It is often said that if humans didn't have thumbs we'd still be living in trees. We should add, if humans didn't have ring fingers (leaving eight) we might have colonized Mars by now. Why? Many common math operations become much more obvious when you have a 'natural' binary base, such as 2, 4, 8, 16... The most interesting and the most useless design problem in the world today may well be finding a better way of writing numbers. We already have numbers. I realize that. I did say useless.
Maybe we already have numbers -- but with the advent of computers the earth has shifted a bit in the math and counting world. The base for computers isn't the unnatural 10. 'Simplify' no longer means 'reduce and obfuscate' -- that was just to make the hand calculations easier. Simplify now means 'expand and clarify'. With a good binary based numbering system, the clarity and transparency of numeric operations can be inherent in the numbers themselves. Think multiplying Roman numerals vs multiplying decimal numbers - it doesn't have to end there.
I've fooled around with this idea for a few years on and off, and now have a few weak designs and a mostly incomplete and contradictory 'goals' spec. Fortuitously, your threshold for entertainment is clearly low today, so please read on. In a perfect world this would spur you to submit a better design, then we could have grand sweeping conversations about it -- hands waving dramatically, brows furrowed jaws clenched, stabbing the air with our stinky pipes! Oh sweet thou passioned retort!!
Our brains have a great capacity to map symbols to meaning. Writing is nothing more that a series of agreed upon shapes that have mutually understood meanings. In fact language itself is pretty much the same thing, using sounds instead of shapes. Our ability to map things is so good that we are often happy to settle for broken representations of our ideas and use context to patch things up. Our daily sounds and shapes have almost always been adapted from something else, prodded just enough to be workable, and then left alone - until the next crisis of understanding at least. Look no further than English for examples of stretched mappings - most roots hardly make sense anymore, combinations of adopted words can twist the tounge, the spelling can be random or incorrect. There is also plenty of listener confusion amongst even common words like "fifty" and "fifteen", or "can" and "can't" (which we try to sidestep by using "FIF-ty" and "fif-TEEN", or "kin" and "kan't"). Eventually the pressure builds enough that we change the way we speak and spell (we don't say "fiveteen" after all) - but we tend to stop once things are 'barely workable' again.
A few tricky words are not the point here though. More interesting is how far we will go, how many flaws we can tolerate, and how rarely we even notice a terribly flawed mapping. Take the Roman alphabet as an example. Certainly inspiration for it came from elsewhere, but it was created to map directly to Latin, one symbol per sound. And it still fits pretty well onto latin languages (as anyone who's tried to learn Spanish can tell you -- a native English speaker can probably spell better in Spanish than English after a few days of study). However, look at how it gets used in other languages. In Europe you see all kinds of accents, tails and dots, trying to map the Roman glyphs to sounds its creators had never heard. In Vietnamese, or Pinyin (Romanized Chinese), you see tone accents, and some letters change sounds ('c' becomes 'ts', 'x' becomes 'sh'). English, due to its patchwork of origins and wide dispersal has more different sounds than could ever be thought of as resonable. Thus it has pretty much given up on extra markings. Instead it goes for multiple sounds per letter ('g'eorge vs 'g'orge), and letter combinations, like 'ch', 'sh', 'oo', or the ever troublesome 'ie' vs 'ei'. Move forward a few hundred years, and many of the sounds no longer even match the spellings, giving us bizzare spellings, like 'night', 'gnome', 'knife' - or even more bizzare - 'enough', 'through', 'though'. There is indeed pressure on these spellings (from those not inclined to view flaws as culture), but change always drags behind fact. This slow and selective evolution allows us to absorb tried and true improvements, and more importantly, move beyond cultural disasters like the seventies. The take-away point is the Latin alphabet doesn't really map very well to any languages other than the Latin ones, yet it is still quite useable (even with complex and broken mappings). In fact, we usually don't even notice that our alphabet has been hacked in. The wasteful side effects of these bad mappings are just considered normal, or at least acceptable. Think about it - virtually no one is able to spell all English words properly, despite having courses in 'Spelling' right through elementry school. This fact just isn't an issue in many other languages -- but so what right?
Hell hath many levels, and English may well not be on the bottom one. The first prize for mapping round pegs to banana cream pies might go to Kanji - the Chinese characters in Japanese. In Chinese, each character is spoken with one sound. They don't always have a single meaning, and certainly a combination of two characters can mean something new (like 'electric words' meaning telephone), but, key idea, *a character is always consistant in the way it is pronounced*. Each dialect or language in China may have a different way to pronouce the character 'big', but it will be pronounced the same way within the dialect where ever it appears - either alone or in compound words. This means that if a person sees an unfamiliar word, made up of two familiar characters, they will know how to pronounce it. It also means that people from different languages can communicate via writing, because they both will know that the characters for 'electric words' will mean telephone, regardless of how they say it. The cost of this is that the characters can have no information in them that tell you how to pronounce them - it is a pure mapping system much like our numbers are today (123 can be one two three, or bir icki uch, there is nothing about the digit '1' that suggests it starts with a 'wa' sound). And yeah, unlike digits, Chinese has lots of characters (though not one per word) - something like 10,000 of them depending who you ask.
Knowing this, it should be possible to map Chinese characters to English words, and write English in 'Chinese'. There are problems though, because English doesn't map to Mandarin nearly as well as say, Cantonese does. We say 'tele' and 'phone' (far and sound), not 'electric' and 'words'. We say 'computer' (something that does computations) rather than 'electric thought', and tele-vision (far see) instead of 'electric watch'. We would have a few options. We could give up on direct sound mapping and pronounce these character combinations as we normally do (telephone etc). We could change the language itself and use the same meanings (start saying 'electric words' instead of telephone). We could also adopt the Chinese words (say 'dian hwa' instead of telephone). This is the problem Japan faced when importing Chinese characters, and ended up doing all of the above, and more. Each Kanji character in Japan can be pronounced in a variety of ways - a japanese word, a synonym, a Chinese loan word... There are also two phonetic alphabets in use (phonetic meaning one glyph per syllable, not per sound), one for 'spelled' Japanese words and one for import words. Finally there is the adapted Roman alphabet. Wow. I noticed one curious side affect of this when we moved into a new neighborhood. My wife (who is Japanese, and very literate) wasn't sure how to pronounce parts of our address until asking a neighbor. Then phoning her mother, she spent a long time telling her how it was written, having to describe how some characters looked. Less funny was just after the Kobe earthquake, where the annoucers (in Tokyo) were unable to relay emergency messages for different neighborhoods, as they couldn't pronounce many of them. Yet they muddle through just like we do in English. They look at kids still learning writing in the seventh grade much like we view the same group learning spelling (the difference being they master it eventually ;). Humans have a near infinite capacity to tolerate near random mapping systems - in fact we seem to take a sick pleasure from it.
One can argue that having an imperfect mapping in a language, writing system, or even number system, is a great mental excersie. The flip argument is that a crappy tool makes for a crappy job. Who knows, though I suspect a perfectly mapped writing system is less important than it might seem. Writing represents language, which unlike numbers, doesn't really have such a deep underlying truth and logic that needs to be revealed. A more perfect writing system has little bearing on the understanding, because understanding comes mostly from context, emotion, and all those hundreds of tiny little judgements we make while listening or reading. It is only at its most mechanical that the writing of a language has an impact - how do you pronounce this word, how do you spell receive, why isn't sugar 'sh'... And who gives a shit about that part really. What we can't deny is that English people expend a lot of energy learning to spell, and that Japanese (and Chinese) people expend a lot learning to read. Time that could either be spent winning the nobel prize for cleverness, or more likely sitting infront of the TV eating chips.
Hopefully at this point you are convinced we can map things very well, and our mapping systems aren't always the most sensible in spite of working. If not, you should learn to speak !Kung and write Chinese before continuing.
The numbers we write are also a sort of written language, but there is one big difference - they represent something that doesn't bubble up from within us - they represent unemotional, rigid, quantities. Another very important point is, unlike language, quantities relate to each other in very, well, mathematical ways. So while it isn't so useful to embed pronounciation or contextual information into written numbers, there are many mathematical properties inherent in a number. A very simple example would be the size of the number could be reflected in the size of the glyph. Most people will just think you are being an arse if you say 0 is bigger than 1, but when you look at it, clearly it is - same height but wider. If you made a numbering system out of smaller to larger boxes, no one would wonder about the order - even the first time they saw it. So clearly numbers have meanings and relations that are the same in all situations, across cultures and aren't about to evolve any time soon. And clearly our ten digits capture none of these - patterns only emerge from groups of digits, and not even very naturally then. Math is clearly easier and more obvious with our current numbering system than it was with Roman numerals or Sumerian glyphs. This has opened the door to major and rapid advancements - culminating in no less than three faked moon landings. The tool is much more important with math than with language.
So what is a numbering system made up of? If numbers are quantities, numbering systems are ways of sequencing these quantities. To represent this, there are essentially three parts. First, and most important, is a way of counting. This includes how you increment, group and scale things. Then there is the 'pictoral' representation - usually 0123456789 - but not always (it is different in Arabic for example, which is strange because in English they are called Arabic numbers). Lastly, there is the way you say them - which changes the most across cultures (and matters the least).
Ten is not a very natural number for anything but humping starfish. Sure it was a better choice (for the base of our numbering system) than seven or thirteen, but things still don't fit into each other nearly as well as they could. Half of ten is five, and half of five is a wounded sea creature. This obscures and abstracts many simple mathematical truths. We learn addition by memorizing seemingly arbitrary patterns and facts, multiplication by rote, and for trig we just write the answers on our shoe. The shape of the digits, and the numbering system itself hide some of the simple additive and algebraic patterns numbers naturally have.
It wasn't always base ten. Most of our history has been one-two-many. Then came simple tallies (stroke, stroke, stroke, stroke, crossout). The first known numbering system, created by those fabulous Sumerians (of wheel fame), was base 60 (or base 'six groups of ten' to be more accurate). Now if you think of clocks and circles a bit, you are lead to the inescapbable conclusion that old habbits die hard. One nice thing about base 60 though, is it can be quartered. Yeah, that is useful. Every chance we get to quarter something we do - quarter to six, a buck and a quarter, drawn and quartered... Notice that you can't quarter a nickel, dime, or even a quarter. That is why they are made out of metal. The things you learn. 60 can be more than quartered of course - it is the first number that can be divided by 1, 2, 3, 4, 5 and 6. It is like magic if you do a lot of those little divisions, which is why it fits circle math and clocks so well. It is also good to know that we can exist comfortably with more than one numeric base in our lives - it isn't like marriage or anything.
Base twelve was also fairly popular (and still is), though actually more as just a measure than a base. Twelve is an interesting number, as it is divisible by one, two, three and four making it quite handy with things that need to be sectioned (without being hard to count, like 60 pebbles are). Easily factored numbers like these make low number math easier too, kind of like how we find division by five or ten to be easier than division by 8. This isn't just abstract math though - if a dozen beer were suddenly to be sold in the harder to divy up 'elevens' or 'thirteens', you would see a violent spike in alcohol related homocide. Clearly this is important.
Base twenty was used by the Mayans, although that (very advanced btw) numbering system seems to have faded soon after the Spanish introduced shoes (or was it smallpox?). A few early male dominated cultures even used base eleven, however this made it difficult to count past 10 without losing one's concentration. By far the most popular was, and still is, base ten. This is favoured because it is the square of two plus the square of two plus two. There are also exactly ten months if you don't count the last two, so it was pretty much a shoe in.
Our concept of a numbering system is based on a line, we think of amounts stretching to the left and right of zero along this line, with numbers at regular intervals. While this isn't the only way to think of numbers by any means (for example months are not regular), steady intervals work best for most math. If we were to birth a numbering system from this numberline concept - history aside - what would we do? First thing we would need is a unit. Now it doesn't really matter how long this unit is, because we care only about the numbers and their relations - we always change the units to suit the task at hand (2 dollars vs 2 hours). Next we add units to the right and left until we run out of patience. We see very quickly that a pictographic system (one symbol for each number) isn't going to work, or at least isn't going to get finished any time soon. We have to start grouping things somehow. The first big question is, what is the most mathematically useful way to cut this up into groups? If you said by tens, what is it about ten that is mathmatically better than the other choices? Does it matter? Not if you are just counting, but if you are transforming numbers (doing calculations) then you should prefer a system that made this easier. This leads us to look for numbers that are more symetrical, more divisible, and generally 'rounder'.
If you follow the numberline from one, rightwards, the first noteable event has to be 'two'. No matter how far right you go, this has to be the most significant event you'll find after one ("My God, there's more of these!"). It is the next whole number, it is the result of adding the unit to itself, and the result of doubling the unit. Fireworks. This is binary, and indeed base 2 is at the heart of virtually all mechanical calculation. The downside, is it is hard for people to read or talk about binary, because the numbers get so long and blurry. So tip the hat, but scratch two unless you have no soul.
The next significant event is either three or four. Three is what happens if you are counting, but we've ruled out counting as a method of grouping already. Four is what happens if you double two, so it is a continuation of the doubling idea that two had. It is very even, 2X2, 1X4, and it quarters nicely. The problem is that four digits still make pretty long numbers, so it again quickly leads to counting long sequences of blurry digits. That would make them slower to say as well.
The next event is five, good for nothing - though at least it is still pretty easy to visualize five objects at a glance. Then six, which is pretty round, divisible by 2 and 3 but not the fairly natural 4. Seven is perfect for beauty and would make an interesting 'anti-symetric' numbering system (it is probably the most 'odd' number), but it divides like mushy cereal. Eight is very even. Like four and two, it follows the doubling pattern - so it divides by two evenly forever: 4, 2, 1, 1/2, 1/4... Like four, it is a very significant measure in music, and for good reasons. It is awful for divisions by three, but to be fair, divisions by three probably aren't as mathematically 'pure' as by two or four - don't ask me why. Ok, you asked. Think cutting a cake (or even a round pie) into three sections, rather than two or four - why is that harder? Three isn't geometrically even, as it breaks up straight lines. As we experience it, things like gravity, direction, inertia, and even lots of abstract math ideas are measured in terms of a conceptual straight line. Even with clocks, which can easily be sectioned into thirds, people stick with quarters and halfs for rounding. Looking the other way, three is awful at accommodating any even numbers, and most odd ones. Don't agree? Well fair enough, some people are born to waltz, but I'm going to forget three anyway. So forget nine too, because it's big strength is it is a perfect three (three sets of three).
That brings us to ten - two fives and not much else. Seven is the only number up until now with less redeming properties for a base - which at least makes it a perfect opposite. Of course it seems very round to us, but that is just the illusion created by the hundreds of top ten lists out there. Mathematically it is about as symetrical as 14. The next two significantly round numbers are 12 and 16, and 16 is very very round (four groups of four - and four is very even itself as we know). It deliverance for we the blighted, who have spent our lives searching for the 'perfect four'.
Note that if we choose something other than a numberline as a base concept of numbers, the above ideas on numeric symmetry still hold. For example, we could use 2D area instead of a line, where we start with a unit square's width (or height - its a square), and count units every time the area of the square increases by one unit. 16 units would still be thought of as 'even' or 'natural' here, and seven units still wouldn't. The same system could be applied to cubes (the units based on volume), or 4D for that matter. Whether the increments are based on unit circles, spheres, spirals, or bunches of bananas - the 'eveness' of 16 comes from how amounts are related to each other, not from anything inherent in the numbering system.
Base ten is acceptable, much like the Roman alphabet in English, as long as only humans are involved (which why it has been accepted for so long). Of course children find it annoying and clunky to learn and use, but what do they know. Computers however, can not be made to shutup and study, yet they are disturbingly good at math. They use binary (base 2 - digits 0 and 1). This should tell us something. To humans binary usually seems like just a glorified tallying system without the crossouts. 1111001001000110111110. Etc. So we prefer to read it in slightly bigger chunks. Base 4 units are still a little small making for long numbers (33021012332). Base 8 (octal) was fairly popular for a while (17110676), but base 16 seems to be what we've settled on. It is actually more 'even' than base 8, because it is four to the power of four, rather than four to the power of three. Not surprisingly, base 16 is very common when working with computers. Once a person is used to it, this base becomes a much more natural way to group numbers. The problems mostly come from the unhelpful way we write it, which fuels our need to translate everything back into base ten rather than just think in base 16 (and translating to base 10 is an especially unnatural operation - ick!).
The next thing to do after choosing a base (the answer is 16 in case you missed it - base 16 is king - News at 0x0B) is to figure out a visual way of representing it. The simplest systems just use counters - 37 is represented by 37 pebbles. Of course inflation can really break your back here, but if your counting system doesn't go past twenty it is probably fine. Next up is a different symbol for each new set of ticks. This is like having a second person keep track of how many times you've counted to ten on their fingers. And a third person to count the times the second person counted to ten (hunderds). So 354 becomes ^^^*****&&&&, or something. This system makes it easy to add (order of the symbols is irrelevant so you just combine them - ^^*&&*^***&& would also be 354), but tricky to multiply or do anything else really. It is also missing the obvious, obvious, how could this have taken 100,000 years people, ZERO. Once this was discovered, number systems based on place value became common. As far as I can find, zero based systems have all (historically) been base ten. These postional systems make numbers easy to multiply, to have decimals, and in general do good math (easy to transform). Counting is no harder, and counting backwards can be much easier, especially if you start at one. So until something better comes along, a positional system is probably the way to go.
To state the obvious, a postitional system is represented by glyphs. For base ten we use 0-9, or the equvilent in your culture. If you use base 10, you then need ten glyphs, for binary (base 2) you need two glyphs, for base 7 you need seven glyphs, for base 13, thirteen glyphs... If you look at enough bases, and check the required digits, you start to see a pattern emerge. So for a base 16 positional system, we will need 16 glyphs.
Lets stop calling this base 16. Our wonderful base sixteen system is actually has a name already - it's called 'hexadecimal'. That is a linguistic slam dunk - it means one group of ten and six left over -- described in base 10! No self respecting numbering system would ever willingly describe itself with another system's base. It's like Canadians calling themselves 'Our neighbors to the north', except this kind of thing bothers a number. It gets worse. The digits we use for hexadecimal are, drum roll please, 0123456789ABCDEF. Yeah, there are the ten digits from another base, and then hey we'll just chuck in a few letters until we get enough, up to the letter 'F'. And we all know what the letter 'F' stands for. That is as confusing as it is fouled up. Even 01234567-ABCDEFGH would have been better - at least it gives us a clue that eight is significant. Sure, as a species we could have done better, but you just weren't worth it Madame Hexadecimal. I propose we at least call these numbers 'BIOCTAL' until we can think of a better name. I will only refer to hexadecimal numbers as 'bioctal' from here on, except in the places that I forget. And that last sentance doesn't count.
The nice thing about having a totally awful name and a totally awful set of glyphs for a totally wonderful numbering system, is there will be less resistance to chucking it wholesale and starting over. Ha ha ha ha, I had to say that somewhere. Clearly that would be an uphill battle, however it is important to note that this isn't really about a struggle to throw out decimal numbers - that has already happened where it needed to happen. It is really easy to cast off thousands of years of history when the old tool doesn't do the job, and a new tool works so well (as any cordless drill owner can attest). So the trick here is to get a glyph set that is so much better than the current one that the transition to using it becomes a no-brainer. Ok, back to being depressed.
Before tackling the glyphs, the last thing to consider is names for the digits and numbers. It wouldn't really make sense to use 'one, two, three', because the bases don't match after nine. The last thing we want to do is convert to base ten every time we mention a number. Ideally the sounds would be universally the same - then we could get ripped off in every street market in the world without being shown a calculator. The sounds should map to the same ideas that the glyphs map to, and should be made of universially common sounds as far as possible. Saying the sounds of long numbers shouldn't trip you up. Oh, and 4233 shouldn't sound like "butthole sniff sniff" in any language - as far as that is possible at least. More on that later, for now lets just keep in mind that these glyphs and glyph combinations will need sounds associated with them. Ultimately there would need to even be braille versions, Unicode indexes, etc. Oh, and T-Shirts. Lots of cool T-Shirts for those of us in the know.
This section is mostly geared at younger students, because 1) they are the future, 2) their brains run faster, and 3) they are forced to do this kind of stuff, we aren't. Man is it nice to be an adult.
Properties of Numbers
To best understand some of the special properties of numbers, it is easier if we go right back to the very basics. We all know how to add, and hopefully multiply, but do we really 'know' how, or do we just know the pattern? We'll first try to jar those painful memories of your third grade teacher trying to explain to you why those patterns worked (remember? That was just before she said, "well nevermind then, just do it like this...").
Would it be easier to do math if numbers were written vertically? For some reason we have a much stronger sense of up and down than we do of left and right (or is it right and left..?). This exercise will help us examine a long solved problem with fresh eyes. 438 + 497 could look something like this :
4 4| 1 8 | 9 3 9| 1 2 | 3 +8 7| 5 | 5
The only difference is higher place values are on higher lines, rather than 'further left'. You can now easily see where 8 and 7 is 15, and where 3 and 9 is 12 (vertically). You don't really 'carry' the one here, it just naturally is on the tens line when you write 15. It is probably easier to see that the coorelation between higher numbers and increased place value. So why don't we write numbers like this? Well because we aren't Mongolian - however if you want to see a really cool writing system, look at theirs!
So what are some 'addition truths' here? 'Carry the 1' is actually just shorthand for writing 15. It clearly doesn't make a difference if we put the carried one back on the 3 9 side, or in the interm result, but putting it on the left side saves us doing the second set of additions (like we do above). In fact it doesn't make a difference if we switch the order of anything that is added, because 1+2 equals 2+1 (that one is important!). The zeros under the numbers are implied, but filling them in would make the place value even clearer. You don't have to carry when the total is less than 10. To solve the middle row, you must first solve the bottom row. Flipping the whole thing vertically would make it untrue, but only if there were carries. Rotating the 9's to make 6's would always be wrong (but that doesn't really seem like an addition 'truth' does it?).
For each of these obvious things, there is a little mathematical fact at work. They may be so trivial as to not even seem worth noting, but when going to a different numbering system they are about the only thing familiar we will have to cling to. Math is built on the little truths of course, not magic templates that you fill numbers into, so these 'obvious' little things should always be the focus. Improved numbering systems always make these things clearer, much like beer doesn't.
Binary numbers are exactly like our numbers, just they run out of digits sooner. We start at 0 and count until we are 'maxed out' -- at 9. Adding 1 more causes us to bundle everything up into a single package and move it into the column on the left. The column to the left always counts the number of times the column to the right was full. So 325 means the middle column was full 3 times (the middle column moves one 'marker' to the left column when it is full), the rightmost column was full 2 times, and there are 5 extra.
With binary numbers there can only be 0 and 1 - off or on, empty or maxed out. Each slot acts like either your bank account or your credit card. To picture how binary counting works, first imagine a kindergarten class sitting on little kindergarten chairs, in a nice row. The teacher continously hands the child on the right, Mary, a small peice of plasticine. Each child is told that if they get a peice of plasticine they should keep it - unless they are already holding a peice. In that case, smack the two peices together, and pass it along to their right. The more mathematically inclined of you probably already have the horrifing image of little Johnny, seat 26, being crushed flat by a 20,000 pound chunk of oil byproduct (in 2168 AD that is). For the rest of us, we can see that this will generate binary counting. The order is 0001, 0010, 0011, 0100, 0101, 0110, 0111... The children holding something are ones, the children without are zeros. Not only that, but the first child always holds 1 peice when full, the second 2 smacked together peices, the third 4 peices, the fourth 8 peices, the fifth 16 peices... If all the peices of the binary number 1011 were put together they would be the weight of that number (1011 is eleven) multiplied by the weight single peice. This last bit is obvious, eleven peices have the weigth of eleven peices, nothing was created or destroyed, thus the world continues to exist. It was explained poorly, but it reaffirms that the counting system is actually counting. Errm. So if there were 10 students and they all were 'full', what number would that be (binary 1111111111)? How about 20 students? Tricky, but never too tricky to try.
Ok, let's say those kiddies in seats 10 to 20 got very bored (we can assume 1 to 10 would be loving every moment of this, and no one had to pee). So the teacher made two rows of 'counting' students. They went at different speeds, and eventually arrived at two different numbers. If you matched them chair for chair, and wanted to add the numbers together by having one side hand their plasticine to the other, what new rules would you have to add? Would they have to pass what they had across in a paticular order? Would this still work with three rows of kids? If both rows had ten kids, all holding plasticine, how many extra chairs would you need to hold the result of the merge? If everyone got up, and moved one chair to their right, what would that do to the value? How does this affect the 'weight' measure of all the plasticine? How would you subtract?
Seeing as kindergarten students are way too busy doing serious work, we'll try to represent these things using animation. Do try to think those questions through first though, it really really helps. No one is sure what it helps, but it does help.
Depending on the numbers, adding can be simple or very simple. If there are no overlapping ones, then adding is just a matter of merging the two numbers. For example 10001 + 01110 = 11111 (17+14=31).
If there is overlap, the very same rules apply - no slot can hold two numbers, so smack them together and pass the value up (carry one).
One consideration is when both numbers have full slots and they are passed a value from a carry as well -- this leaves three in one slot. Here we have to clarify that 'smack together' always means 'smack two together' rather than 'smack them all together'. In fact, uber technically it means 'smack the amount of the base together' -- in decimal numbers we always smack together 10 before carrying, not two. Anyway in this case, binary, when there are three peices, a set of two will carry and one will remain. This is like having 5+5+5 in decimal - the two fives cause a carry and the last five remains (because it still fits). As always, this is easier to watch happen than explain -- easier for you that is, its a terrible amount of work at this end. Man it must be nice to be a student.
Adding Many Numbers
As you can guess, adding a bunch of numbers isn't that different than adding two. There are many approaches you can take, but it still boils down to the rule of passing on all the sets of two (or whatever the base is) to the next level. You can add two numbers at a time, adding the running total to the next number. Like before this will never leave more than three to deal with at once (a pair of ones and a carry). Alternately, you could add them all at once, leaving a (potentially) large result in each column, and then just squirt off pairs until you have fewer than two left. Of course there are many other options here. Again, lets visualize this with those bobbin' blobs.
(rest of this part in progress, sorry)
<columns vs two by two>
... maybe now start boxed 16's?
Oh yes, lots of problems. Sketchbooks full in fact.
(oh so rough, but the logic of it - maybe last letter optional per language)
nili noti nevi nayi bilo boto bevo bayo dile dote deve daye zila zota zeva zaya
and the for groupings (our thousands, millions etc):
na, ba, ta, za, ma, ka, da
or maybe only one syllable needed per number, and an 'end of number' marker sound:
ni no ne na bi bo be ba di do de da zi zo ze za
1) Quick and dirty visualizations of different sequences -- inc by constant value, primes, fibonacci, mersenne, mersenne primes, amicable, digital roots, mult, add, pi (you can substitute your own glyphs here)
2) Practice typing using four simultaneous keys (x.c.;.', and z for zero)
Download Typing Flash File (you can substitute your own glyphs in here)
(working on these using the new C#>swf compiler, may be a while ; )