What type of number is this $\frac{\sqrt2}{2}$? $$\frac{\sqrt{2}}{2}$$
In this monomial, an irrational number is divided by a rational number. However this is not a general case but can any one tell me that when we divide an irrational number or multiply an irrational number or its multiplicative inverse by a rational number then what type of the number we get in output? Either rational or irrational?

If $p$ is a rational (non-zero) number, and $x$ is irrational, then
$$
px, \quad \frac px, \quad \frac xp, \quad p+x, \quad p-x
$$
are all irrational. This can be proven by contradiction. Assume, for instance, that $px$ is a rational number, call it $q$. That is, set $q = px$. Now divide through by $p$, and we get $$\frac{q}{p} = x$$
The left-hand side is a fraction of two rational numbers, so it is rational. The right-hand side is $x$, so it is irrational. These two can therefore not be equal, and therefore the assumption that $px$ is rational must be false.
The others can be proven the same way. Be careful, however, because for each of the five expressions
$$
xy, \quad \frac xy, \quad \frac yx, \quad x+y, \quad y-x
$$there are some irrational numbers $x, y$ such that makes it rational (although not all at once).
