Question

1. Prove that √5 is not rational numbers,​

Answer 1

we have to prove √5 is an irrational number .

Frist , We have to assume that √5 is an rational number .

so, this √5 can be written as

$\frac{p}{q}$

If you do squaring of both sides .

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

Answer 2

Answer:

Let us assume that √5 is a rational number.

then, as we know a rational number should be in the form of p/q

where p and q are co- prime number.

So,

√5 = p/q { where p and q are co- prime}

√5q = p

Now, by squaring both the side

we get,

(√5q)² = p²

5q² = p² ........ ( i )

So,

if 5 is the factor of p²

then, 5 is also a factor of p ..... ( ii )

=> Let p = 5m { where m is any integer }

squaring both sides

p² = (5m)²

p² = 25m²

putting the value of p² in equation ( i )

5q² = p²

5q² = 25m²

q² = 5m²

So,

if 5 is factor of q²

then, 5 is also factor of q

Since

5 is factor of p & q both

So, our assumption that p & q are co- prime is wrong

hence,. √5 is an irrational number

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