Question

prove that root 3 irrational​

Answer 1

Answer:

The square root of 3 is irrational. It cannot be simplified further in its radical form and hence it is considered as a surd. Now let us take a look at the detailed discussion and prove that root 3 is irrational.

...

Prove that Root 3 is Irrational Number

Answer 2

Answer:

Let √3 be a rational number of form p/q where p and q are co - primes.

$\sqrt{3} = \frac{p}{q} \\ squaring \: both \: the \: sides \: \\ { (\sqrt{3}) }^{2} = { (\frac{p}{q} )}^{2} \\ 3 {q}^{2} = {p}^{2} \\ {q}^{2} = \frac{ {p}^{2} }{3} \: \: \: \: \: \: (1)$

Now, if a prime number divides a number's square then it also divides the number.

So let,

$\frac{p}{3} = c \\ p = 3c \: \: \: \: \: \: \: \: \: (2)$

From (1) and (2)

${q}^{2} = \frac{ {(3c)}^{2} }{3} \\ {q}^{2} = \frac{9 {c}^{2} }{3} \\ {q}^{2} = 3 {c}^{2} \\ {c}^{2} = \frac{ {q}^{2} }{3}$

Now again as 3 divides q^2 so it also divides q

So it is a contradictory because p and q were co - primes so both can't be divided by 3

Thus our assumption was wrong,

So √3 is irrational

Hence Proved