An Impossible Ratio I'm facing a bit of a difficulty thinking about the aspect ratio of A4 paper.
The beauty of this paper size is that when it is folded in half along the longer side, it becomes A5 paper which has dimensions proportional to A4 paper. (ie, the aspect ratio of all A series paper is constant)
Let $a$ be the longer side of my A4 paper and $b$ be the other. If I fold it in half along the width $a$, then that side becomes $\frac{1}{2}a$ which is now less than the other side $b$. 
Now, $b$ is the long side and $\frac{1}{2}a$ is the short side.
So, since the length to width ratio of these rectangles are the same because of what I've mentioned as the beauty of A4 paper :
$$\frac{a}{b} = \frac{b}{\frac{1}{2} a} \implies \left|\frac{a}{b}\right| = \sqrt{2}$$
How is it possible an incommensurable quantity be expressible in the real world units?
Edit: I was under the assumption that $a,b \in \Bbb Z$ (set of integers)
because in the real world we are not so accurate as to have irrational lengths. But upon further reflection, we can't have any non-terminating decimal exist in real world units either because of limitations of precision.
(Question inspired by Numberphiles)

Why wouldn't it be possible?
It is well known that in a plane that satisfies the usual axioms of plane geometry, there are lines that have exactly this ratio -- for example the ratio of the diagonal of a square to its side. And it is easily possible to put two such lines at right angles to each other.
Now in actual industrial fact, the dimensions of the A-series paper formats are defined by numbers rounded to whole millimeters (with tolerances measured in millimeters too), so the "all formats have the same ratio" slogan is only approximately true. But that's more or less independent of the underlying mathematical model which has no problems with incommensurable ratios whatsoever.
