Question

1/2-√5 is irrational number​

Answer 1

Answer:

Yes it is an irrational no. as if any no. is divided,multiplied,added or subtracted from any irrational no the result is always irrational

Answer 2

Answer:-

To prove:-

1/2-√5 is a irrational number...

Solution:-

we know that,

√5 is a irrational number.

now,

Let us assume that 1/2-√5 is a rational number, we can write in the p/q form and also q≠0

first rationalize the denominator,

$= \frac{1}{2 - \sqrt{5} } \times \frac{2 + \sqrt{5} }{2 + \sqrt{5} } \\ \\ = \frac{2 - \sqrt{5} }{4 - 5} \\ \\ = \frac{2 - \sqrt{5} }{ - 1} \\ \\ = \sqrt{5} + 2$

according to condition,

$\sqrt{5} + 2 = \frac{p}{q} \\ \\ \sqrt{5} = \frac{p}{q} - 2 \\ \\ \sqrt{5 } = \frac{p - 2q}{q}$

here,

p-2q/q is a rational number but we know that√5 is a irrational number.

so our assumption is wrong.

1/2-√5 is a irrational number.

Hence Proved...