Question

Check whether $\dfrac{5}{\sqrt{2}}$ is an irrational number or not.

Answer 1

Hint: We need to find the relation between the binary operations on rational and irrational numbers. The different outcomes in different cases will help to find whether $\dfrac{5}{\sqrt{2}}$ is an irrational number or not.

Complete step-by-step solution
We use the concept of binary operations between two rational or irrational numbers.
Here we need to talk about the concept of divisibility between two rational or irrational numbers.
If we take any two rational numbers the division of those two numbers will be rational.
Like, let’s assume two rational numbers p and q. then the division form is $\dfrac{p}{q}$ which is already in the form of rational numbers. For example, we take $\dfrac{2}{3}$ and $\dfrac{5}{7}$.
Here divisional value is $\dfrac{{}^{2}/{}_{3}}{{}^{5}/{}_{7}}=\dfrac{2\times 7}{5\times 3}=\dfrac{14}{15}$ which is in $\dfrac{p}{q}$ form. So, the term is rational.
Now we take any two irrational numbers the division of those two numbers will be rational or irrational depending on the numerator and denominator value. If those two are same then the value will be 1 which is rational and if they are unequal then the result will be irrational.
Like let’s assume two rational numbers a and b where $a=b$, then the division form is $\dfrac{a}{b}=\dfrac{a}{a}=1$ which is already in the form of rational number. But if \[a\ne b\], then the division form is $\dfrac{p}{q}$ where none of them are in integer form to get the result as rational number. For example, we take $\sqrt{5}$ and $\sqrt{7}$.
Here divisional value is $\dfrac{\sqrt{5}}{\sqrt{7}}$ where none of denominator or numerator is integer form. So, the term is irrational.
But if the terms are both $\sqrt{5}$ then the result is $\dfrac{\sqrt{5}}{\sqrt{5}}=1$ which is rational.
At last for mixed values where we use 1 rational and 1 rational in the denominator and numerator, irrespective of their positions we get only irrational values.
This case is given in our problem.
Here $\dfrac{5}{\sqrt{2}}$ have 5, a rational number in its numerator and $\sqrt{2}$, an irrational in its denominator. It can’t be expressed in the form of $\dfrac{p}{q}$ where p and q both are integers and \[q\ne 0\]. So, the expression also can’t be expressed in the simplest form.
So, $\dfrac{5}{\sqrt{2}}$ is irrational number.

Note: The case whether we used similar irrational numbers to get rational value 1, will also work for the binary operation of multiplication. So, that part is an exceptional case which we need to keep in mind.