡) and (¡¡) we get

5 is a common factor of p and q

so, our assumption is wrong.

√5 is an irrational number.

Proved

Answer 2

Let √5 be Rational number.

√5 = p/q

Squaring Both sides

(√5)² = (p/q)²

5 = p²/q²

5q² = p²

Here,

5 divide p ,and

5 also divide p².......(¡)

Let p be 5

5q² = (5)²

5q² = 25

q² = 5

Here,

5 divide q ,and

5 also divide q² ......(¡¡)

from (¡) and (¡¡)

5 is a common factor of p and q

so, our assumption is wrong.

√5 is an irrational number.

Here is your answer