Base-ic math

Cole Ditzler
6 min readOct 29, 2019

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Despite its ubiquitous usage in the modern world, the base-ten numeral system with which we are all familiar [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], is both arbitrary and flawed. Humans typically have ten fingers, which really seems to be the reason why a base-ten numeral system was adopted in the first place. This makes obvious sense at first glance, but there is an underlying problem.

“Two, four, six, eight! Who do we appreciate?..”

An even numbered base has at least one advantage, any number represented with a terminal digit (the last number, the one in the 1’s spot) that is itself even will mean that the whole number is also even. Contrast this with base three, in which the quantity four is written “11” as in one ‘three’ and one. This advantage is enormous, but the trend also holds.

For any base, the multiples of prime factors of that base will have a simple relationship with the way they are written. In short, its immediately obvious that a number ending in 0 or 5 in base-ten is divisible by five (the prime factors of 10 are 2 and 5). Another example would be something like the number 427 in base-21, which would be written (20)7, as in twenty ‘twenty-ones’ and 7. As 21 is divisible by 7 and “(20)7” ends in 7, we know right-off-the-bat that this number is divisible by 7, and if you know your base-21 multiplication tables, there are three sevens per 21, meaning 60 sevens and seven, or 61*7 = 427.

If thats a lot of math, buckle up.

“And I can’t help but think that you’ve seen the mistake, but you let it go…”

The glaring flaw in base-ten is how we are forced to deal with numbers which are co-prime to all of the prime factors of the base i.e every prime except 2 and 5. A readily salient example is the term 1/3. It fundamentally cannot be expressed in complete decimal form in base-ten. 0.33333… etc. I get it, but why? Given the common response of “ok, theres a problem with three, but we can just use base-twelve right?” Well sure, but then we lose five right? 5/12 is a repeating decimal in base 12. Its approximate form is “0.435186(10)35186(10)… What has been gained in base twelve when we lose something so basic. Not to mention the fact that a quantity of fifteen would be represented as ‘13’ in base twelve, with five losing its ‘quick-factoring’ property that it has in base ten.

“If you’ve seen one…”

Now it seems reasonable to me, to try to move to a system that encompasses the benefits of both base-ten and base-twelve. Their highly superior extension, base-60. Base-60 has outstanding benefits over both base-ten and base-twelve in that it has ‘quick-factoring’ for every number that ends in a digit which is a multiple of any of the following: 2, 3, 4, 5, 6, 10, 12, 15, 20 and technically 30. Whereas for ten its a measly 2, and 5, and for twelve it has 2, 3, 4 and 6.

Even over base-twelve it is clear that base-60 has several times the human usability. In addition to all these quick tricks, there is the fact that many numbers like 1/3 in base-60 are non-repeating being written in a single digit ‘0.(20)’ (remembering that in base-60, 20 is a digit).

In groups of 10, the digits 0 through 59 in base 60

The above image contains the sexigesimal system I would like to see implemented. It is a font I constructed (and can be found here)using alt-codes to encode the additional characters (e.g alt-31 is the character for 31)

This system is built to be internally consistent and self-referential. By which i mean that a six is shaped like two ‘bred’ with three, and twelve is shaped like two ‘bred’ with six. Every number attempts to be shaped like its own prime factorization (thus primes get their own family of symbols) a number like 48 is made to look like a ‘four-twelve’ so to speak. 46 is accented 23, and numbers like 59 (the highest single digit prime in the base) is a unique symbol. It is something I think that anyone can learn and its easy too! If you know six, you know twelve, and eighteen, and twenty-four, and so on. Learning a small family of primes teaches you the entire system. It helps with math involving time and angles, but it can be used to do any other kind of math.

I’ve put a lot of thought into base-60, I’ll admit, more than was reasonable. I went so far as to perform number theory in base-60, and was able to publish a trivial series on the OEIS, but I would be remiss not to point out the flaws of base-60 which are sadly equally glaring as base-ten and base-twelve. Firstly the numbers 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 and 59 are both coprime to all factors of 60, and less than 60 themselves. this means that multiples of these numbers do not follow the pattern of ‘quick-factoring’ they may pair with a factor of 60, but they do not give themselves away with the last digit as the other numbers do. Also these same numbers and their multiples, when used as the denominator of a fraction do not terminate. They repeat infinitely, in the same way as 1/3 in base-ten.

“Where does it end?”

A number system that would capture the most primes seems best, but it quickly gets out of control. A nice idea like a base that contains 2, 3, 5, 7, and 11, quickly becomes infeasible when you realize that would be base-2310. Perhaps it would be nice, but learning and differentiating 2310 symbols at 10-pt font is out of my wheelhouse. I think base-60 is a happy medium, it hits enough primes, and is highly-composite number. Its size is just shy of unwieldy, but not outlandishly so.

“A familiar face in a crowd”

For programmers, binary and hexidecimal are common enough, occasionally octal and ternary make their appearances. I have rarely met someone who advocates for their primary usage over base-ten.

“oh, and one more thing”

In base-ten any number multiplied by 9 yields a number from which the sum of the digits is congruent to 9, (6*9 = 54, 5+4 = 9). In base-60, this is true with 59 and 59, (59*4 = 3(56), 3 + (56) = (59). Also I feel the need to share an interesting fact I learned in my base-jumping. There are no 4-digit palindromic primes in any base. For tat matter there can only be one palindromic prime with an even number of digits in any base, and even then it isn’t always there. in base 60, the number is 61 ‘11’ and in base-ten it is eleven or ‘11’, see a pattern? Once ‘11’ comes around, it divides every single palindromic number with an even number of digits in that base. Given that the quirks carry over, and the increased precision for many human-relevant calculations, why not make a mental transition to whatever base you prefer?

I’ll be sticking with base-60.

I even went so far as to build a phone calculator for big math.

Maybe someday we will have three arms on each side, with ten fingers each, or maybe two arms on each side with five fingers each with three finglets at the end of each finger, articulated and strong. But I think thats a hard sell.

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