I once read an appendix in a math textbook on mathematical bases. A base is how many numbers you use to count. We typically use 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. But we could use one number less, by counting 1, 2, 3, 4, 5, 6, 7, 8, 10 (missing the 9). In this latter system, 100 would be our equivalent of 90. Alternatively, we could also add numbers. So if we took the usual 1 to 9 and added A, B, C, D, E and F (and pretended they were numbers to) then “EF” would be equivalent to our 239. These are called bases. They are an alternative way of counting, and are fun to play with.
Here are some of my favourite ways of counting (some of which are bases):
Unary
If you keep removing numbers from our system, you eventually get to binary: Just 0’s and 1’s. In binary, 1101 is equivalent to 13 in decimal (normal counting). However, you can actually go one more, and get rid of zero. Now you’re left with just 1. So 11 is 2, 111 is 3, 1111 is 4, etc. Of course, this is actually very simplistic. It’s as good as counting by dropping pebbles in a jar.
Imagine a sheperd, who for every sheep he counted, he would add a pebble to his jar. Assuming he didn’t make a mistake, the pebbles in that jar would be a direct mapping of his sheep, a pure representation of information – which isn’t so much a material thing, as a composition. That’s why I like unary.
Dozenal
Is just another base. Get the usual 0 to 9, and add two more numbers: D, and E. So E is 11, 10 is 12, 90 is 108, etc. This sounds all very normal, except it’s actually a slighlty better system than the base 10 we all know and love. A dozen is easier to divide than 10. A quarter of a dozen is 3: A whole number, whereas a quarter of 10 is 2.5. A third of a dozen is four, but in decimal that’s 3.33.. If we all counted in decimal, counting would be a little simpler.
Old british money
Speaking of dozenal, old british pounds were 240 pence. Now there’s a very divisible number. There was a special coin for 12 pence (shilling), 6 pence, etc. Admittedly, I’m not so much a fan of old british money as a counting system for these features, but simply for the stories that I enjoy that use it (for the romantic value). In the last chapter of ‘A Christmas Carol’, Scrooge offers the street boy half a crown. Perhaps the most human system might now be the most mathematically perfect one. Feet, inches and miles are a wonky system, but measuring a person in feet is somehow more relatable. When I was a kid, my dad would drive a van, and as he was reversing, he’d ask me how far he had. I initially started telling him in meters. One point twenty five meters. Then I realised this was really silly, and it was much easier and more intuitive to measure that distance in feet!
Binary
Of course, the language of computers. but what made me continue enjoying binary was binary finger counting, a superior method to the usual count to ten with your fingers. This is better explained with a video though, linked below.
Clocking
Clocking is the idea of counting, but resetting the counter when you get to a certain number. It’s called clocking, because that’s exactly what we do with a clock. Nobody wants to know that it’s 33 hours past yesterday’s midnight, but rather that it’s seven hours past today’s. You drop the 24 (or the 12) every time you get to it.
There’s a mathematical operation called modulo (represented by a percent % symbol) which involves clocking. It’s like simple division, except instead of discarding the remainder, you discard everything else. So while 16 divided 5 is approximately 3 (we care less about the remainder 2), 16 % 5 is 1… This is obviously a bit weird, and one is less likely to apply it in real life, though a simplistic form of hashing could help:
Say you and your friend knew the name of the winner of a race. You both wanted to talk about it, but only if the other already knew the name of the winner. However you don’t want to tell the other the name of the winner. This, if you were into such things as a kid, is where modding might help. Instead of giving the name away, you give away the remainder of the sum of the letters of the name. So if the name is Bob – 2, 15, 2 – and you just keep the units (modding by 10) you get 9. So you can prove that you probably know the name without giving it away. If you didn’t mod the result, i.e. you told him the sum of the letters was 19, it would be a lot easier to deduce the name.
Of course, all this counting isn’t much use unless one has something to count. And sadly, clocks and calendars take care of a lot the important counting for us. However, in case they’re not around, perhaps it’s worth learning binary finger counting, how to comfortably count all the way to 255 with just your two hands, and that’s not even including your thumbs! Other things worth looking up if you like these ideas are:
- Soroban – the Japanese manual calculator (superior abacus)
- A Dinotopian abacus
- Terra Mystica – bowls of power
Links
- A story that involves using pebbles to count: https://www.greaterwrong.com/posts/X3HpE8tMXz4m4w6Rz/the-simple-truth
- Numberphile’s video on the dozenal system:
https://www.youtube.com/watch?v=U6xJfP7-HCc?list=FLZi-o2oxhoS4LyfPHeEJ7rg - Binary finger counting: https://www.youtube.com/watch?v=I8V4kVSO5Ns
Joe Grimer 