So, because this blog still needs some more things that might actually get me writing in it, I’m re-establishing one of the perennial favourite segments from my previous journalling. In this feature, I will provide a brief but rigorous explanation for some of the aspects of my degree subject that have come up in conversation with my many, many, many fans, for the purpose of informing and enlightening the tragically uninformed and unenlightened mass that is my readership.

Many of you, it appears, in your boundless ignorance and lamentable misconception, still fail helplessly to understand, and in some cases actually fear, even the most approachable and easily grasped concepts: circles, binary numbers, imaginary square roots of things that don’t exist, and many other such delights and wonders of formalised logic to which you are so blind and irredeemably oblivious.

But there is still hope. You might still be saved from your stinking pit of innumeracy and ineptitude. And it is I who can save you.

Welcome to the Happy FunTime^{1} Maths Hour.

Today: Numerical systems using place-value notation and positional bases. (Don’t worry, you know a lot of this already.)

These days, when most people use numbers (disregarding some awkward foreigners with their own systems, which I find unfamiliar and intimidating, and am thus choosing to disregard), we use *place-value notation*. This is the system by which the *position* of an individual symbol in a number affects its *value*.

For instance, the digit 8 in 836 represents a value of eight *hundreds*, but in 386 it represents eight *tens* (also known among some academic circles as “eighty”). It’s still an eight, but the question of eight *whats* is resolved differently in each case. Because we’re working here in base ten, the 8 might just mean eight, or eight tens, or eight tens-of-tens, or eight tens-of-*those*, and so on, depending on where in the number the digit occurs.

Some of this probably sounds familiar to you. You may have used these numbers, or others like them, in your own life or work. Maybe you have an amusing anecdote about seven, or a moving tale about how you were personally affected by nineteen point four. If so, I’d love to hear about it.

It so happens that we have ten digits in common usage in our numeral system. They’re the ones just above the letters on your keyboard, the ones that don’t spell words in any language other than “obnoxious internet teenager”. In case this needs further elaboration, they look like this: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Nifty.

But why are there exactly that many of them? Who’s to say we wouldn’t be better off with another digit or two stuck on the end there, aside from the practical difficulty of my keyboard not being able to type them? Or alternately, why do we need so many? When was the last time that poncey upside-down 6 on the end there did us any good? I’ve always said he was no use to anyone. 7 should hurry up and eat the bastard^{2}, if you ask me.

There’s absolutely no reason to have ten digits rather than any other number. Well, there are reasons why we do, of course, but none related to mathematical coherency. We use ten digits essentially because we have ten fingers, which were themselves often useful to keep track of things we wanted to count, back in the days before Maths came along and rendered the whole concept of fingers unnecessary and obsolete.

We also use base sixty, as part of our system of measuring time, because of a Sumerian counting method involving three knuckles, on each of four fingers (twelve knuckles total), being counted off five times on the other hand (five lots of twelve knuckles = sixty. True story. Those Sumerians are crazy). And it’s not that hard to imagine a system with fewer, or more, digits involved.

Imagine seeing inside a mileometer on a car, reading 000000. For each digit, there’s a little loop of numbers, laid out sort of like a conveyor belt, looking like 0 – 1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 0 , joining up with itself at either end. The far-right position on the display is constantly rotating very slowly as you drive, and moves round one digit for every mile you travel. So, the first 9 miles are counted out easily by just that digit on its own, making almost a complete rotation around its loop, and giving us a final reading of 000009.

Obviously if this digit keeps moving, it’s going to go back to 0 next, so as it does this, the second place along, in the tens column, also moves around to show the next digit, so the display now reads 000010. The right-most digit continues moving on steadily, and once it gets back to 9, the process repeats, and after twenty miles the second digit ticks over again to show 000020. Once the total mileage reaches 000099, the right-most completed revolution causes the second positon to also complete a revolution back to 0, so now the *third* digit makes its first move around, and we’ve gone 000100 miles.

You may be following this all quite easily and making impatient gestures with your hands to encourage me to get to the point, if you’re familiar with this material – say, if you’ve ever seen a car, or can already count up to one hundred.

But now, let’s say that this mileometer can be easily taken apart and reassembled. Imagine that we pull each of these loops apart, snip off the parts with the digit 9, and reattach them, all with the 8 now adjacent to the 0. They’re still all completed loops, just slightly smaller. Now what’s going to happen as we go for a drive?

For the first eight miles, it’ll be business as usual, but after that, the digit on the far right doesn’t have a 9 to turn to – it’s got to loop back to 0, one mile earlier than it would’ve done before we took a screwdriver^{3} to the inside of our car. This means that the second digit has had to tick over a mile earlier too, so the read-out is showing us 000010 after we’ve done only nine miles.

Another nine miles later, we see the same thing. It goes from 000018 to 000020, as we drive from mile seventeen to mile eighteen. The second position is now a count of how many *nines* of miles we’ve driven, rather than how many tens. It carries on like this up to 000088, the biggest number that can be represented using two digits when you’ve taken away the 9’s, at which point we’ve actually only gone eighty miles (eight nines, plus eight). Then the third digit ticks over, and we’re at 000100 after doing eighty-one (nine lots of nine) miles.

The upshot is that numbers now get bigger by a factor of nine for every zero you stick on the end, instead of ten (and what we’re used to calling ten now looks like 11). And it works the same way on the other side of the ~~decimal~~ nonary^{4} point, too: 0.1 is one ninth, 0.01 is one ninth of one ninth, and so on.

So that’s pretty much how other bases work. With higher bases it’d be the same idea – insert a few extra symbols (generally just the first few letters of the alphabet, but you could choose whatever you like) after the 9 before it loops back to 0, and you’ll get a display of 000010 after you’ve gone, say, thirteen miles. *Hexadecimal*, or base sixteen, is often used in computer code, where counting goes 0-9, A-F, then 10 (representing sixteen), and so on up to FF (fifteen lots of sixteen, plus fifteen), and then 100 (sixteen lots of sixteen, or what we would normally write as 256 in base ten).

Of course, in terms of the mathematical construction, any one of these is as valid as any other. We happen to group things together in terms of tens, hundreds, thousands, millions, and so on, but it’s only *because* we’re so used to thinking in base ten that these are defined as being “round” numbers. Nine thousand, two hundred and sixty-one might be wonderfully neat and round and easy to work with, depending on how many symbols are in your numerical system.

You may, bewildered wretches that you are, be torn up inside with wrenching anxiety, even now, from the one true mystery that still remains – that of why I care whether you know or understand any of this stuff in the first place. Well, it’s primarily because I enjoy talking about it (and you may draw from this whatever perverse conclusions about my depraved and bestial predelictions that you wish). But there are actually some interesting repercussions as well.

I won’t get all post-modern on you – I fear that if I did, my head might explode, either from the same kind of concentrated bafflement that always ensued whenever my former housemates (for whose benefit these essays were first created) tried to explain post-modernism to me, or from the large and unforgiving gun I would surely feel morally obliged to swallow if I turned into the sort of person who deliberately provides post-modern explanations of subjects of interest.

That said, it can be worth appreciating just how artificial and arbitrary many of the expressions of reality, which we take entirely for granted, actually are. Under a different base system, the way we think about and describe any numbers would be totally different. You could be forty-twelve or eleventy-one years of age. Clocks might strike thirteen, as in that famous George Orwell novel, 1544.

The fraction one-third is conventionally represented as an endlessly repeating decimal, 0.333333…, but in base twelve, it is simply 0.4; the infinite becomes finite. People would speculate in their online journals about bizarre alternate number systems where nobody uses the number ¬ at all, yet somehow they still manage to count up to it. Someone would write a lengthy, rambling article in his blog, to explain how such counting systems could exist with only a scant ten digits, and nobody would really be interested.

The Ultimate Question of Life, The Universe, And Everything really could be “What do you get when you multiply six by nine?” – if we just worked in base thirteen, as many Douglas Adams fans have observed before me, the universe could actually make some fundamental sense.

Everything would be different, with – I would hazard a pitifully uninformed guess of my own – some fascinating anthropological repercussions. And maybe pi really does contain a secret message from God, answering all our questions about the meaning of life, and how to achieve eternal happiness, somewhere within its unending numeral expression – if only we’d invented another number or two.

If you’ve read this far, well done. If you’ve understood some of it, even better. If your brain now requires some recuperation, I suggest you go and make yourself a nice cup of tea, and enjoy some relaxing pornography or hard drugs to help you wind down.

Tune in to the next Happy FunTime Maths Hour for: “Infinity – it’s bigger than it looks.”

on February 17, 2008 at 6:08 pm |DuskI did not know we had the Sumerians to thank for 60 but it makes a kind of sense and thus I feel edumacated. [applauds] Thank you, sir.

on February 18, 2008 at 9:56 pm |cubiksrubeI was going to attribute it to the Babylonians, but apparently Sumeria got there first. Yay edumacation!

on February 19, 2008 at 6:17 pm |DuskSumeria FTW!

on March 4, 2008 at 3:23 pm |Is God the source of human morality? « Cubik’s Rube[…] I’ve just had a totally great idea for a future Happy FunTime Maths Hour post, but I won’t go into all the number-y stuff in any depth here. Basically, he could win […]

on August 5, 2009 at 3:35 pm |fropome“And maybe pi really does contain a secret message from God, answering all our questions about the meaning of life, and how to achieve eternal happiness, somewhere within its unending numeral expression – if only we’d invented another number or two.”

Have you read Contact?

See the last part of the plot summary here:

http://en.wikipedia.org/wiki/Contact_(novel)