In the artform of worldbuilding, one facet of design that I particularly enjoy is the design of numeral systems and mathematical notation. The familiar decimal system is an example of just one type of system, that being positional notation, with some popular alternative positional systems being those such as binary, dozenal, hexadecimal, or even vigesimal. The full extent of what can be done with numeral systems, even just positional ones, has far more flexibility than just these different bases display. For example, many cultures in the past have used mixed radix systems, switching between bases for each digit, often with complicated rules governing what each position represents.
I recently read an article written by jan Misali exploring some advantages that could be had from using decadozenal, or base-120, in a mixed radix system. I have some of my own thoughts about this, and I find the system to be rather interesting, so here's some findings from my own independent "research" on the matter.

A basic way to demonstrate how "good" a base is is to examine how it expresses simple ratios. For example, decimal writes $1/5$ as $0.2$, while $1/7$ requires six repeating digits, those being $0.\overline{142857}$. For now, I'm only going to focus on which prime fractions can be expressed simply in a given base, as composite fractions can be derived from those. In base $n$, a prime fraction $1/p$ can be expressed in a single terminating digit if and only if $p$ is a factor of $n$. If $p$ is not a factor of $n$, it can be expressed with $k$ repeating digits where $k$ is the least whole number such that $p|(n^{k}-1)$. For example, in decimal, $1/7$ is $0.\overline{142857}$ because $1/7=\frac{142857}{10^{6}-1}$. That is, $7|(10^{6}-1)$.
Which prime fractions will be at least somewhat simple in a given base $n$ can quickly be seen by looking at the prime factors of $n-1$, $n$, and $n+1$, as the fractions for the factors of $n$ can be expressed with a terminating digit, for $n-1$ with a single repeating digit, and for $n+1$ with two repeating digits, as a consequence of the fact that $(n+1)(n-1)=n^{2}-1$.

Lets call a prime fraction $1/p$ "convenient" in a base if either $p|(n-1)$, $p|n$, or $p|(n+1)$. Below is a table that can be used to quickly see which bases have a lot of convenient prime fractions:

Multiples of $2$ and $3$ are not marked as $1/2$ and $1/3$ would be convenient in any base, though because of how common they are, the only bases $n$ that could really be practical are those such that $6|n$ (shown in bold) or, if more flexibility is needed, $2|n$ and $3|(n-1)$ (shown in italics).

The smallest base for which both $1/5$ and $1/7$ are convenient is seximal, base-6. For $1/5$, $1/7$, and $1/11$, the smallest is triseptimal, base-21. However, if the search is restricted only to bases divisible by six, dozagesimal, base-120. For $1/13$ as well? While base-155 could work, the smallest multiple of six is base-714, far too large even with a mixed radix system.[1]Here's a few other bases worth mentioning:
Base-76: Italicized, convenient up to 1/11
Base-274: Italicized, convenient up to 1/13
Base-210: Up to 1/7 terminate, 1/11 is a single repeating
Base-105: 1/3, 1/5, and 1/7 terminate, and 1/8 is a single repeating digit, making it a theoretically usable mixed-radix odd base
This is why I say dozagesimal is uniquely good at fractions. $1/2$, $1/3$, and $1/5$ all terminate, $1/7$ and $1/17$ use a single repeating digit, $1/11$ uses two repeating digits, and even $1/13$ only needs three repeating digits, a consequence of the fact that $13|(120^{3}-1)$. $1/19$ is the first prime fraction to be truly unusable, requiring nine repeating digits. No base smaller than 120 is nearly as good, and no significantly better base is within even an order of magnitude of its size.

If you were paying close attention, you may have noticed that while I've been calling base-120 dozagesimal, jan Misali refers to it as decadozenal on his website. This is actually an important distinction, and it's something that I want to look at further than jan Misali did in his article. In short, while dozagesimal can be seen as base-12*10, decadozenal would be base-10*12. To explain why that's relevant, here's a quick rundown of mixed radix notation:

In a sense, the way time is written can be seen as mixed radix base-60, also known as sexagesimal. One second less than ten hours can be written as 9:59:59, where the colons separate digits in base-60, if "59" is viewed as a single digit. However, you can also view the digits as a mix of base-10 and base-6, with the digits in blue rolling over once they reach 10 and the digits in green rolling over once they reach 6.
For dozagesimal, the same principle holds. To keep the notation from looking like time, I decided to use apostrophes instead of colons, so $83*120^{3}+94*120^{2}+105*120+116$ would be written as 83’94’Χ5’Ɛ6. The blue digits are base-10, while the green digits are base-12, with Χ standing in for ten and Ɛ for eleven.
Decadozenal is the other way around, such that $13*120^{3}+12*120^{2}+11*120+10$ would be written 11’10’0Ɛ’0Χ. In dozagesimal, 1’00’00-1=Ɛ9’Ɛ9, while in decadozenal, 1’00’00-1=9Ɛ’9Ɛ.

To see how these systems differ, here's a table showing some basic fractions in both:

Fraction	Dozagesimal	Decadozenal
$1/2$	$0.60$	$0.50$
$1/3$	$0.40$	$0.34$
$1/4$	$0.30$	$0.26$
$1/5$	$0.24$	$0.20$
$1/6$	$0.20$	$0.18$
$1/7$	$0.\overline{17}$	$0.\overline{15}$
$1/8$	$0.15$	$0.13$
$1/9$	$0.13‘40$	$0.11‘34$
$1/10$	$0.12$	$0.10$
$1/11$	$0.\overline{10‘Χ9}$	$0.\overline{0Χ‘91}$
$1/12$	$0.10$	$0.0Χ$
$1/13$	$0.\overline{09‘27‘83}$	$0.\overline{09‘23‘6Ɛ}$
$1/14$	$0.08‘\overline{68}$	$0.08‘\overline{58}$
$1/14$	$0.08‘\overline{68}$	$0.08‘\overline{58}$
$1/15$	$0.08$	$0.08$
$1/16$	$0.07‘60$	$0.07‘50$
$1/17$	$0.\overline{07}$	$0.\overline{07}$
$1/18$	$0.06‘80$	$0.06‘68$
$1/19$	$0.\overline{06‘37‘Χ7‘44‘25‘31‘69‘56‘Χ1}$	$0.\overline{06‘31‘8Ɛ‘38‘21‘27‘59‘48‘85}$
$1/20$	$0.05$	$0.05$

Overall, the fractions in dozagesimal are slightly simpler than in decadozenal, with five terminating in one digit in dozagesimal and only three in decadozenal. Jan Misali accounts for this by using what he refers to as "symmetric decadozenal", using decadozenal for the integer part of the number but switching to dozagesimal for the fractional part, such that 1’00-0.01=9Ɛ.Ɛ9. The reason he gives for doing this is that "dozagesimal is just as good at writing expansions of rational numbers, but makes it a bit harder to do divisibility tests". To assess this claim, here's a table of divisibility tests for each system:

Test	Dozagesimal	Decadozenal
One	Yes
Two	Ends in 0, 2, 4, 6, or 8	Ends in 0, 2, 4, 6, 8, or X
Three	Sum of tens place and ones place is a multiple of 3	Ends in 0, 3, 6, or 9
Four	Tens place is even and ends in 0, 4, or 8, or tens place is odd and ends in 2 or 6	Ends in 0, 4, or 8
Five	Ends in 0 or 5	Sum of double twelves place and ones place is a multiple of five
Six	Passes tests for two and three	Ends in 0 or 6
Seven	Sum of groups is a multiple of seven
Eight	Sum of double tens place and ones place is a multiple of eight	Twelves place is even and end in 0 or 8 or tens place is odd and ends in 4
Nine	Sum of digits in second to last group plus a third of sum of digits in last group is multiple of three	Sum of fourth to last digit and a third of the sum of the third to last digit, second to last digit, and a third of the last digit is a multiple of three
Ten	Ends in 0	Sum of double twelves place and ones place is a multiple of ten
Eleven	Alternatingly add and subtract groups, repeat until there's a single group, check if digits match	Alternatingly add and subtract groups, repeat until there's a single group, check if sum of digits is multiple of eleven
Twelve	Passes tests for three and four	Ends in 0

While more of the divisibility tests for decadozenal are very simple, it's less clear cut overall. It's subjective, but I personally feel that the decadozenal test for five is significantly less simple, and thus slower to do in your head, than the dozagesimal tests for either three or four. I also find the test for nine more practical in dozagesimal, especially for larger numbers, and the test for eleven is significantly more practical for smaller numbers. However, taking into consideration the fact that divisibility tests for three and four are used decently often in daily life, and that dividing an integer by three or four can be made far easier with simple tests, just having those may be enough to make decadozenal worth it.

As the difference in theoretical practicality between these two systems I've been describing is rather minute, perhaps it's more interesting to examine how a consistent and sensible set of notation could be designed for a mixed radix system. Before I mentioned the idea from jan Misali's article of symmetric decadozenal, in which the fractional part of the number is treated as dozagesimal. This attempts to preserve the good of both systems, but I personally feel that this somewhat misses the point of why things such as simple representations of fractions and easy divisibility tests are beneficial, and it tends to break down for the purpose of changing the scale of numbers. Following this system, the number 45’45.45‘45 when divided by 1’00 becomes 45.53‘45‘45, and when multiplied by 1’00 becomes 45’45’39.45.
However, while I believe this would be rather inconvenient for something like scientific work, this kind of change of base isn't too strange compared to notational systems that have been in use before. An example of this is the way that small angles are written, where in 242° 42' 42", the arcminute and arcsecond measurements act as sexagesimal groups, while the degree amount acts as a plain decimal system. Though, to be clear, I do find that this is rather inconvenient in scientific work.
The way the ancient romans wrote fractions can be viewed in a similar way. Integers were written in base ten, while fractions used a division of twelths, though this was a rather rudimentary system for writing fractions. Other systems can be viewed as sorta between the ideas of writing a rational number as a fraction and as a decimal expansion. Ancient egyptian fractions relied on sums of unit fractions, and most other systems around that time used a similar method of writing values less than one with symbols or systems of symbols that represented specific unit fractions.[2]Egyptian fractions are actually incredibly interesting. Every positive rational number can be written as a sum of distinct unit fractions, but doing this with the least possible is far from trivial.

The final point I want to make is in regards to seperating groups. Though I've been using apostrophes as a way to seperate groups, jan Misali did not, and simply placed all the digits together. I don't really like this way of doing things for a few reasons. Not only does having seperators makes a number easier to read and understand, it also points to a meaningful symbolism of the notation used. The dozagesimal integer 12’34 can almost be read as a decimal integer, only with the apostrophe acting as a symbol that represents the value of decimal 120. This is similar to the way that many other numerical systems function, such as chinese numerals, where the characters "⼆十三" directly represent the idea of "two tens plus three". A purely positional system, while cartainly useful, is actually relatively rare among historical numbering systems, and shouldn't be taken as the default for the purposes of worldbuilding.