Can we represent a line of length equal to irrational numbers? From Euclid's postulate we know that "A straight line segment can be drawn using any two points ". Now  Let us suppose, we take a unit base and a unit perpendicular, so we have two definite points on a plane. From Euclid's postulate we can draw a line segment, but pythagoras theorem tells us that the hypotensue should be of length root 2 which is a irrational number. 
So how is it possible that to draw a line segment when we don't know where the other point lies. By taking approximation we will make the number rational. 
Can we say that in such scenarios, a line doesn't connect two points rather there is a ray which tends to a hypothetical point. It looks analogous to taking limit where a tangent is approximation of a secant and here a line is an approximation of a ray.
 A: Assume you don't know anything about rational vs. irrational, and are given the following problem: "Draw a line $\ell$, choose an arbitrary point $0\in\ell$ and another point $1\in\ell$. Where is the point that should get the label $\sqrt{2}$?" You would go ahead and find this point with a little construction, and without having any doubts that this is exactly the point you wanted. 
Of course this construction lives in an ideal mathematical world, and we all know that the lines drawn with a pencil have a certain thickness, etc. But this is a completely other matter. 
Concerning scaling: Given any two points $A$, $B\in\ell$ you can scale their distance $|AB|$ by a factor $\sqrt{2}$ using the three points you already have and drawing a few parallels.
A: From the way you've described it, it sounds like you're choosing an abstract line segment of a certain length, placing it on the plane between two points $P$ and $Q$, and hoping that it fits. This is a problem if you don't already have a line segment of length $\sqrt2$.
But the real construction is the opposite of this: starting with $P, Q$, we simply draw the line segment between them. Axiomatically, the line segment exists. Its length is $\sqrt2$.
A: Say that the segment in question is supposed to be between $A$ and $B$. Say your segment starts at $A$ and its length is some rational number $q < \sqrt{2}$. Then there will be space between the endpoint and $B$, and that doesn't make sense. You can't keep changing the length of the segment to make it bigger: each point either definitely belongs to it or definitely doesn't.
A: No, according to Euclids geometry the line do connect the two points. That it certainly says.
What it doesn't say and known not to be true is that two lines does need to be comensurable (that is you can take multiples of them and end up with equally long line). For example the diagonal in a square and the sides of the squares aren't.
Note that the numbers used then were only the natural numbers, so the problem with "knowing" where the points were was not a matter of numbers since the numbers available was definitely inadequate for the task anyway.
A: You can definitively represent some irrational numbers for example to represent $\sqrt{n^2+1}$ where $n$ is an integer, draw a rectangle whose sides has length $1$ and $n$, the diagonal has length $\sqrt{n^2+1}$.
To represent $\pi$, draw a circle of diameter $1$.
