Number having sexagesimal expansion end with infinitely many zeros? I am looking for all the real numbers whose sexagesimal expansion (base $60$) ends in infinite tail of zeros. Does they really exist?

It seems absurd to me or mm thinking it in a wrong manner?
A number $N$ has a finite sexagesimal expansion if it can be written as
$$
N=a_0+\frac{a_1}{60}+\frac{a_2}{60^2}+\dots+\frac{a_n}{60^n}
$$
with $a_0$ any integer and $0\le a_i&lt;60$ for $i=1,2,\dots,n$.

Then we can write
$$
N=\frac{m}{60^n}
$$
The converse is obviously also true: take $a_0=\lfloor N\rfloor$ and write
$$
N-a_0=\frac{m'}{60^n}
$$
which is true for a unique $m'$ with $0\le b&lt;60^n$. Then write the base $60$ expansion of $b$ and you're done.

You can notice that base $60$ has nothing special. A number $N$ has a finite base $b$ expansion if and only if it has the form
$$
N=\frac{m}{b^n}
$$
for some integer $m$ and nonnegative integer $n$.
$$ \forall n,a \in N, 60 \nmid n \lor a = 0 $$ $$ \frac {n} {60^a} $$

Example:
$$ \frac {175371} {60^2} $$