Question

Identify whether -√0.4 is rational or irrational number with justification.

Answer 1

We have to write -√0.4 in fractional form.

$\displaystyle \Longrightarrow\ -\sqrt{0.4} \\ \\ \\ \Longrightarrow\ -\sqrt{\frac{4}{10}} \\ \\ \\ \Longrightarrow\ -\frac{2}{\sqrt{10}}$

As √10 is irrational, then so will be -2/√10, because a rational number divided by an irrational number gives irrational.

Thus we can say that -√0.4 is irrational.

Or let me prefer you another method.

Assume that -√0.4 is rational, and let it be x where x is a rational number.

$\displaystyle \Longrightarrow\ x=-\sqrt{0.4} \\ \\ \\ \Longrightarrow\ x=-\frac{2}{\sqrt{10}} \\ \\ \\ \Longrightarrow\ \sqrt{10}=-\frac{2}{x}$

A contradiction occurs at the last step as it seems that √10 can be written as a fraction, but it's actually irrational.

This contradiction breaks our earlier assumption that -√0.4 is rational.

Hence we can also say that -√0.4 is irrational.

Answer 2

Answer:

We have to write -√0.4 in fractional form.

⟹ −0.4⟹ −104⟹ −102

As √10 is irrational, then so will be -2/√10, because a rational number divided by an irrational number gives irrational.

Thus we can say that -√0.4 is irrational