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Definition: (from Wikipedia)

In number theory, a Liouville number is a real number $x$ with the property that, for every positive integer $n$, there exist integers $p$ and $q$ with $q > 1$ and such that

$$
{displaystyle 0<left|x-{frac {p}{q}}right|<{frac {1}{q^{n}}}.}
$$

A Liouville number can thus be approximated "quite closely" by a sequence of rational numbers. [....]

My question: How can I convince myself that the above definition is not arbitrary. In other words, how nice is to know that a given number $alpha$ is a Liouville number?

Not to be that guy, but all definitions are arbitrary. A better question to ask would be "Are there any real numbers that satisfy my definition?"

Thankfully the definition of a Liouville number is "good" in the sense that there are real numbers which are Liouville numbers. Perhaps the most famous one is Liouville's Constant:
$$
lambda = sum_{k=1}^infty 10^{-k!} = 0.1100010000000000000000010ldots
$$
This number has a $1$ at every place in its decimal expansion that is equal to a factorial, and $0$'s everywhere else. You can verify that this number satisfies the definition of a Liouville number directly.

Once we know that the definition is "good" in the sense that there are examples of objects that satisfy the definition, we can ask further questions. Do these objects all belong to some well studied, larger class of objects (are they algebraic or transcendental)? How many objects satisfy the definition? If they live in some ambient set with structure, can we say anything about how they fit in that universe (like do the Liouville numbers form a set of zero measure in $mathbb{R}$)? Are these objects "fundamental" in some way (like can every real number be written as the sum of two Liouville numbers)?

However, as much as I love transcendental number theory, we can also ask the question "Do I really care that these things exists?" And I unfortunately have to concede that 99% of mathematicians, and therefore 99.99999$cdots$% of human beings, have absolutely no use for Liouville numbers on a year to year, let alone day to day, basis. I think their value is far more apparent from an educational and historical perspective than it is from a working mathematician's perspective. And in that sense, you could say that it doesn't really matter if you know that any given number $alpha$ is a Liouville number.

Liouville numbers are in fact the most irrational numbers in one sense known as irrationality measure, so hence "nicest" in this sense.

Let us define an irrationality measure as the least upper bound of $mu$ where $mu$ satisfies $$ 0 < |x - frac{p}{q}| < frac{1}{q^mu}$$ for an infinite number of pairs $(p,q)$ with $q > 0$. As one notices, this has similarities with the definition of the Liouville numbers.

A Theorem known as the Thue-Siegel-Roth Theorem states that $mu(alpha) = 2$ if $alpha$ is an algebraic integer. We have that $mu(e)= 2$, $mu(pi) < 8$ and many others. The Liouville numbers $beta$ can be shown to be those that satisfy $mu(beta) = infty$.

To be fair, one could object that this measure is still somewhat arbitrary, but the Thue-Siegel-Roth Theorem shows that it does contain some relevance to the algebraic-transcendental distinction.