Question

Division of two irrational numbers is:
A) always a rational
B) always an irrational
C) not an irrational
D) either rational or irrational

Answer 1

Hint: In this given question, we are required to find that if two irrational numbers are divided, what would the resultant be. We will try to do this with the help of examples. We will show with the help of examples that the resultant will be a rational number, an irrational number, or either rational or irrational number.

Complete step-by-step solution:
Here we are asked to find that if two irrational numbers are divided, what the resultant would be? We will try to find it by the use of some examples.
Ex. \[1\]: Let, we have to divide two irrational numbers \[4\sqrt 2 \] and \[\sqrt 2 \]. We do it as,
\[\dfrac{{4\sqrt 2 }}{{\sqrt 2 }} = 4\]
We get the result as \[4\] which is a rational number.
Now again we take example as,
Ex. \[2\]: We have to divide two irrational numbers \[4\sqrt 6 \] and \[\sqrt 2 \] as,
\[ \dfrac{{4\sqrt 6 }}{{\sqrt 2 }} = \dfrac{{4\sqrt 3 \times \sqrt 2 }}{{\sqrt 2 }} \\
   \Rightarrow \dfrac{{4\sqrt 6 }}{{\sqrt 2 }} = 4\sqrt 3 \]
We get the result as an irrational number.
From the above two examples, we see that we can get either a rational number or irrational number by dividing two irrational numbers. Hence we get the answer as D).

Note: We could have also proved the above given question theoretically, but this is always the easiest way to prove such questions. Find just a single example of one kind to prove your result. This kind of technique can also be used to contradict a statement. Just find an example that shows the result against the statement and that statement is contradicted.