Base 60: digits and terminology

Ideally, every number (in its abstractness) should have its own symbol. However, since there are infinitely many natural numbers and our minds are limited, various symbolical methods to denote large numbers using only a finite set of symbols were developed. Because of its usefulness in applications, the positional number system (which we use today) was at last adopted by many civilizations, including the modern “western” civilization. What differed among civilizations was the base – the standard base now is 10, but in the Maya civilization, for example, the base 20 was used, and some native Americans used base-4 system.

Probably the most interesting base which humans are able to effectively comprehend is 60. In fact, it’s used even today (in principle), although its numbers are written using base 10. For example, we have 60 seconds in a minute, 60 minutes in an hour, 360 degrees (a multiple of 60) to measure angles, etc. People found the number 60 so important that in many languages it has it’s own name (apart from “five dozens”). What’s so special about this number? It’s its number of divisors. 60 can be divided by 2, 3, 4, 5, 6 and some greater numbers. This makes it in fact a better candidate for the base of “percent notation” – because 1/3 and 1/6 of 60 are integers, but this isn’t true for the “base” 100. The more divisors the base of a positional system has, the more conveniently division and multiplication can be performed in the system.

Old Babylonians are probably the only people known to use base 60 as their primary counting system. However, their digits were very large and hence unusable in practice. My goal was to create a new digit system in which the digits would be suitable for practical purposes and would manifest some sort of mathematical logic. And here they are:

The principle is simple – the symbol of a prime doesn’t contain any “middle” part, all compound numbers contain all symbols for prime numbers present in their prime decomposition. The vertical line, circle and slanted lines denote the second to the fifth power (of the prime they are attached to).

Symbols of prime numbers

I divide the prime prime numbers into five groups (to remember them easier). It’s convenient imagine the numbers somehow to be able to count in base 60 independently on base 10. A good way is to imagine dots in plane. That’s why I list the numbers together with a “decomposition” as a sum of a square and a small number. The syllable next to the symbol is the name of a prime.

Small and large primes share many similarities. They are all twin primes and the symbols of their twins can be found by reflecting their symbols horizontally. In fact, all twin primes share this logic (not only those in the two mentioned groups) and all primes that don’t have a twin have a horizontally symmetric symbol. Inserted primes are “dual” in the sense that their sum is 60. Beginning and end primes also share many similarities. Tu and re, as well as gu and ge aren’t twin primes (and hence their symbol is symmetric). The symbol for gi suggests that it has its twin – and indeed it has, but 61 is not a digit.

Names of primes and pronunciation

The structure of names of primes follows the division into groups described in the previous section. The ending vowels of the primes in 4-member groups are -u, -e, -i, -a. Inserted primes and non primes end with the vowel -o. The consonants are chosen in such way, so that all primes are easily recognizable from each other. Thus, they are different for all primes, apart from the rarely used end primes for which “g” symbolizes that they are big (great, grand, giga, giant…).

Powers of primes are determined by suffixes: -k for the second, -l for the third, -b for the fourth and -j for the fifth (and they are chosen so that there’s no possibility for a letter to appear twice in a row when creating longer names of numbers). So, for example, rek = 9 (re = 3, -k = 2), tub = 16 (tu = 2, -b = 4). Powers are denoted by adding a vertical line for the second power, a circle for the third, and slanted lines for the fourth and fifth as you will see in the table of all numbers.

The pronunciation is completely regular. “g” is always pronounced as in “get”. Vowels are always pronounced as in “put”, “get”, “fit”, “car” and “hot”. So, for example, “be”, as a number, is pronounced as the first two letters in “bed”, “fi” as the first two letters in “fit” and “no” as the first two letters in “not”.

Compound numbers

Symbols for compound numbers contain all symbols of its prime components with an appropriate symbol for their power. The symbols are not created completely regularly, but if you follow the rule that the power of a prime is always (with an exception for rel) in the bottom of its symbol, you will find, there’s usually only one natural way to put the symbols into a contiguous larger symbol. All numbers from 0 to 60 are:

If we want follow the traditional way to call larger numbers, that is “digit order digit order” (as in “three thousand seven hundred”), we have to create names also for the orders, that is, 60, 3600, …, or 10, 100, … in base 60. I’ve chosen the following names:

All consonants use different manners of articulation (nasal, fricative, sibilant, voiceless plosive, and voiced plosive) and the position of articulation is order from front to back (bilabial, labiodental, alveolar and velar). They also don’t collide with the letter denoting powers of primes. The letter “y” in is pronounced the same way as in the English word “my”.

Numbers above 60 follow completely regularly the form “digit order digit order…”. If the first or last digit is 1, it can be dropped, so, for example, myno, mytu and myre is sixty one, sixty two and sixty three. Tumy is one hundred twenty, tumyse is one hundred twenty seven etc. (for example = tufyremyfi = seven thousand three hundred eighty five).

By following this algorithm we created names for numbers up to gigygitygisygifygimygi = 46 655 999 999 = utterly long name in English. Then, we use the following notation: A-B = A x 606 + B, so, for example, no-zo is “(no times 606) + 0″, i.e. 606. The largest number having its own name is = gigygitygisygifygimygi-gigygitygisygifygimygi = 36 279 705 599 999 999 999 = thirty six quintillion two hundred seventy nine quadrillion seven hundred five trillion five hundred ninety nine billion nine hundred ninety nine million nine hundred ninety nine thousand nine hundred ninety nine.

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Jakub Marian is a student of mathematics at Berlin Mathematical School. In addition to that, he's a passionate learner of foreign languages, musician, and nutrition and cognitive science enthusiast. Apart from just doing these things, he also likes to share his insights and opinions on these fields through his blog to help others to learn more easily.