Question

prove that √3 is irrational​

Answer 1

Answer:

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Step-by-step explanation:

Let $\sqrt {3} = \frac {p}{q}$ (where p and q is a co-prime)

$\sqrt {3 \times q} = p$

Squaring both the side in above equation

3 × q² = p²

If 3 is a factor of p²

Then, 3 will also be a factor of p

Let p=3m ----- (where m is a integer)

Squaring both sides we get

p² = (3m)²

p² = 9m²

Substitute the value of p²

3q² = p²

3q² = 9m²

Cube rooting both sides

q² = 3m²

If 3 is a factor of q²

Then, 3 will also be factor of q

Hence, 3 is a factor of p & q both

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

So, √3 is an "irrational number". Hence proved.