Quick Argument For Seximal

Intro

Quick explainer of what bases are: we use base ten, or decimal normally. This means the first digit of a number has a multiplier of one, the second digit of a number has a multiplier of 10, the third digit has a multiplier of 100, and so on. Ten to the power of the digit, starting from zero. This works the other way too. The digit before the first digit has a multiplier of one tenth. But what if we use a different value, like 6? Well then we have a ones, sixes, and thirty sixes (or nifs) column, and before the radix point (base neutral term for decimal point) we count by sixths. So 0.1 in seximal is one sixth.

Every base has that many digits. Base 10 has 10 digits, and base 6 has 6 digits. Those digits are usually 0 to b-1, and we won't be talking about anything unusual like that. So seximal only uses the digits 0-5, since there's no need for 6 if 6 can be written as <10>.

NOTE: I will be using seximal for the rest of the article, unless I'm showing the way another base notates things! So just keep in mind that the second digit of a number is multiplied by six instead of ten. I'll also show the word name for the number so you don't get lost. The word and the number are not in conflict.

Size, Factors, Neighbors

Size

Some people prefer larger bases, but I think that in general, smaller bases work better, since they are more informationally efficient and the powers of the base don't get out of hand too fast. (That's why I like binary, but that's a topic for a previous time.) This is hard to argue over, as it is subjective.

People typically try to find a balance. The bigger a base is, the more digits you have to memorize and the more complicated math gets. Also, you lose the advantage of the powers of the base. But if it's too small, there's too many digits to remember per number rather than per system. I think that binary is a great base, but it is a bit tricky to use for everyday things, since it handles everything as well as decimal handles thirds, while seximal handles the most important things really well. In other words, seximal is very conventional in comparison to binary, even if binary is "technically" superior.

Factors

14 (ten)'s factors are 2 and 5, so doing divisibility tests by 2 and 5 in decimal is easy, and the fractions for 1/2 and 1/5 are clean too. But 10 (six) has the factors 2 and 3, so divisibility tests by 2 and 3 are easy, and the fractions 1/2 and 1/3 are clean.

Neighbors

One less than ten (14) is nine (13), and nine is divisible by three (you can tell because I'm using seximal and it ends in a three!) so threes and nines in decimal aren't terrible to work with. But one less than six (10) is five (5), so seximal isn't terrible with fives, despite not having 5 as a factor. Doesn't it make sense to make 2 and 3 the easiest, and 5 the second easiest? As opposed to making 2 and 5 the easiest, and 3 the second easiest? Lastly is seven (11). Decimal is very bad with sevens, there's no divisibility tests* and the fraction is 10 (six) digits long and recurring. In seximal, the fraction is 0.05 with an infinite number of 05s afterward. Pretty simple!

*Okay, so, it turns out there kind of is. I don't know why no one else has mentioned this yet, but there's actually a bunch. Just google it. They're all a bit convoluted though, but I just thought I should mention that I was wrong about divisibility tests here.

Factors Continued

Powers of the Base & Composites

Seximal handles 2 and 3 perfectly, 5 pretty well, and 11 (seven) decently. Decimal handles 2 and 5 perfectly, 3 and 13 (nine) pretty well, and 15 (eleven) decently. But what about composites? Well six squared is pretty small compared to ten squared, and this is a good thing! To test if a decimal number is divisible by four, you have to memorize 41 (twenty five) two digit pairs! If it ends in one of those digit pairs, it's divisible by four. This is because pairs of digits act as the base's square, and 244 (one hundred) is divisible by four. Nif (six squared) is also divisible by four, but 100/4 (thirty six divided by four) is only 13 (nine), much more manageable! This also works with testing divisibility by nine, and you only have to memorize 4 digit pairs for that. Then you can also do divisibility by 20 (twelve) if it ends with 00, 20, or 40, but that one's pretty self explanatory. Same with divisibility by 30 (eightteen).

Fractions

Seximal's fractions are very nice and simple. one half is 0.3 (three sixths), one third is 0.2 (two sixths), one fourth is 0.13 (one and a half sixths), one fifth is 0.1, one sixth is 0.1, and one seventh is 0.05. Decimal's fractions aren't too bad, but they're not as good as seximal's. One half is 0.5, one third is 0.3, one fourth is 0.25, one fifth is 0.2, one sixth is 0.16, and one seventh is... well I have to look it up now. If that's not proof enough I don't know what it is. Ah, it's 0.142857. Yikes!

Divisibility Tests

I mentioned divisibility tests earlier, but I didn't go into much detail. I'll continue. In seximal, if a number ends in a multiple of two it's even, if it ends in a multiple of three the whole number is a multiple of three. To test four, you have to look at the last two digits. If it's 00, 04, 12, 20, 24, 32, 40, 44, or 52, it's divisible by four! Alternatively, to avoid memorization, look at the second to last digit. If it's even, make sure the last digit is 0 or 4. If it is, the number is divisible by four. If the second to last digit is odd, the number must end in 2 to be divisible by four.

To check by five, add up all the digits and check if that's a multiple of five. To check by seven, add up all the pairs of digits, so for example 3,3403, 3 + 34 + 03 is 44, which is a multiple of seven! This works because one less than nif is divisible by seven. Alternatively, you can do an alternating sum thing. Add the first, third, fifth, and so on digits. Then add the second, fourth, sixth, and so on digits. Subtract the evens from the odds. In this case 14 - 3 is 11, which is seven.

Dozenal

Lots of people like base 20 (twelve). I think it's bad for two main reasons: it's bad with fives and sevens, and it's large. One fifth in dozenal is 0.2497, and one seventh is 0.186X35 (I'm using X for the ten symbol and E for the eleven symbol). Also, single digit quarters are very gimmicky. You don't need repeats in the prime factorization. A base should try to be smaller and then let the powers of the base repeat those primes for them.

Conclusion

That concludes my argument for seximal, I think. I may have forgot something. In conclusion, seximal is pretty cool and conventional enough to be actually used sometimes. If this interests you, check out jan misali's seximal website! It inspired this webpage and is the only reason I know about seximal. (Well, the video he made that links to his website, but still) It's like this webpage but higher quality and longer. Lastly, I designed my own seximal symbol set. Here are the digits in order, 0-5: , , , , , . They're not particulalry good or anything, but I think they're pretty and I like using them.

I made a dice set with them! (And a 3D printer)