A day after my delayed, 2am arriving plane and I have enough energy to write again. Hooray!
So I’ve just started a preliminary course before I start my PGCE, which is a little scary considering I have to make sure that I get 30 hours per week of work documented. So, what does this mean? Well, naturally it means that I’m going to take it out on all of you, because I most certainly have a sadistic streak. Wednesdays are my independent study day, though apparently those will increase as time goes on (maybe, probably, perhaps?), but until then Wednesdays will be my “I write a maths post” day, along with weekends, where I will likely find a mathematical concept or topic and just go to town on it like the seagull that attacked me for my food on my first day back. I’ve never swung at a seagull before, not in the 17 years I’ve spent in Brighton over my life, but it invaded my personal space as if it was going to try and take my sandwich right from my hand. Cheeky bugger.
Anyway, the mathematical topic for today will be number systems. “Why number systems?” I hear you ask along with “what are you on?” and “why are you pushing numbers at me?” as I watch my follower count diminish (also, off topic here but… Yay! 100! You guys and gals are awesome people. Hugs!). Number systems are an interesting topic, but first we will visit the concept of the abacus to get an idea for how we will construct and look at number systems (also, it’s the topic that I covered on Monday so it’s still fresh in my mind. I built an abacus out of dice. You can always spot the role-player in the room when they start pulling out dice that aren’t necessarily 6 sided).
The abacus has been around for… thousands of years. It was a tool used for calculations back to ancient times, with the Egyptians, Romans, Greeks, Persians, Chinese and even the Old Babylonians using forms of abaci before we even start counting years in the right direction (AD as opposed to BC). Since the flip from BC to AD (or BCE to CE if you go that way) we have seen the abacus used by many more, such as the Japanese, Koreans, Chinese (continued from BC/BCE), Indians, Native Americans and Russians, but what we will look at for now is a simple, base 10 abacus.
We have all seen one of these at some point in our lives (right? Right) even if we have never used one. I remember when I was a little tyke waiting in the aptly named waiting room at the doctor or the dentist and seeing an abacus in the box of items which every good waiting room has to stop children from getting bored and start playing a rousing game of parent annoyance. I will readily admit that I had no idea how to use one, nor did I really care as a child, far more interested in Lego and… more Lego, but I would always pick out the abacus and play with the beads with not a care in the world, oblivious to the fact that I was about to get a needle jabbed in my arm or something equally uncomfortable because I never learned the pattern. Anyway, childhood experiences aside, the abacus is actually used today as a means of calculation. When I was told that it would become easy after a while to perform addition, subtraction and even multiplication and division on an abacus I laughed and went back to thinking about what I was going to build later on in my evening Garry’s Mod session or how nobody loved me or something equally as nerdy teenagery. Then a few years later the abacus was brought up again, and now as a post-graduate it has come up again.
And you know what? It comes as almost second nature now. Why? I unwittingly learned how to use an abacus when I studied my Computer Science degree (*shudder*) and learned how to perform binary operations without the use of a computer. The only difference was that I used an abacus with 2 beads per line instead of 10 beads. It was also all in my head. I’m not crazy, just a mathematician. Wait… Don’t you dare say anything!
At this point you are likely screaming at me to just explain how the damn thing works, or you’re rolling your eyes as you already know all of this so I will continue on. An abacus is simple. Take the picture above as an example. We will start at the bottom. There are ten beads on the row and these will represent units (0, 1, 2, 3, 4 . . . 8, 9). Then the above line will represent the tens (00, 10, 20, 30, . . . 80, 90) and the line above that will be hundreds (000, 100, 200, 300 . . . 800, 900) and so on and so forth. Taking this into account we can see that the above abacus can display in an easily understandable way up to 10,999,999,999 as there are 10 rows, each with 10 beads where 9 is the largest number each row can hold apart from the last which could be 10. Now I can hear you screaming (you’re screaming at me a lot today. You’re going to damage my fragile emotions!) that you can get higher by putting all 10 in each column, but that will get weird quickly as 10 in a row equals 1 in the above row and reverts back to 0 in its own row. It will get weird and you know it. Deep down. Unless you know how to use an abacus with more proficiency with me (very possible) but the direction I would like to go with this article is more for the educated layman such as myself; I only have another year and a half to categorise myself as such!
Addition is simple. You put all the beads on the left (or right, it doesn’t matter) then move the beads representing your first number to the opposite side. Then you simply add the units, add the tens and add the hundreds. Subtraction is the same idea but the other way around. Multiplication is a bit weirder, but still fairly simple. You do the same as addition, but instead of adding the numbers you multiply the units by the second number, then the tens, then the hundreds etc. Remember to add one to the next row every time you hit 10! Division is icky and sticky to explain without a visual aid. Sorry! I’m just not confident that I could explain it without grabbing a pen and paper (or an abacus), sitting down at a table with you and pushing beads around. This is how we do arithmetic using the decimal system which we all know and were taught at school (though not using an abacus for most I would assume). Units are 10^0 (^ being “to the power of”), tens are 10^1, hundreds are 10^2 etc.
So what happens when we change the abacus. I will give you a hint by going back to binary arithmetic, one of two things I could actually do on my Computer Science degree (the other being logic). Binary arithmetic, whilst I will not give you a lesson on how to do it as it is, again, far easier to demonstrate than explain how it works, I will tell you that it is done in the exact same way as decimal arithmetic on the abacus, just with 2 beads per row instead of 10. First row is 2^0, second is 2^1, then 2^2 and so on and so forth.
With this knowledge we can effectively make an abacus for any number base and then perform arithmetic in the same way as we have with decimal numbers. So, let’s get to inventing a number system! All of the numbers I would normally have chosen are apparently already an invented number system (4, 5, 8, 36) so let’s go with… base 40. The first column is for 40^0, second is 40^1, third is 40^2 etc. As each column can only be labelled with a single digit we will have to go from 0 – 9, then we will go from A – Z for 10 to 35 and then we will use… \, /, # and ~ for 36 to 39. Sorted! So to write… 1850 in this new number format (a good number because it’s a good points value to play Warhammer 40k at!) we would have to write it like… 16A (1 lot of 40^2 (1600), 6 lots of 40^1 (40 * 6 = 240) and 10 lots of 40^0 (10 * 1 = 10) for 1600 + 240 + 10 = 1850). What about 1500? That’s also a good points value for Warhammer 40k. For that we need no 40^2 as that is 1600 in itself so the next step is to see how many 40s go into 1500. The answer is 37.5 (hooray calculators!) so we will need to round down to 37 (37 * 40 = 1480) then make up the remaining difference with 40^0, where we need another 20. 37 was represented by a / sign and 20 was represented by a K (I had to use my fingers to count that one up. Such tools are always helpful through life!) so 1500 is represented by /K, which honestly looks more like a chat shortcut in an MMO, but it works! I would love to call this number system Lexicon’s Ludicrously Labelled Number System, but I don’t think I’m the first person to consider a base 40 numbering system and that it would likely have a boring name like quadresimal or quadesimal. We could, however, build an abacus that runs on base 40 but we’d need a lot of beads as each row would need 40 of them each!
You can apply this method to any number and make a number system, though I find that the larger the number the more unwieldy it becomes for us, though larger number bases become easier for computers to handle as they don’t have to process such long values. Tetrasexagesimal is base 64 and is used in computing, for example. However, I would stick to lower number bases. That could be personal preference, but I would have to start inventing symbols pretty soon after base 40 (and my artistic skills are on par with an eight year old).
I want to wrap this up as this is already longer than the word limit for my first assignment that will be coming up, so I won’t bore you with too much more about number systems, but they are a very powerful system and the abacus is a very powerful calculating tool if you get used to it. Those exam papers that say no calculators say nothing about skills with an air-abacus (doing calculations on an imaginary abacus. People do it! See? I’m not the only one who’s crazy!). However, if you decide to do an entire maths exam in binary, please remember to convert to decimal afterwards; the mark scheme does not account for answers in other numerical systems!