Comparing number bases and their coexistence with decimal

all numbers decimal, except for the fractions’ values

We’ll be comparing bases six, twelve and ten, ranking their mathematical simplicity in dealing with fractions;

Ranking will be weighted: the more simple, that less digits a fraction requires to be written and memorised, the more points this gets to the base; types of fractions are themselves ranked: terminating fractions > recurring fractions > decimal percentages;

We’re considering decimal percentages because that’s how we most encounter proportions, in the sense that we have to deal with them everyday in our daily lives: taxes, interest rates, economic, demographic analysis charts, academic studies, all use prominently decimal proportions, so, if you want to use other number base, it helps to be able to mentally and conversationally convert between decimal and your base proportion;

Finally, we’ll compare, though not rank, another common use of fractions for bases six and twelve, that is fractions of the day, sezimal and dozenal time keeping and managing, and how well you can transpose you’re currently standard (not decimal) time notions to your preferred base;

Terminating fractions

Clear winner here is twelve, followed by six, both are significantly better at dealing with terminating fractions than decimal, after all, decimal, as we all know, wasn’t made to be divided :);

Recurring fractions

digits on the recurring pattern after ‍󱹮‍ or ‍󱹯‍ represent amount of digits in that position

Here, suprisingly, decimal is by far the best base to deal with recurring fractions; when they do occur in decimal, they have less digits and a simpler recurring pattern than those of bases six and twelve, on average;

Twelve and ten are opposite here, unsurprisingly, while six, more balanced, outperforms both ten and twelve;

In the number bases’ race, twelve and ten are sprinters, though ten is running backwards :) , while six is a distance runner;

Frequently encountered fractions #1

5, 10, 20, 25, 50, 100 – decimal proportions and money

In here, base twelve takes a toll for it’s disregard of anything five related; but decimal isn’t going anywhere, anytime soon.

The only case where base twelve fared better than six was terminating fractions, but not by much, for all other instances, six is more adequate than both twelve and ten, by quite a lot;

Pattern recognition for fractions of 5, 10, 20, 25, 50 and 100:

On the table above the fractions are written in a way that highlights the fact that, from 55% up, they mirror their −50 counterparts, just adding the base equivalent for a half, 0‍󱹮‍3 or 0.6; the fifths have no fixed digit in either base, actually;

The way base six works with factors involving five is far easier, having just one single recurring digit for all the “landmarks” of decimal proportion, this single digit repeating in a predictable pattern, for each group, fifths (1‐2‐3‐4), tenths (3‐4‐1‐2), twentieths and overall (just 4‐3‐2-1);

Twelve, though it has a clear pattern of four digits in alternation, that alternates four times, the exact pattern of alternation of the four digits themselves, within their group, is not clear, since it’s not a simple right or left shift of one digit;

Lastly on this fractions analysis, of course we can express proportions in any base, and learn how to use them, but the fact is, that we’ll encounter far more decimal proportions in life, and fractions involving some factor of 5, and if we like our base best, there’s the natural tendency to mentally convert them; in doing this, six is indisputably better than twelve, and it would be overall easier to explain to a decimal-only person how to work with it.

Frequently encountered fractions #2

civil time divisions of the day

Time managing is, second to proportions and money, probably the most common way people interact with numbers and math on a daily basis;

There would be a clear advantage to use a “pure base” time accounting, meaning fractions of the day in a given base, regardless of base, than the current mixed base system we use today;

That being said, as it happens with base ten, mixed base time keeping will not vanish overnight, and both bases six and twelve would probably coexist with it for a very long time, so, we see on the table above how close time keeping in base six and twelve are to the current system;

The reasoning for this analysis is that, in solving the mixed base time problem, the closer we get to the current system, the less confusion and less difficult it is for people to transition to a pure base time system;

For base twelve, both the notation and the prefixes used with “day” come from the Primel Metrology and it’s showcase clock app UncialClock Deluxe; the same time format can also be found on the Dozenal Now page;

For base six, the same notation as regular civil time notation makes sense, where the : colon is intended to show a change of base, from 24 to 60 and then 60, since each two digits in base six correspond to one digit on base thirty‐six, though a notation like 5555.55 could also work; prefixes shown here come from the Shastadari Prefixes, and a working clock example can be found on the Sezimal Now page;

Both six and twelve deviate from the way we current understand civil time, but six has some advantages, compared to twelve, in making the transition easier:

• it can use the same amount of digits to represent roughly the same amount of time passed for the same positions;
• the “named positions”, hours, minute and second, have much less deviation from the current standard than twelve;
• tritiday and pantiday positions allow only up to the digit 5 in their positions, the same as standard time, so the sense of “almost at the end” is the same;
• 40 min time is a very useful time span in real life: for work life, it’s close to the time span of a meeting, a business lunch, a regular lunch time (in the USA, at least), a doctor’s appointment, a physical therapy session etc, and, for school life, it’s a time close to a class, half an lunch interval (for kids), nap time for toddlers, study sessions etc.;

Apart from the pure deviation analysis, those are not all objective reasons, of course, but they do reinforce what the numbers on the table represent: time kept in base six solves the “pure base” time keeping without sacrificing familiarity and practicality;

In essence, base twelve does improve fractions representation, at the expense of familiarity and coexistence with decimal proportions and money and the current standard time keeping;

Base six, on the other hand, improves every aspect analysed, compared to base ten and standard time keeping, with more familiarity, that translates into a less painful transition, and better coexistence with decimal proportions and money;