“I can’t do it” – Words that every teacher will hear

I was working with a class, as part of my teacher training, that was doing an assessment the other day and the immortal words were uttered.

“I can’t do it.”

This is a phrase that I hear so much as a trainee maths teacher and as I had very little to do during this assessment lesson I got thinking. Why do you hear this so much in a maths classroom and I think I started to get on to something. Well, at least I would like to think that I started to make some progress, though this is nowhere near the level of a university dissertation which I imagine would be a very viable topic for such a thing.

So what did I think about it? I think that it might stem from the idea that maths and sciences are very factual, whereas arts and humanities are more open to interpretation. From a young age we are taught the idea that in sciences and mathematics there are the answers and the answer is right or wrong. For example, 2 + 3 = 5 (in the decimal number system, the system that we all use every day), whereas your interpretation of how the magic school bus goes about its crazy adventures is largely opinion based; one person may make some observations, but someone else may make completely different observations, both of which may be right. But as I said, in mathematics, you are right or wrong.

This will undoubtedly cause some insecurities in the minds of a young student. Fear, for one, will cause them to be afraid to answer questions, on fear of getting it wrong and thus they will not get as much practice in mathematics as they may in English, where you’re never “wrong” and everything is down to interpretation. Generations of fear have led to one of the many stigma that is now associated with mathematics: mathematics is difficult and complicated. Mathematics is very, very simple at the basic level, much like English, and is arguably easier even at the highest levels of study. Working out what a Victorian era author is trying to put across in his or her work of fiction is will never be as accurate as the solution to a question on complex numbers. People may argue, again, that the difference between the two is unfair as one is largely interpretation and one follows a set of rules, but the truth of the matter is that most marks in exam papers are for your working out, not the actual answer.

As an aspiring writer on a mathematics teacher training course, I would like to think that I have a fairly valid opinion on the differences and similarities between English and mathematics, though this is by no means a hard fact, set in stone and immortalised in the annals of history, however I believe that there is some truth to the next statement: writing fiction follows rules just the same as mathematics does. Poorly done mathematics is like the majority of the drivel on fanfiction.net, only less laughable. Poorly done mathematics implies that the person is trying their best, but I’ve found that when looking upon poorly done mathematics I laugh a lot less than I do when reading badly written fan fiction (some is good…). Poorly done mathematics can be remedied far easier than poorly written fan fiction. I would go as far as to liken it to incorrectly done mathematics as well, as poor fiction is, I would argue, fiction done wrong. In both cases you can help people get better through constructive criticism, which I have had plenty of thanks to the lovely friends who read my work and then tell me where to improve; you guys are the best! However, when it comes to incorrectly done mathematics, people seem far more receptive to the constructive criticism needed to improve, as long as they do not fall into a pit of “I can’t do it” despair, whereas practitioners of the arts are far less receptive to such and, at least from what I’ve seen, can take constructive criticism as a personal attack because obviously their work is perfection and they are the next Shakespeare or something. This, one would think, would counter-act the fear and stigma surrounding mathematics and it does to some extent, at least at the higher levels; though even at school level I have found that students are willing to listen as I try to explain things to them.

However, the thoughts that mathematics is complex and only smart people can do it has also led to another form of stigma: the “uncool” stigma. I’m sure that I do not need to go into details about this, as we all know that at school the mathematics geeks are largely surrounded by the unwashed, nerdy perception as opposed to the gloriously attractive and amazing P.E sports team or art crowd (I’m not bitter. No, I’m really not…). Recent advances in our culture have helped combat this, with shows such as the Big bang Theory showcasing nerd culture as entertaining and fun as opposed to smelly and anti-social (I mean, now we even have hipsters trying to pretend that they’re nerds, which is super adorable. It’s like me pretending to be a sports fan. Ha!). But at the younger levels such interpretations aren’t around as much, with sports personalities and air-head celebrities being role-models as opposed to political leaders (Churchill was one of mine. As well as Thatcher (the author was brutally murdered by angry readers shortly after writing this)), historical idols (Queen Elizabeth I) or at least classical authors (Charlotte Brontë for me). Being the smart child in the class makes you a pariah, a plebeian among the footballer-worshipping masters. Well, do you know what’s I find really “uncool” as an adult in a developed country? Being uneducated in a place where you have all the opportunities to get a good education, such as in England.

However, this is in itself grounds for a giant essay, which I do not want to get into here. This is just a brief look at my views and opinions on why mathematics is perceived as difficult and scary by students, as opposed to at the very least doable. So, to summarise:

Fear of being incorrect: well, do you know one really amazing way to learn? Making mistakes and learning from them. Do you know what’s even better? When you have ample opportunities to make these mistakes without it costing you anything but time that is already there for exactly that purpose (school).

Perceived difficulty due to facts as opposed to opinions: mathematics is what you make of it. Most marks are made through working out the answer, as opposed to the answer itself being correct. If you go about a mathematical problem in an appropriate way, you are as correct as a person who interprets what they believe an author is trying to get across in their fiction (and don’t even get me started on historical source accuracy!).

Uncool mathematics and “clever subjects”: oh please. Do I really have to say how this is ridiculous? The idea that being an achiever is uncool? Just… What? Since when was success uncool? Yes, it really is that silly.

I am going to try and dissolve some of these notions through my own teaching career, but how much can one person do, eh? Well, at least maybe, just maybe, I can influence a few people to not view mathematics as scary or “uncool”. But who knows? I’m just a trainee, after all, with very little experience thus far.

Why I Love Geogebra

So… This week I was introduced to Geogebra as a part of my mathematics course. In short, it’s a computer program that allows you to do graph functions and constructions, though I have only skimmed the surface of what it can do. Apparently I can also do 3d geometry, but I have only spent a few hours on it so far.

So why do I love it?

Well, some of you would know that I absolutely hate geometry and I hate constructions; I’ve always been a numbers and algebra kind of mathematician. The reason why I absolutely love Geogebra is because it makes constructions an absolute walk in the park. Instead of reaching for a compass and staring at a problem for ages I can start playing about in Geogebra and it makes my life easy. And I love it when my life is made easy.

Secondly, it’s totally free. Totally. Free. And it works on Linux as well as Windows and Apple iOS (ewww). It also has an incredibly large array of languages available. I am trying to improve my French, for example, so my Geogebra is set in French, but I could easily put it in Japanese or Russian or Spanish or three different kinds of English (UK, US and AUS). Also, it can be run from a USB stick. You don’t need to install it if you don’t want to. You can literally run it from an external device just as easily as you would play a video.

However, this got me thinking. My teacher really loves geometry, but pushes algebra aside a little, whereas I do the exact opposite. What I am wondering is that if there is a connection; if numbers and algebra are linked, but geometry is a completely new ball park. I am already a living counter argument of stereotypes. Apparently if you’re good at mathematics, you’re also good at sciences. I hated sciences, but I loved humanities and languages alongside mathematics. I can’t do physics or chemistry. My brain just doesn’t work that way. However, I love getting into a tough mathematical problem, as long as it’s not geometry. Co-ordinate geometry is different, I can get that as it’s a combination of algebra and geometry, all I need to do is hold onto the algebra tightly and go on an adventure. The general idea that I am getting is that geometry is more innate, where algebra is more about problem solving and crunching out procedures. Also, it really does seem that there is a massive divide between geometry and algebra and I am not the only person who finds one easy whilst the other one is incredibly difficult.

I never got the attraction of geometry, how it all worked and often boiled down to a simple solution. It doesn’t excite me to spot something and then write a line or two to show why something is right. It doesn’t stimulate me to draw shapes that fit together, even when I consider the mathematical properties that dictate just why that happens. I feel academically accomplished when I work hard, not when I happen to spot something that works. Now, whilst some people may say that geometry does get complicated, the problem there is that by that point I’ve spent years subconsciously  giving it a lower priority whilst my algebra and number skills get all the practice that they need and then some more. Now, I’m not saying that geometry isn’t a challenge (quite the opposite for me. I can’t do it!) or that it isn’t important. It is an incredibly important part of mathematics and I understand that at the age of twenty three, but when I was younger I did not and so I did not care enough to try hard at it. My A* at GCSE was only earned because I aced everything and then scraped by on geometry, however I didn’t care as an A* was an A* at the end of the day.

I don’t have much to write about this week as we’ve done a small mountain of geometry (I hated it…) which is really difficult to demonstrate on a computer, but also partially because I’ve been really busy with my short stories for the Black Library which I managed to send off. I sent both off, one from my own personal email and one from my alias’s email, which have sucked up a huge amount of time. Hopefully they reply to at least one saying that they want to publish the whole short story, but if they reply to both then… Well… I may have bitten off a bit more than I can chew, but we’ll see. I’m confident in my ability to do it! Fingers firmly crossed.

Equivalence and its Uses in Fractional Arithmetic

This week we started looking at equivalence. The first question that was asked was, undoubtedly, “what is the first thing that comes to mind when you think of equivalence?”

Well, as you may or may not know, I’m taking a pseudo-vow-of-silence this week. This means that I did not answer the question, though I did answer it in my head. Naturally, the first thing that came into my head was logical equivalence, an expression in propositional logic that translates into English as “if and only if” and is shown by what can be explained as a long “equals” sign with arrowheads on each side pointing away from each other. For example P is logically equivalent to Q (I don’t have the symbol on my keyboard because I’m surprisingly normal in that regard) means that P and Q must have the same logical content, which is where we get all of our logical equivalence laws from, such as De Morgan’s laws, associativity, commutativity, double negation and so on and so forth. However, this is not an article on logical equivalence, this is an article on mathematical equivalence.

Which is a very similar concept. Now, we were not looking at anything complex; the course that I am on is designed for people who have not done A-level mathematics, which still comes as a surprise to me, much like how my course mates on Computer Science as an undergraduate hadn’t done A-level mathematics, despite it being listed as necessary on the prospectus. We were looking at basic equivalences. I feel an example is warranted:

I go into a super market and buy something for half price, because sales, offers and reduced to clear items are how I live. I get to the checkout and get chatting with the cashier and comment on how I only paid 50% of the original price of the item. I then decide to work out what my saving on the item was (because I didn’t look at the reduced price, only the original and the fact that it was half price. I get impulsive around food…). I type into my calculator the original price (which we’ll say was £3.50) and multiply it by 0.5.

Deconstructing the example we have the concept of half (1/2), a percentage (50%) and a decimal value (0.5). All of which are equivalent (but not equal!).

These are not equal because whilst you can write 1/2 = 0.5, you can’t write 1/2 = 50%. 50% of what? 50% of 1, sure. What about 50% of 10564? That is definitely not equal to 1/2! 50% requires context, so it is not necessarily equal to 0.5 or 1/2, however it is equivalent. We, or at least I, use equivalence without considering that we are using it. For example, if I were working out what 25% of a number is, I will default to a quarter and just divide it by 4. If I wanted to work out what 43.8% of a number (without using my calculator) I will, again start using fractions which are equivalent to what I need (43.8 -> 43 4/5, 43.8% -> 43 4/5 / 100) as I find them easier to use than playing about with decimal numbers and place value.

The largest area in the maths syllabus where equivalence comes up is, with no surprise from me, in fractions. Equivalent fractions are a large topic. Then again, it’s important to be able to spot equivalent fractions or work them out. From a basic point of view, it means that you don’t have to deal with such out of control numbers. What would you rather deal with? 1/3 or 17/51? I know that I am lazy and love to keep my life simple, so I would take 1/3. You can have 17/51 if you really want. It’s yours. Go nuts.

I came up with a way of explaining why simplifying, or inversely simplifying if you need to match denominators for an addition or subtraction, works the way it does. So, first a general question to you, the reader. What happens when you multiply any number by 1?

I really, really hope that you said “it stays the same” or “nothing changes with the original number” because, when considering equivalence, that is totally true. Multiply anything by 1 and it will not change. However, consider this: take 1/2 + 1/8. We need to get 1/2 in terms of eighths (that’s a really odd word to spell…) so we multiply it by 1.

“But hang on…” I hear you say, “1/2 multiplied by 1 is still 1/2.” You are totally right about that, but that’s what you get if you multiply 1/2 by 1/1. 1/1 is just a fancy fractional way of writing 1. So is 2/2. And 3/3. And even 1/2 / 1/2. So, to get 1/2 in terms of eighths, we multiply it by 1, otherwise written as 4/4 (as we divide the larger denominator by the smaller one to know what number we need to multiply the smaller one by to get the larger number. Remember, division is reverse multiplication). 1/2 * 4/4 = 1*4 / 2*4 = 4/8. Therefore we can write 1/2 as 4/8 and thus add 1/8 to it to give us 5/8. Not 5/16. In addition and subtraction we do nothing to the denominator once we have made them the same. Nothing. We don’t touch them. It’s like they suddenly got a bad flu and we want nothing to do with them. Or it peed itself and now has a fairly anti-social smell of urine about it.

Similarly when we’re simplifying fractions all we are doing is multiplying by 1. You find a number that is a factor of both the numerator and denominator (either by deconstructing both to their prime factors or just seeing it/testing numbers) and then make a fraction of it as a unit fraction over itself. This is incredibly awkward to explain without paper and pencil, so I will give an example. With 17/51, we can see that 17 is a common factor of both numbers (17 * 1 and 17 * 3). So to reduce it we simply multiply it (or divide it. Multiplying by 1 and dividing by 1 give the same answer!) 1/17 / 1/17. 17/51 * (1/17 / 1/17) = (17 * 1/17) / (51 * 1/17) = 17/17 / 51/17 = 1/3.

Of course most of us do this in a much more simple looking way, but the roundabout explanation of it is this. And it all revolves around the idea of equivalence. Equivalence to solve equivalence.

Similarly we can use this concept to explain how to divide fractions without simply saying “you turn the right hand one upside down and multiply!” by again changing the question to “dividing by 1.” What we have to remember is that a fraction is just another way of saying the numerator divided by the denominator; we even use / to represent this when typing on a keyboard as I am right now. Therefore 1/3 divided by 5/12 is the same as 1/3 / 5/12, where 1/3 is the numerator and 5/12 is the denominator. This concept of fractions as a part of a fraction is a very important concept to understand, in my opinion, as it provides a very strong base from which to springboard into fractional arithmetic and understanding. So, now that we have our fraction built up of fractions, how do we simplify it? Well, we want the denominator to equal 1 and the tools at our disposal are multiplication and division, as addition and subtraction will change the number (or involve adding or subtracting 0, which is not useful), unlike multiplying or dividing by 1! So, breaking it down… What do we multiply 5/12 by to get 1? Well, if we multiply it by 12 we will get 5, which is not what we want. So how do we get from 5 to 1 using multiplication or division? Well, we divide by 5 as 5/5 = 1. So, we’ve got there! How did we get there? We multiplied by 12 and divided by 5. So we multiplied by 12 divided by 5, or 12/5. However, we need to remember the original question and the fact that we’re using the idea of multiplying by 1. We’re multiplying (or dividing) the original fraction built of fractions by 1 as to not change what the original number is, so whatever we do to the bottom (denominator) we have to do to the top (numerator). So we’ve multiplied the bottom (5/12) by 12/5 to get 1, so we must multiply the top (1/3) by 12/5 as well. This gives us (1*12 / 3*5) / 1 = 12/15 / 1 = 4/5 / 1.

4/5 divided by 1 is simply 4/5. Acquired because of 2 simple concepts. x*1 = x and equivalence (2/2 = 5/5 = 1043/1043 = 1)

Now let us look at what happens if you simply turn the right hand fraction upside down. 1/3 divided by 5/12 becomes 1/3 * 12/5, the exact calculation we were doing for our numerator in the previous paragraph. 1/3 * 5/12 = 12/15 = 4/5. We reach the same answer, but skip the excessive number manipulation.

Also, to highlight my point on fractions being easier to use than decimal numbers: try doing 1/3 / 5/12 as decimal numbers without a calculator. So that’s… 1.333333333… recurring 3s, divided by 0.4166666666666666666… recurring 6s. I’ll wait over here and continue to sing and dance along to Bad Romance by Lady Gaga. Take your time. You’ll need it.

You remember when I said that I’d wait for you to calculate it?

I lied.

(And I really wanted to make an Arnold Schwarzenegger reference).

There isn’t actually much more I wanted to write about in this article. I wanted to touch base on equivalence and show that it’s largely taken for granted. I also wanted to show the power of multiplication and division by 1 using equivalence to change what it looks like you are doing. In algebra we write x + 3y = 8z, not 1x + 3y = 8z, because x is the same thing as 1*x. Also, to finish on an interesting observation… Take 36 divided by 9. We know this is 4, yes? Now, multiply 36 and 9 by 218 and divide them again. The answer is still 4.

I wonder if that’s been named yet. I bet it has. My quest to have a mathematical concept named after me is, as usual, still not being completed.

Weekend Mathematics Article and Personal Updates!

It’s that time of the week again. I will be writing on calculations and mental mathematics today, but first I need to get some personal stuff off my chest. If you want to skip me crying all over my keyboard, click here to go to the maths.

So, rounding off the week is a bout of delicious paranoia. Wonderful, I know, but we’re old friends; we’ve known each other for about 10 years now.

And you know what? I wish we didn’t. I wish paranoia would stop calling me up, contacting me on Skype, walking up to me in the street as if we’re best friends or peering over my shoulder whilst I work. It’s annoying. More than annoying. This time, paranoia is here to let me know two things:

1 – I’m utterly useless. So, apparently the January crowd on my course (it’s split so that the first wave join in January, then the second wave are added in March and finally the third wave are added in June) is for people who have not done an A-level in Mathematics. It feels like Paranoia (who we will capitalise now like a proper noun as he has, by now, taken on a human-like form. And yes, it’s a he, because I’m also obviously a man-hating feminist who believes that anyone bearing a Y chromosome should be enslaved. Obviously) is holding this sentence up to my face and laughing hysterically. Am I really so crap? As someone who did both an A-level in Mathematics -and- an A-level in Further Mathematics, this is really weighing on my thoughts. Heck, I more than just did an A-level in Mathematics; I managed to get an A grade whilst also managing crippling depression, insecurities and identity issues. It was a metaphorical cake-walk that only took up a relatively small percentage of my brain power to do, though Further Mathematics required more than I had available and I would be interested to see what I would get now that I am (mostly) over these hurdles. So why the hell am I in the “you haven’t done an A-level in Maths” group? Why am I not in the March group of people who have done an A-level in Maths? Am I so useless that the University feels that I need to be put in the group where I am, quite clearly, above the mathematical par? This isn’t just me being big headed, this is what I’ve observed. But am I? I feel like I’m trying to prove myself to myself in class, commenting on things that don’t necessarily need commenting on and asking questions that I already know the answer to in order to look intelligent. Like I’ve considered more than just what we’ve been shown.

2 – That leads me on to the other piece of paper that Paranoia is holding up to my face (he is a total bastard) which reads “everybody hates you. Even your instructor hates you. Because you’re an annoying little self-important ‘princess’.” I get the feeling that, because of my behavioural patterns derived from point number 1, my fellow trainee teachers hate me, as well as our instructor. I get the feeling that they think I’m aloof or self-important or that I am trying to prove that I am better than them (which is totally not me. I would never go out of my way to put other people down, having been bullied myself for the entirety of my days in primary school, and for two years afterwards as well). I’m really scared that my above feelings are making me do things that I would not normally do and that these are the first impressions that I’m making on what will be my peer group for the next 21 months. I feel as though I am going to alienate myself through no fault of my own, but because Paranoia is repeatedly punching me in the stomach and reminding me that I’m useless and everyone hates me.

This has me in a slight panic, so I’m going to go radio silent for the next week in class. I will speak when spoken to and no more. I can complete this work. I can complete this work with ease. I am learning from my time here and, as it is funded by the government, it is affordable (I get nothing from parents and couldn’t do a full time job alongside what is a pesudo-full time job) and preferable to spending another 3 months in Spain, going insane in a different way. It’s just… I have these thoughts in my brain that I really need to get rid of. Emotions: such a love hate relationship I have with them.

Anyway, that’s enough of that. Time to do some mathematics.

Picture courtesy of Wikipedia. The 4 basic numerical operations!

Picture courtesy of Wikipedia. The 4 basic numerical operations!

This is a maths article first and foremost, so let’s get down to the nitty gritty of calculations. I want you to ask yourself how many times every day, on average, you do mathematics. Got a number in your head? Good. Well, that was once at least, so we’re making progress. Here’s the part where I tell you the truth: it’s likely a larger number than what you just thought of.

The four basic mathematical operations are addition, subtraction, multiplication and division. I’m hoping that everyone reading this knows the basic premise of what these do. This isn’t calculus, this is simple mathematics. The first things that we learn and the cornerstone of everything to come. Basic arithmetic is incredibly important, and I can’t stress that enough. Arithmetic is used when we go shopping (fun!), when we order lunch, when we tell the time, when we cook, when we withdraw money from the cash machine, when we pay rent, when we gamble (well, those who do it), when we play any board game involving dice. I think you get the point I am trying to put across. We use arithmetic in our day to day lives and don’t even consider the fact that every time we do so we are doing maths. We take for granted how easily we can add two numbers, or multiply basic values (or at the very least estimate, another important mathematical skill), but over the past week I’ve had to look in detail into how we do it, and thus how to teach it to primary school children who have not learned how to do so yet. We start with counting. Addition is simply counting. Subtraction is reverse counting. Multiplication is repeated addition and division is repeated subtraction. It’s all linked.

I know that mathematics is not for everyone. Some people find numbers difficult, but may weave the most wonderful poems or create spectacular sculptures and paintings, so I’ve come up with 3 basic strategies to improve basic numerical skills that I have used through my life, considering that this is the cornerstone of mathematics, the foundations of the mathematical manor (and we all know what happens when you build a manor on shoddy foundations. It involves lots of costs and crumbling). You don’t even have to pay for this, it’s like witchcraft!

1 – Number games. Looking at the clock. It’s 23:05. What can I do with 2, 3, 0 and 5? Well, I can make 10 (2 + 3 + 5 + 0); I can make 0 (2 * 3 * 5 * 0 or 0 / (2 * 3 + 5) or 2 + 3 – 5 – 0). You can also introduce more complex operators into this number games if you’re bored of the basic 4, such as factorial or indices. I’ve done this myself since I could tell the time (which I believe I learned at about the age of 4(?)) and it really helped me grasp and manipulate numbers, even from a young age.

2 – Find a hobby that uses lots of arithmetic. For me, this is Warhammer 40k. Those of you who know me will know that I tend to theorycraft a lot. What happens if a unit of x fires at a unit of y? How many Guardsmen can I swamp the board with in 1500pts whilst maintaining tactical viability? What’s the average MEQ kill rate of Warp Spiders? The best part about this one is that it can be enjoyable. I love crunching the numbers to work out how streamlined I can make lists and how much I can expect a unit to do on the tabletop; it’s an extension of a hobby that I have loved for years. This has really helped me work out more complex multiplications (17 times table? That’s a unit of Striking Scorpions! 19? That’s Warp Spiders) and even juggle fractions in my head (1 boltgun shot from a Battle Sister will kill 1/9 Space Marines (2/3 to hit * 1/2 to wound * 1/3 to get through the armour save), so you need 9 to kill 1. This is probably the one that has helped me the most, because where the first was an idle action that I did when I looked at the clock, this one was an activity that I sought out in my own time. A suggestion for if you want a hobby with slightly more advanced mathematics would be AD&D dungeon creation. GCSE level circle theorems used to be a necessity for DMs!

3 – Practice. The most boring of the three. There are plenty of resources out there (don’t feel like an idiot for doing Primary level activities. I’m doing them as part of my teacher training and I don’t feel like an idiot for doing so!) to help with number skills. Use them. However, headbutting a wall until it falls over will leave a headache. The two above methods are akin to bringing a climbing rope and a rocket launcher to the wall first.

I’m just throwing ideas out there for this. Basic arithmetic is everywhere in our lives and is, in my opinion, an essential skill; an undervalued skill. As always, you can choose to try it or not. I was notoriously lazy as a child and lacked motivation for anything academic, so this was merely my way of getting around the fact that I never wanted to do anything. A shorter mathematical article today, but I am not madly in love with the idea of basic arithmetic. I may be a sub-par mathematician, but I’m slightly above basic arithmetic. Mainly because I’m a Warhammer 40k and tabletop RPG player. Got to have the best stats and everything has to be perfect!

New Developments and the Return to Studying

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.

Image courtesy of Amazon. I just typed "abacus" into Google, okay?

Image courtesy of Amazon. I just typed “abacus” into Google, okay?

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!