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?
 A: Based on what other people said. Let $q_n = 1 + 1/1! + 1/2! + ... + 1/n!$. This is a rational number, thus the associated cut $q_n^* = \{ x \in \mathbb{Q} ~ | ~ x < q_n \}$. The number $e$ is the supremum of all of these. Thus, we can say, 
$$ e = \bigcup_{n\geq 1} q_n^* $$
A: 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 $\pi$.
A: If one feels uneasy to use series to define the Dedekind cut for $e$, we can instead take the set $Q_e$ of every rational approximations of $e$ and define the cut as the set $R_e$ of every rational $q$ such as $q < p$ for some $p \in Q_e$.
The series provides a method to compute some members of the cut, but you don't need the series to define the cut itself. The same goes for $\pi$ or any real number.
A: André Nicolas posted a very natural Dedekind cut for e, but lamented that the equivalent for $\pi$ is less natural. So this answer covers a "natural" Dedekind cut for $\pi$.
The left set consists of all rational numbers which are less than the perimeter of some polygon inscribed in a circle of unit diameter. The right set is the complement of the left set with respect to $\mathbb{Q}$. Equivalently, but more usefully, the right set is the set of all rational numbers which are greater than or equal to the perimeter of some polygon circumscribed about a circle of unit diameter.
It is not hard to prove that this satisfies the requirements of the Dedekind cut axiom:

*

*The left set is nonempty, because it contains all the negative rationals.

*The right set is nonempty, because you can construct an arbitrary circumscribed polygon, which must be of finite perimeter and thus shorter than at least one rational number.

*The left set is closed downwards, because if it contains $x \in \mathbb{Q}$, then $x$ is less than the perimeter of some polygon $P$, which is also true of every rational number less than $x$.

*The left set lacks a greatest element, because, returning to our example of $x$ and $P$, we require that $x$ is strictly less than the perimeter of $P$, and so there must be another rational number which is closer to the perimeter than $x$ is. $x$ cannot be the greatest element, and so there can be no greatest element at all.

*Proving that our definition of the right set does indeed produce the complement of the left set is a bit trickier, but as discussed below, the general idea has been understood since Archimedes (albeit not in the specific context of set theory).

Furthermore, we even have an algorithm for classifying rationals (which is not required, as the Dedekind cut axiom does not say that the sets must be recursively enumerable, but merely that they must exist). To determine whether a given rational belongs to the left set or the right set, iteratively construct regular polygons both inscribed in and circumscribed about the circle, with an increasing number of sides each iteration. Eventually, the rational number will either exceed the perimeter of a circumscribed polygon, or be exceeded by the perimeter of an inscribed polygon, and then you have your answer.
Unfortunately, the perimeter of a polygon is not necessarily rational. So on its face, this definition is in terms of other irrational numbers, which may feel less pure. On the other hand, you don't strictly need to work in the reals, so long as you can determine whether a given rational number is larger or smaller than a given polygon's perimeter, which means you just need an arbitrarily-accurate numerical approximation of a large-enough set of polygons  (e.g. just the regular ones) to classify all rationals. The exact formulas, and methods for their rational approximation, were described by Archimedes, and this was apparently considered a reasonable method of approximating $\pi$ as recently as 1630.
Overall, this definition has the advantage of being extremely well-studied, and "obviously correct." However, we should acknowledge that modern numerical methods converge faster and are easier to calculate to a higher degree of precision. This definition is great if you want an easy proof that $\pi \in \mathbb{R}$, or if you want an intuitive feel of "what $\pi$'s Dedekind cut actually looks like," but this is not necessarily the best way of calculating $\pi$'s digits.
A: We invoke the power of abstraction. If we construct the real numbers as Dedekind cuts of the rationals, then we use this method to show that the methods of calculus and real analysis work properly.
Then, we use our considerable experience in calculus to construct $e$ and $\pi$.
If we were so inclined, we could take the entire construction in terms of calculus, and rewrite every individual part in terms of Dedekind cuts. However, nobody ever does this: there is no benefit in carrying out such a tedious exercise.
A: 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.
