Dedekind cuts for $\pi$ and $e$ I tried to search in the internet about this but did not get any exciting answers. So my question is: How is construction of transcendental numbers like  $\pi$ and $e$ explained via Dedekind cuts?
Here is a simple description for $e$. The left set consists of all rationals $r$ such that 
$$r\lt 1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots +\frac{1}{n!}$$
for some $n$. This description is close in spirit to one of the many definitions of $e$.

One can give a similar description for $\pi$, though there is nothing as natural. We could use the following variant of the "Leibniz" series, using for the left set all rationals $r$ such that 
$$r\lt 4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+ \cdots +\frac{4}{4n+1}-\frac{4}{4n+3}$$
for some $n$. Note that we stop with a "$-$" because we want to make sure we are below $\pi$.
The real numbers are really troublesome due to the infinities that enter the whole game. It is possible to show that $\sqrt2$ is a cut, because you can easily go to a rational by squaring. With transcendental numbers this is much more difficult, and actually impossible. You simply have to believe that transcendentals are good numbers. It is a choice so to say. 

What is done above actually, is that $e$ and $\pi$ are represented by some finite rational interval that can be chosen arbitrary small. But rational intervals do not obey the field rules, since multiplication is not distributive over addition in an unambiguous way no matter how small you choose them. Also the interval that contains zero causes serious problems in elaborate calculations. Calculations based on intervals always can result in an interval that contains zero. Personally I would rather consider real values instead of real numbers, since there is no clarity whether real values are good numbers in general. It is still debated among mathematicians whether the reals are good numbers.