plot any irrational number on number line. I have a basic question:
Can we plot any irrational number on a number line? I can plot all integers and rational numbers on it but how can I plot any irrational number, such as $\sqrt2$,$\sqrt3$ etc.?
Choosing a unit length, you can use the spiral of Theodorus to find any length of the form $\sqrt{n}$ in which n is an integer:



As a matter of fact, as shown in the picture below, using a similarity argument we can deduce that
$CH^2=AH\times BH$, which means by choosing AH to be the unit lenght, we can construct $\sqrt{BH}$. You just have to find a way to construct such a triangel (Hint: Use a semicircle!).



For other irrational numbers, it depends on the number AND the tools you are allowed to use. For example, it has been proved that these numbers are not constructible  using just a straight edge and compass: $2^{\frac{1}{3}}$, $\pi$, $e$. But, if you're allowd to use a tool to construct a hyperbola and another for a parabola, then you can construct $2^{\frac{1}{3}}$.
First of all draw the number line. Mark point 'A' as 0 and ponit 'B' as 1. This means A,B = 1 unit.Now, at B, draw BX perpendicular to AB. Cut off BC = 1 unit. Join AC. By Pythagoras Theorem in right triangle ABC, we get AC= root 2 units. Now with radius AC and center A, mark a point on the number line. Let the marked point be M.M represents root 2 on the number line.