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?

Solution 1:

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 $\pi$.

Solution 2:

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, $$ e = \bigcup_{n\geq 1} q_n^* $$