tag:blogger.com,1999:blog-28818233679847057512017-07-21T14:46:17.170-07:00Beyond GA blog about the mathematics, science, and art of Large Numbers outside of Academia ...Sbiis Saibianhttp://www.blogger.com/profile/12289340153316642062noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-2881823367984705751.post-21648994131459535012012-02-02T13:16:00.000-08:002012-02-03T23:51:46.500-08:00Very Small Very Large NumbersOne googologist's quest for the smallest member of his trade<br />(Written by Sbiis Saibian)<br /><br />What do you consider to be the <span style="font-style: italic;">smallest</span> Very Large Number? Yes you read that correctly. Today we will be exploring some <i>very small</i> very large numbers for a change. If that sounds like an oxymoron, bare with me; there is a point to this odd juxtaposition. Although the term "large numbers" is thrown around a lot now days it is a vague term to say the least. What precisely IS a large number? If there is a definite answer than you should be able to point to any number and determine whether it is or is not a large number. It should also be possible to determine then what is the <i>smallest</i> large number, since clearly there are some numbers which aren't large. Look, we all know that there is no end to large numbers, but where exactly do they begin? To draw on an old metaphor, when does a heap cease to be a heap?<br /><br />If you have any familiarity with googology (the study of large numbers) you know that googologist's don't worry too much about where their subject begins, because they are more concerned with making numbers larger and larger, and since there is no end to this process they've got no time to consider such inconsequential notions like "what is the smallest large number". They've got more serious mathematics to do ... :)<br /><br />It seems whenever you ask a professional mathematician for the largest number, they are quick to respond that there is no such thing. But how would they respond to a question about the <span style="font-style: italic;">smallest</span> large number? Perhaps they'd say that's like talking about <span style="font-style: italic;">millikilometers</span>. It's just silly. Or more seriously they might argue that there is no such thing either, because numbers go infinitely in both directions on the real number line. On the surface this seems reasonable, but what do we mean when we say a number is small?<br /><br />Would you say that 0.5 is a small number? Incontestably, right? 0.1 is even smaller, and 0.01 smaller still. How about a REALLY small number: a <span style="font-style: italic;">googolminex</span> is defined by Conway & Guy to be 1 over a <span style="font-style: italic;">googolplex</span>. So if a <span style="font-style: italic;">googolminex</span> is a really really small number, what can we say about zero? It would seem that zero would be infinitely small, just as infinity is "infinitely large". In this way of thinking does it make sense to have a number "smaller" than zero. No. Intuitively speaking, zero <span style="font-style: italic;">is </span>"the smallest number". Negative numbers should not be considered along a "large-small" axis. Instead Negative Numbers should only be considered "less than zero". <br /><br />In any case it would be silly to consider Negative Numbers as "large numbers" of any kind: small, tiny or otherwise! In fact, at the outset, let's ban all negative reals from consideration to avoid this discussion slipping into absurdity.<br /><br />What about Complex Numbers? Quaternions? Octonions? Well ask yourself, how do you compare numbers like <span style="font-style: italic;">i</span> , or <span style="font-style: italic;">3+2i</span>, to other real numbers, or each other. It seems like an exercise in the ridiculous. Numbers which must be defined with more than one dimension can't really be compared in such a way that every number is unique, and there is a definite order to them. There is a way however to compare them, if we are willing to accept that different numbers will be of the same size. By taking the "absolute value" of a complex number, quaternion, octonion, etc. we can assign a non-negative real value to any such number. This is equivalent to finding the numbers distance from zero. Under this system however, numbers like 1, <span style="font-style: italic;">i</span> , -1 , <span style="font-style: italic;">-i</span>, are equal because they are all 1 unit from the origin (zero). It stands to reason however, that if we are going to reduce complex numbers to non-negative reals anyway, why not just stick with comparing non-negative reals and skip all the unnecessary work. Besides, a "large complex number", in the end is just a complex number for which one or both of it's parts (real and imaginary) is large. There are no large complex numbers, without first having large reals, so it kind of just ends up being a very round about way to define a "large number". <i>The only way I can think of generating large complex numbers without first constructing large reals would be to use a recursive formula like the one used for the mandelbrot set and specify a large number of iterations</i> ... but let's face it; it's simpler to work with integers and the result is the same, so whats the point. Basically we're interested in comparing numbers, and the most "natural" kinds of numbers for comparison seem to be those used to represent quantity. There is no such thing as having -1 apples, or <span style="font-style: italic;">i</span> gallons of milk. There is only one direction that quantities move away from zero: along the positive real numbers. So let's eliminate complex numbers, and everything else beyond that, too!<br /><br />So we are left with zero, and all the positive real numbers, as possible candidates. Then again, should "0" really be eligible for being a "large number". Clearly not, since there is no "quantity" that is smaller. I think we can also agree that all the positive reals less than 1 are small. So the question is among the remaining positive reals which is the <span style="font-style: italic;">smallest </span>large number?<br /><br />Obviously numbers at the forefront of the large number race, like <span style="font-style: italic;">meameamealokkapoowa oompa</span> and <span style="font-style: italic;">Rayo's Number</span>, are large numbers by default just because we don't know of anything much larger. These are incontestably large, in the same way that numbers between 1 and 0 are incontestably small.<br /><br />But these are the behemoths that resulted from countless hours of mathematical research (<span style="font-style: italic;">meameamealokkapoowa oompa</span> as a result of Bowers time spent defining array notation, and <span style="font-style: italic;">Rayo's Number</span> would not have been possible without the countless hours put into developing first order set theory, even though first order set theory wasn't developed for that purpose). Surely we can do better than to call these the <span style="font-style: italic;">smallest</span> large numbers! Of coarse from a certain point of view, they are VERY small large numbers, on account of the fact that almost all large integers are larger still. That is, there is an infinite number of integers larger than them, but only a finite number that is smaller. As you can see, in the large number field, size is very relative. Numbers that are a lot larger than another can be, just "a little larger" to a googologist, whose logarithmic number sense doesn't work with ratios but with power towers, chain arrows, and beyond. Numbers can be simultanteously "insanely large" when compared to already insanely large numbers like a <span style="font-style: italic;">googolplex</span> and <span style="font-style: italic;">Graham's Number</span>, and still be "googol-scopic" compared to still larger numbers. Perhaps this is making you dizzy. That's all the more reason to find some much smaller large numbers.<br /><br />Clearly, when we say a number is large, we don't mean with respect to all numbers! If we we're to agree to that then EVERY NUMBER would be SMALL! There would be no large numbers, and no reason to have this discussion, since you can't have small examples of something that doesn't even exist. What do we mean then? We mean large relative to our sense of things. Most of the large numbers that googologists study are much larger than the largest finite numbers used in even the most speculative theoretical physics. Clearly, if we're talking about a number bigger than the known and unkown universe, more than the possible number of parallel dimensions, and longer than how long you'd have to wait for history to repeat itself exactly, then <span style="font-style: italic;">we are talking about a large number</span>!<br /><br />Hmm. In that case, we can certainly get a lot smaller while still remaining outside of reality. How about <span style="font-style: italic;">Graham's Number</span>? That's WAY WAY beyond anything known to be real! Is this the <span style="font-style: italic;">smallest</span> large number? Nah, it's still too big. We can do better than that. Let's go even <span style="font-style: italic;">smaller</span> ... a <span style="font-style: italic;">Moser</span>. Surely. This number is known to be somewhere between G(1) and G(2). These kinds of numbers defy ordinary description already. Surely these are large numbers! What about G(1)? Is that the <span style="font-style: italic;">smallest</span>? G(1) = 3^^^^3, or 3 hexated to the 3rd. That's also 3^^^3^^^3. This is a tricky number to explain in ordinary language. Think of it this way. Let "3" be Stage 1. Let "3^3^3" or 7,625,597,484,987 be Stage 2. Let 3^3^3^ ... ^3 w/7,625,597,484,987 3s be Stage 3! For every succeeding stage have a power tower of 3s with as many terms as the previous Stage! G(1) is equivalent to Stage "3^3^3^...^3 w/7,625,597,484,987 3s". Woah?! That is clearly something way beyond our understanding! How about a smaller large number, like 3^^^3. That would be 3^3^...^3 w/7,625,597,484,987 3s. That largest numbers in Physics only amount to power towers with a handful of terms, so even this<span style="font-style: italic;"> very very small </span>very very large number is way beyond reality!<br /><br />Alright then, how about a <span style="font-style: italic;">googolplex</span> (A classic). Certainly a large number, right? But there's a problem. Just a minute ago we defined a large number as something that transcends known reality. However there are probably more than a googolplex parallel universes (or would be if they existed). So a <span style="font-style: italic;">googolplex</span> would be "too small" to be "a large number"!<br /><br />That's laughable though, because a<span style="font-style: italic;"> googolplex IS</span> a really large number. Afterall, it's what started the whole googology trend to begin with, right? Perhaps a googolplex should be pronounced the smallest large number then. If we go any lower we are going to be plunging back into known reality, so perhaps this is it? Maybe the smallest large number should be one more than the largest number with any scientific use. In that case we've already gone too far, because the largest number used in science is Don Page's massive "10^10^10^10^10^1.1". Hmm.<br /><br />Most large number discussions don't even get this far! Surely a <span style="font-style: italic;">googolplex</span> is a large number, even if it has been overtaken by speculative science. Remember all that talk about how you couldn't even write out it's decimal expansion if you tried. So let's revise our definition of the large. Perhaps a large number is just something that boggles the human mind. Something that is overwhelming from our point of view. In other words, "large" is a subjective human term, and that's exactly as it should be. "Large" only has meaning in comparison to ourselves. So when we are talking about "large numbers", we are talking about what we as humans find staggeringly large.<br /><br />With that in mind, even a <span style="font-style: italic;">googol</span>, or 1 followed by a 100 zeroes, is unfathomably large. If it wasn't so easy to construct you might almost doubt it exists. Although there is only 10^80 particles in the observable universe (the part of the universe we can actually see), there is no reason why there might not be a <span style="font-style: italic;">googol</span> or more particles in the entire universe. Scientists don't even have a definite fix on how large the universe is, and some have even estimated it to be as large as 10^10^12 meters across. Woah! So a <span style="font-style: italic;">googol</span> is most likely really out there in the universe. But even the observable portion of the universe is BIG, and particles are unimaginably small compared to us, so even 10^80 is really something we can barely imagine. Surely we can do better, though we do seem to be homing in on a definite value.<br /><br />Let's see, if 10^80 is large, so is 10^70, or 10^60, or 10^50, 10^40, 10^30. Hmm. Let's tread cautiously now. 10^27 is what is called an <span style="font-style: italic;">octillion</span>. Most people probably haven't even heard of an<span style="font-style: italic;"> octillion</span>. It's a thousand to the ninth power! Pretty HUGE for a smallish large number. There aren't even an <span style="font-style: italic;">octillion</span> gallons of water on the whole earth! Come to think of it, just how many gallons of water are on the earth? That's still got to be really really huge, right? One source says there are about 326 <span style="font-style: italic;">quintillion</span> gallons of water on the earth. We've really reduced the size of the numbers considerably, but isn't there smaller numbers we would still find "staggeringly large"? If so, we haven't located the smallest really large number yet.<br /><br />What about a <span style="font-style: italic;">trillion</span>. There is a large number. Everyone (except the googologist's) seem to think this number is really really large, from the economists to the scientists. Ordinary people are often surprised at how large it is, and have a tendency to underestimate it. Would this be an ideal place to begin any large number discussion. I don't know. A <span style="font-style: italic;">billion</span> is still very large by human standards. Even if you counted for your entire life you'd have a very slim chance of counting to a <span style="font-style: italic;">billion</span>. You'd have to devote your every waking moment to counting for 95 years at least and you'd still have a slim chance as you'd have to count a new number, on average once every 2 seconds. Some numbers from 1 to a <span style="font-style: italic;">billion</span> are really long and hard to say under 2 seconds. Try saying 567,895,234 without skipping a syllable. Say "<span style="font-style: italic;">five sixty seven million eight ninety five thousand two thirty four</span>" as fast as you can. I can't seem to get it under 3 seconds. Keep in mind that 90% of the numbers will be like this. Guess I'd never make it! That's pretty staggering in itself: A number you couldn't hope to count to in a life time. Wow! We must be really close now, so let's get just "a little" smaller.<br /><br />How about a <span style="font-style: italic;">million</span>. Perhaps this is the<span style="font-style: italic;"> smallest </span>large number. It's certainly a good place to start the large number conversation. It is the base on which the -illion numbers were formed, so much like the<span style="font-style: italic;"> googol</span> and <span style="font-style: italic;">googolplex</span>, it began a trend of larger and larger numbers in the same vein. A <span style="font-style: italic;">million</span> really is the quintessential large number; the one that started it all. It borders nicely between unfathomable, and just barely obtainable. Is it still <span style="font-style: italic;">staggering</span> enough? Well consider this illustration. Most faucets are gauged so that at full force they only allow 2 gallons of water through per minute. How long would it take a million gallons to pour through your kitchen at this rate? Well 500,000 minutes of coarse, which amounts to roughly 347 days! You'd literally have to let your faucet run at full force for about a year to waste a <span style="font-style: italic;">million </span>gallons of water. Safe to say, it be really hard to do that accidentally. The only likely scenario I can think of is if you forgot to turn it off just before you left for a year long vacation. Here's another figure. To live for a million hours you'd have to live to the advanced age of a 114! It's hard to fathom every day in our lives, let alone every hour, so clearly a million is a staggeringly large number. But then, if a million is staggeringly large, isn't 999,999 as well? Hmm. Clearly we can still get smaller.<br /><br />Certainly 900,000 is still large, as is 800,000 or 700,000, or 600,000. How about a 100,000. Can you visualize a 100,000 in your mind? Can you be aware of 100,000 things simultaneously? Probably not. The fact is, that we've become somewhat jaded to numbers like this because we've become use to numbers like a <span style="font-style: italic;">million, billion</span>, and <span style="font-style: italic;">trillion</span>. But when you stop to think of it, even a number as "small" as a 100,000 is still difficult for us to really imagine. Just counting to a 100,000 would take you a few days, even under the best assumptions. Sure 100,000 is "large" but is it "staggeringly large"? Well consider this:<br /><br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo <br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br 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/>oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo<br />oooooooooo oooooooooo oooooooooo oooooooooo oooooooooo <br /><br />That's only a string of 10,000 consecutive o's. You'd need 10 times that to have a 100,000. So a 100,000 is still much more than we can really wrap our brain around. In fact baseball stadiums can only house about half this many (around 50,000 or so). Okay, so what about a number like 10,000? It was known to the greeks as a myriad, and represented the base of their large number system, in the same way that a thousand is the base of ours. I just presented 10,000 objects, so clearly it's something very tangible and close to our level of experience. It's "staggering" but in a much more limited sense. What about a <span style="font-style: italic;">thousand</span>? That's still a large number right? Maybe not a "very large" number, but it's still large. Hmm.<br /><br />Perhaps a number doesn't have to be staggering to be "large". Afterall, even a thousand is something we can't completely wrap our brains around. Perhaps it just needs to be something we can't perceive all at once. What about a <span style="font-style: italic;">hundred</span>? This is considered to be a "large number" by most kids. In fact it is usually the first number we are ever introduced to as a "large number". If you look at how number words are constructed you discover that a "million" simply means a "great thousand", and a "thousand" means a "strong hundred". So a hundred is a kind of foundational block for large numbers. There are lots of numbers, like a googol, that use a 100 in their construction, so clearly even googologists seem to think it's "pretty big". As a matter of fact, "googology wiki" has a policy regarding large number submissions, that the number can be no smaller than 100. Thus, according to googology wiki, "100" <span style="font-style: italic;">IS</span> the smallest large number. This is probably the answer that would satisfy most people. But I can't help asking... isn't 90 still a big number? Even 80, or something like 50 seems like a big number. I can't keep track of 50 things at once, so clearly 50 is big in that sense. Let's see where this thinking leads...<br /><br />If a number is big simply because it's outside our ability to perceive directly (this is known as our <span style="font-style: italic;">number sense</span>) than clearly a lot of numbers in this range are still big. 40,30, 20. How about <span style="font-style: italic;">ten</span>? Maybe THAT's the<span style="font-style: italic;"> smallest</span> large number. After all, <span style="font-style: italic;">ten</span> is the base of our decimal system, and therefore the base of numbers like <span style="font-style: italic;">hundred, thousand, million, googol, googolplex</span>, etc. What kind of conversation about large numbers could we have if we couldn't talk about <span style="font-style: italic;">ten</span>?<br /><br />Not convinced that <span style="font-style: italic;">ten</span> is large? Well how about a couple having <span style="font-style: italic;">ten</span> kids, or worse a mother giving birth to decuplets! That's a lot then. Eating<span style="font-style: italic;"> ten</span> pancakes or waffles is likely to make you very full. Try juggling <span style="font-style: italic;">ten</span> balls, or counting <span style="font-style: italic;">ten</span> objects constantly scurrying about! Could you imagine if popular movie series had<span style="font-style: italic;"> ten</span> parts! Harry Potter had<span style="font-style: italic;"> eight</span>, Star Wars had <span style="font-style: italic;">six</span>. Could you imagine having something like Rocky 10, or Superman 10?! There is now <span style="font-style: italic;">ten</span> numbered Megaman games, but that's more an exception than a rule, and it really is insane that something like "Megaman 10" is something real, and not some knock off pirated title. Seen this way,<span style="font-style: italic;"> ten is large</span>. Just think ... there was a time when there wasn't even a name for this number and the only number words were <span style="font-style: italic;">one</span>, <span style="font-style: italic;">two</span>, and many. Okay. So maybe 10 should be designated the "smallest large number". If not, we are clearly running out of options fast!<br /><br />If a number is large simply because we can't perceive it all at once, than <span style="font-style: italic;">five</span> would probably be the smallest large number. It's the smallest number that most people can't perceive without counting. Most people can recognize <span style="font-style: italic;">one, two, three</span>, and <span style="font-style: italic;">four</span> objects on sight, but <span style="font-style: italic;">five</span> tends to make the eyes go cross, and one usually has to split it into a group of three and two to "see" it. So there you have it, the smallest large number. Hmm. There is still something unsatisfying about this answer though, because it's arbitrary.<br /><br />After all, it's only seems large because it's just beyond our number sense capabilities. What about birds though. I suppose<span style="font-style: italic;"> four</span> might seem large to them? Perhaps <span style="font-style: italic;">three</span> seems large to insects, and can a bateria imagine<span style="font-style: italic;"> two</span>? Then there is the fact that<span style="font-style: italic;"> two, three</span>, and <span style="font-style: italic;">four</span> form an important foundation for lots and lots of large numbers. For example, Graham's number begins with 3^^^^3. That is, it begins with three's between four up arrows. So even numbers like these serve some purpose in the large number field. What about <span style="font-style: italic;">two</span>? Remember the Mega and the Moser. A Mega is "<span style="font-style: italic;">two</span> in a pentagon", and a Moser is "<span style="font-style: italic;">two</span> in a megagon". I'm fond of saying that <span style="font-style: italic;">two</span> is the<span style="font-style: italic;"> smallest</span> large number. I have often defined a large number, as any number which is larger than others. Well <span style="font-style: italic;">two</span> is large then, because at least it's larger than <span style="font-style: italic;">one</span>. If we restrict ourselves to the counting numbers (as I normally do) then <span style="font-style: italic;">one</span> can't be a large number, because there is no counting number for it to be larger than! On a more intuitive level, <span style="font-style: italic;">one</span> just can't seem large. After all there is<span style="font-style: italic;"> one</span> person that we're always in the presense of, ... ourselves. So we can't even escape<span style="font-style: italic;"> one</span> if we wanted to. <span style="font-style: italic;">One</span> is so familiar that its status as a number seems somewhat shaky. It makes sense to say that <span style="font-style: italic;">two</span> people form a "group", but it seems silly to call a single person a "group of <span style="font-style: italic;">one</span>". A group is by definition two or more. How about this. Isn't it a bit redundant to speak of "<span style="font-style: italic;">one</span> apple", when it is just an "apple". Seen this way, there is something <span style="font-style: italic;">unique</span> about <span style="font-style: italic;">one</span>. It's the only natural whose every integer power is itself. It's the only number considered to be neither prime nor composite.<br /><br />We've already established at the beginning of the article that any number less than one is small. What about one itself? Well we can't really say it's a small number. Small relative to what? We can't really say it's large either, for the same reason. So I think it's fair to say 1 is neither small or large. It just is. It's just the one and only, average sized number.<br /><br />This brings me to the whole point of this meandering tour through number-space. If we agree that numbers less than <span style="font-style: italic;">one</span> are small, shouldn't it follow that numbers greater than <span style="font-style: italic;">one</span> are big? Perhaps this seems unnatural. However, any other choice does not seem to be mathematically defensible and seems based purely on human subjectivity.<br /><br />So then, is a number like 1.5 big? In most ordinary contexts, no. But consider someone who was 1.5 times taller than you. Wouldn't you consider them to be pretty tall in comparison? What about in the grocery when you pay the same price on a soda bottle that 50% bigger. Isn't that a big savings? Used as a ratio, we can see that 1.5 is a significant change. So yes, in this sense, even 1.5 can be "big" in a certain sense. What about 1.4, or 1.3, 1.2 etc. Think about 1.1. We can imagine having an extra 10% of just about anything and being able to perceive the difference. Perhaps then a number should be considered large as long as we can perceive the difference between it and 1. Again this would be arbitrary. Thus I'm advocating that the most natural mathematically based definition for a large number, is that it simply be larger than 1.<br /><br />The reason I suggest adopting this model has to do with certain considerations of how humans understand numbers. Even though the difference between 2 and 5 Vs. 95 and 98 is the same, we perceive "5" to be much larger than "2" than "98" is larger than "95". This is because we understand numbers on a logarithmic scale in which only their ratios matter. The ratio between 5 and 2 is 2.5. That is a "big" increase in size. But 98 is only 3% larger than 95, so we aren't very impressed. What this means is that size is mostly an issue of scale. If we let "one" become our point of reference, then we can conveniently split the remaining positive reals into the "small numbers" and the "large numbers". In this system every large number has a corresponding small number equal to its recipricol. Also, in this way of thinking there is no smallest number (excluding zero) just as there is no largest number. Lastly, this also means that there is no <span style="font-style: italic;">smallest</span> large number, and also no<span style="font-style: italic;"> largest</span> small number. Thus just as we can have larger and larger numbers without bound, we can now have smaller and smaller large numbers without bound.<br /><br />The fun part is now we can meaningfully talk about some very very small large numbers. For example, 1.00000069 is a large number so amazingly small that you have to raise it to the millionth power to get 2! Better yet we could have a "<span style="font-style: italic;">googolminex and one</span>" as an example of a very very small large number. Or how about a large number so small that you have to raise it to <span style="font-style: italic;">Graham's Number</span> to get a <span style="font-style: italic;">googolminex and one</span>! It's so absurd, and almost as much fun as coming up with "very large" large numbers :)<br /><br />So in a nutshell my whole point is this: A large number should be defined as any real number greater than 1. This seems to be the most mathematically sound way to define them. Best of all we can now give a special name to large numbers as a whole. We can call them the "<span style="font-style: italic;">superuniary numbers</span>" or just the "<span style="font-style: italic;">superuniaries</span>". I'll admit that I really like this term. It sounds more "technical" than to merely saying "large numbers". Now I can say I study the <span style="font-style: italic;">superuniaries</span> instead.<br /><br />To close this absurd journey, I have yet one more absurd riddle to leave you with. So we have determined that a number is "large" so long as it's larger than 1, and that a number can be a "very small" large number, when its very close to 1. But how large does a "very small" large number have to be before its no longer "very small" ... I'll leave that one for you to decide :)Sbiis Saibianhttp://www.blogger.com/profile/12289340153316642062noreply@blogger.com1tag:blogger.com,1999:blog-2881823367984705751.post-16920024854190448002008-12-09T07:16:00.000-08:002012-02-03T22:39:18.033-08:00New Large Numbers Site!I just finished my NEW googlepages (now googlesites) website on large numbers yesterday (2008.12.08). It is now published and viewable online. The site will eventually cover a wide range of topics related to the various orders of large numbers, starting with the smaller ranges ( counting range, exponential range, hyper exponential range, tetrational range , etc. ) and working it's way up to even greater kinds of numbers !<br /><br />Unfortunately the site is "under construction", and as such it only includes "Section I : Fundamentals of the Counting numbers". This section introduces the concept of the counting numbers as an extension of "number sense" and then proceeds to go over various notations, written and spoken, that humans have used throughout the world and through time. Lastly I even propose my own rather unusual naming convention for naming the first 1099 counting numbers. It provides a unique short petname for every counting number from 1 to 1099.<br /><br />Lastly there is the "number catalogs". Here is where I compile lists of numbers within the counting range with certain properties. For example I have a list of all the primes less than 1000, and also a list of numbers generated through binary arithmetic operations such as a+b , axb , a^b.<br /><br />Eventually I want to add even more content and start discussing greater number ranges. The following link will take you to the homepage ...<br /><br /><span style="text-decoration: underline;">http://sites.google.com/site/largenumbers/home</span><br /><br />Feel free to comment on content, offer suggestions, etc.<br /><br />-- Sbiis SaibianSbiis Saibianhttp://www.blogger.com/profile/12289340153316642062noreply@blogger.com0tag:blogger.com,1999:blog-2881823367984705751.post-46291044962203951642008-06-05T18:43:00.000-07:002012-02-02T11:23:59.455-08:001977 is over & so is G<span style="font-family:trebuchet ms;">One googologist's rant against the indeflatable popularity of Graham's Number<br />( written by Sbiis Saibian )</span><br /><span style="font-family:trebuchet ms;"></span><br /><span style="font-family:trebuchet ms;"><span style="font-style: italic;">Graham's Number</span> or simply G has replaced the <span style="font-style: italic;">googolplex</span> as "largest number ever" in the popular imagination. Ronald Graham actually invented <span style="font-style: italic;">Graham's Number</span> back in 1977 as an upper bound to a problem in graph theory. Although the Guinness Book of World records has made the number famous, among professional m</span><span style="font-family:trebuchet ms;">athematicians it's more of a joke. For one, the upper bound is absurdly large, and the actual value is probably a GREAT deal smaller, and secondly the proof has proven to be flawed at best.</span><br /><span style="font-family:Trebuchet MS;"></span><br /><span style="font-family:Trebuchet MS;">While G may be recognized by the Guinness Book, and is certainly the largest number taken seriously, it is NOT by a long shot the "Biggest Number ever defined!". The number was created over 30 years ago, and yet it is treated like it's current and cutting edge. Hardly! It's already lost that title within professional circles as even larger numbers have been used in "serious mathematical proofs" since then. Furthermore, the requirement that a candidate largest number be somehow connected to serious mathematics is of little concern to the amateur mathematician who likes to come up with large numbers just for the sake of it. Accordingly, the largest numbers defined by humans are usually the product of an untamed imagination, having little to do with serious mathematics, even when the untamed imagination happens to be that of a professional mathematician.<br /><br /></span><span style="font-family:Trebuchet MS;">G is popular because it is well known, it takes a little while to explain, any intelligent person can understand it, and it builds on the familiar operator sequence of addition, multiplication, exponentiation, tetration, etc. But if we want to talk about the largest number humans have yet defined than we've got a LONG way to go from <span style="font-style: italic;">Graham's</span> !</span><br /><span style="font-family:Trebuchet MS;"></span><br /><span style="font-family:Trebuchet MS;">First off, you need to completely shift your perspective on this. Yes from a practical standpoint G is absolutely absurdly and unimaginably huge ! But consider this, assuming someone only knew how to add, and how to interpret recursive formulas, <span style="font-style: italic;">Graham's Number </span>can be defined as follows ...</span><br /><br /><span style="font-family:trebuchet ms;"><a,b,1> {a,b,1} = a^b</a,b,1></span><br /><span style="font-family:trebuchet ms;"><a,1,c> {a,1,c} = a</a,1,c></span><br /><span style="font-family:trebuchet ms;"><a,b,c> {a,b,c}= <a,><a,b-1,c>{a,{a,b-1,c},c-1}</a,b-1,c></a,></a,b,c></span><br /><span style="font-family:trebuchet ms;">G(1) = {3,3,4}</span><br /><span style="font-family:trebuchet ms;">G(n) = {3,3,G(n-1)}</span><br /><span style="font-family:trebuchet ms;">G = G(64)</span><br /><br /><span style="font-family:trebuchet ms;">It only requires 6 lines to define it. <span style="font-style: italic;">Graham's</span> is simply an elementary recursive function applied to the operation level in the operator hierarchy. " What ?! simply that ?! That doesn't sound simple at all", you might be thinking. However everything about G is already well understood, and elementary recursion and hierarchies like this can be applied a ridiculous number of times, where G on the other hand simple contains ONE OF EACH !</span><br /><br /><span style="font-family:trebuchet ms;">By this point your probably thinking this is BS, or that I'm talking about trivial stuff like G+1 , G^2 , G^G, G^^G, G(65) , G(100) , G(<span style="font-style: italic;">googolplex</span>) ,G(3^^^^3) , G(<span style="font-style: italic;">Graham's Number</span>) ,</span><br /><span style="font-family:trebuchet ms;">G(G(G(G(...(G))..)) with G G's, etc.</span><br /><br /><span style="font-family:trebuchet ms;">nope, all of that is "trivial" (as mathematicians love to say). It's the 21st century, and all of this is just elementary recursive stuff.</span><br /><br /><span style="font-family:trebuchet ms;">There are MANY new numbers on the block that should get more attention, but are overshadowed by G, not because G is larger, but because it came from professional circles! Some of these numbers are so large, and difficult to describe, that they make G look more elementary than adding <span style="font-style: italic;">two</span> and <span style="font-style: italic;">two</span>! All of these amazing numbers were invented by a little known amateur mathematician by the name of Jonathan Bowers. In 2002 he published his first website where he discussed for the first time his "Super giant Numbers".</span><br /><br /><span style="font-family:trebuchet ms;">To describe his numbers Bowers invented a special notation called "array notation". With it one can easily express numbers much larger than anything which can be expressed using Knuth Up-arrows, Steinhaus Polygons, the G function, and even Conway Chain arrows!</span><br /><br /><span style="font-family:trebuchet ms;">How does it work? Well Bowers first creates a function which can have any number of entries, just like chain arrows, but unlike them they do NOT work on the last two entries, but rather the first two. The effect of this shift is profound and gives Bowers arrays an advantage over alternative notations. We can use the curly braces to contain entries and commas to separate them. When the function is provided with no "entries", it takes on the default value of 1:<br /><br />{ } = 1<br /><br />When a single entry is present, the value of the array is the value of the entry:<br /></span><br /><a face="trebuchet ms"> {a} = a<br /><br />2 entries are equivalent to raising the first entry to the power of the second (in his old notation the first two entries would be added together) ...<br /><br /><a,b> {a,b} = a^b<br /><br />3 entries is equivalent to Knuth's up arrows ...<br /><br /><a,b,c> {a,b,c} = a^^^^....^^^b with c up arrows</a,b,c><br /><br />After this we need to know something about the rules Bowers uses for what he calls "linear arrays". These are the simplest kind of arrays he works with. In linear arrays we can have ANY number of entries, provided they are only separated by commas. If the linear array contains 4 or more entries we need to follow a set of special rules ...<br /><br />1. If the last entry is a 1 then remove it ... {a,b,...,k,1} = {a,b,...,k}<br />2. If the 2nd entry is 1, return 1st entry ... {a,1,c,...,k} = a<br />3. If entries 3 ~ k-1 = 1 & the kth entry does NOT equal 1 ...<br /><br />{a,b,1,1,...,1,1,k,...,z} = {a,a,a,a,...,a,{a,b-1,1,1,...,1,1,k,...,z}, k-1 , ... z }<br /><br />4. If rules 1~3 do NOT apply then ...<br /><br />{a,b,c,...,k}= {a,{a,b-1,c,...,k}, c-1 , .... , k }<br /><br />Now let's compare Bowers arrays to the Graham's function.<br /><br />Firstly ...<br /><br />G(1) = {3,3,4} = 3^^^^3<br />G(2) = {3,3,{3,3,4}}<br />G(3) = {3,3,{3,3,{3,3,4}}}<br />etc.<br /><br />Although arrays may not seem very impressive at the moment, you have to consider this. Bowers has set up his arrays so that every new type of array explodes the old type of array. 4 entries far surpasses 3.<br /><br />for example take the seemingly harmless example ...<br /><br />{3,3,1,2}<br /><br />Using the rules to evaluate we find this ...<br /><br />{3,3,1,2} = {3,3,{3,2,1,2},1} = {3,3,{3,3,{3,1,1,2},1},1} =<br /><br />{3,3,{3,3,3,1},1} = {3,3,{3,3,3},1} = {3,3,{3,3,3}} = 3^^^...^^^3 with 3^^^3 up arrows.<br /><br />Look familiar ? That's right! It's the same growth exhibited by Graham's function. Note that...<br /><br />{3,2,1,2} = 3^^^3 < 3^^^^3 = G(1)<br /><br />Furthermore...<br /><br />{3,3,1,2} = 3^^^...^^^3 w/3^^^3 ^s > 3^^^^3 = G(1)<br /><br />with a little bit of inductive reasoning we can conclude that ...<br /><br />{3,65,1,2} < G(64) < {3,66,1,2}<br /><br />and in general that:<br /><br />{3,n+1,1,2} < G(n) < {3,n+2,1,2}<br /><br />But we've only just begun with Bower's system. What happens if the third entry is a 2? Let's again take a seemingly harmless example:<br /><br />{3,3,2,2}<br /><br />Applying the rules we find that:<br /><br />{3,3,2,2} = {3,{3,2,2,2},1,2} = </a,b></a><a face="trebuchet ms">{3,{3,{3,1,2,2},1,2},1,2} = {3,{3,3,1,2},1,2}<br /><br /></a><a style="font-family: trebuchet ms;"><a,b>The 2nd entry in the outer array is ITSELF an array of {3,3,1,2} which is already greater than G(1) or 3^^^^3.<br /><br />so Now we can say ...<br /><br />{3,{3,3,1,2},1,2} > {3,G(1),1,2} > G(G(1) - 2 ) = G(3^^^^3 - 2 )<br /><br />which is roughly G(G(1))<br /><br />That's right! We are recursively plugging in Graham's function into itself ! But this is only the effect of having a 2 as the 3rd and 4th entry ...<br /><br />{3,3,2,2} > G(G(1))<br /><br />{3,4,2,2} > G(G(G(1)))<br /><br />{3,5,2,2} > G(G(G(G(1))))<br /><br />But this is just the beginning. You see the 2 in the 4th entry is producing 2nd order operations. Think of it like this... Imagine the Graham's function is 2nd order addition. Then Plugging G into itself ( meaning GGG...GG ) is 2nd order multiplication. Now imagine 2nd order exponents , tetration, pentation, and so on.<br /><br />Now you should realize the pattern. We can create a hierarchy of 2nd order operators which are exploded ( just like Graham's function explodes the standard operator hierarchy ) by the 3rd order addition.<br /><br />Now we can have nth order hierarchies. This is the power of ONLY 4 entries !!!<br /><br />Bowers 4 entry arrays already are powerful enough to keep up with chain arrows of arbitrary length.<br /><br />People who know about Chain arrows often scoff at <span style="font-style: italic;">Graham's Number</span> since it's between 3-->64-->1-->2 and 3-->65-->1-->2.<br /><br />Consider however that 3-->3-->3-->3 ( a number that Conway invented using his notation ) is much larger than G, yet is still smaller than Bowers {3,3,3,3}<br /><br />Bowers calls {3,3,3,3} , <span style="font-style: italic;">tetratri</span> , for obvious reasons.<span style="font-style: italic;"> Tetratri</span> uses 3rd order exponents, and is much larger than G.<br /><br />Is <span style="font-style: italic;">tetratri</span> the largest number ? Nope.<br /><br />Bowers is well aware that even this number surpasses anything else out there in the same vein, but he doesn't stop here, but continues into the unknown.<br /><br />5 entry arrays go way beyond chain arrows. To imagine this think of a chain like ...<br /><br />3-->3-->3-->....-->3-->3 with " 3 --> 3 --> 3 --> 3 " threes in the chain<br /><br />This number is still smaller than {3,3,1,1,2} .<br /><br />The 5th entry produces orders of 4 entry arrays in the same way that the 4th entry produces orders of operator hierarchies . YIKES ! It hurts to think about it too much.<br /><br />And example would be {10,10,10,10,10} a number Bowers calls <span style="font-style: italic;">pentadecal</span> .<br /><br />Now Bowers linear arrays are extremely powerful, and Bowers is prepared to use them to the fullest. He defines even larger numbers such as ...<br /><br /><span style="font-style: italic;">Iteral</span> = {10,10,10,10,10,10,10,10,10,10} ( That's 10 <span style="font-style: italic;">tens</span> )<br /><br /><span style="font-style: italic;">Iteralplex</span> = {10,10,10,10,10, ... ... ... ... ... ... , 10,10,10} with <span style="font-style: italic;">Iteral</span> 10's !<br /><br />Now we have just blasted off into new territory.<br /><br />What Bowers does next is even more incredible. Bowers creates "array structures" . He was the first person to discover that if you attempt to extend "entry based algorithms" the most efficient and logical approach is to create a structural hierarchy.<br /><br />Linear arrays are a hierarchy of entries. Every entry is superseded by the entry to it's immediate right.<br /><br />Bowers realized that he could surpass linears and create planar arrays, or 2-D arrays.<br /><br />Actually the dimensionality is superfluous. We can still write such arrays in a linear fashion, as long as we create a new kind of "comma". Bowers uses " (1) " to separate "linear blocks" to form "planar blocks". To show you what I mean, and also show you just how Powerful Bowers notation is look at this:<br /><br />First he creates a new rule ...<br /><br />{b,p (1) 2} = {b,b,b,...,b,b} with p b's<br /><br />The "b , p " is the first linear block , and " 2 " is the 2nd linear block . Together they can form linear arrays of arbitrary length. Now observe how easy this makes it to express <span style="font-style: italic;">iteral</span> and <span style="font-style: italic;">iteralplex</span> ...<br /><br /><span style="font-style: italic;">Iteral</span> = {10,10 (1) 2 }<br /><span style="font-style: italic;">Iteralplex</span> = {10,3,2 (1) 2} = {10,{10,10(1)2} (1) 2 } = {10,<span style="font-style: italic;">iteral</span> (1) 2}<br /><br />Notice the similarity of the rules used to expand <span style="font-style: italic;">iteralplex</span>. That's right. We can use linear arrays to expand linear arrays, thus the 2 at the end represents 2nd order linear arrays. We can then have 3rd order , 4th , ... nth order . But it just keeps going . The 2nd linear block can become 2 entries. For example Bowers has coined the number ...<br /><br /><span style="font-style: italic;">Hyperal</span> = { 10,10 (1) 10,10 }<br /><br />Note that the <span style="font-style: italic;">hyperal</span> is like a 2x2 grid of 10's ( if we imagine it as a 2 dimensional display )<br /><br />Bowers continues his system. He has a number called <span style="font-style: italic;">xappol</span> which is a 10 x 10 grid of 10's , and a <span style="font-style: italic;">xappolplex</span> which is a <span style="font-style: italic;">xappol</span> x <span style="font-style: italic;">xappol</span> grid of 10's.<br /><br />Where could Bowers possibly go from here ?! 3-dimensional arrays, 4-dimensions , heck , arbitrary numbers of dimensions.<br /><br />Here's a REALLY awesome number Bowers coined ...<br /><br />Imagine a 100-dimensional cube where each dimension has a length of 10. Now imagine every entry is a 10. When this array is solved it produces the insane number <span style="font-style: italic;">gongulus</span> ( G's seem to be a popular letter for large numbers )<br /><br />Let's stop for a moment. Consider the <span style="font-style: italic;">googolplex</span> and how it is totally humbled by <span style="font-style: italic;">Graham's Number</span>. Now think about it , <span style="font-style: italic;">gongulus</span> totally mops the floor with <span style="font-style: italic;">Graham's Number</span>. G is only about the size of a 4 entry array with a 2 in the last entry, but a <span style="font-style: italic;">gongulus</span> requires 10^100 entries ( That's a <span style="font-style: italic;">googol</span> entries. Even the number of entries is huge ! ) arrayed in a complex 100-dimensional structure ! In fact, even this comparison doesn't do the <span style="font-style: italic;">gongulus</span> justice, because by the time the 100-dimensional array is solved as a linear array the number of entries would exceed a <span style="font-style: italic;">Graham's Number</span>, G(3^^^^3), G(G(...G(G)...)) w/G Gs, or anything else you could imagine using Graham's function, chain arrows, or even linear arrays, planar arrays, realmic arrays, flunic arrays, ... it's just INSANE! There is absolutely no comparison! <span style="font-style: italic;">Graham's Number</span> was just the very beginning of large numbers all along! These are numbers for the 21st century ! Who would have thought that such a thing was possible.<br /><br />At this point you may consider this too abstract. Actually dimensional arrays use a well defined algorithm. It also uses fairly simple notation. To separate n-dimensional blocks you use the<br />" ( n ) " separator.<br /><br />For example, Bowers has a 3-d array called <span style="font-style: italic;">dimentri</span>. It is a 3x3x3 array of 3's. We can even write it out in full ...<br /><br />{3,3,3(1)3,3,3(1)3,3,3(2)3,3,3(1)3,3,3(1)3,3,3(2)3,3,3(1)3,3,3(1)3,3,3}<br /><br />The commas separate entries inside linear blocks, the (1) separate linear blocks inside planar blocks, and the (2) separate planes inside "realms".( Bowers uses a lot of whimsical terminology. For example he refers to 4-d blocks as flunes )<br /><br />Bowers actually has the dimensional rules worked out fairly well. Unfortunately it requires at LEAST 7 rules ( more rules makes the algorithm a little clearer and neater ). Worse, Bowers uses very wordy and illustrative explanations for the rules. None the less, I have carefully studied dimensional arrays and they do work.<br /><br />If you ask me, a number as impressive as <span style="font-style: italic;">gongulus</span> deserves to be in the Guinness Book of world records for something. True, it's so large that even mathematicians have no use for it !! But that's what makes it so impressive. The Guinness Book should have an entry on Jonathan Bowers as the creator of the worlds fastest growing computable functions, and in the entry <span style="font-style: italic;">gongulus</span> should be mentioned just for kicks ( It would then be the largest number recognized in a professional piece of literature ). Then when bloggers ask "whats the largest number?" instead of<span style="font-style: italic;"> </span>just hearing about<span style="font-style: italic;"> googolplex</span> and <span style="font-style: italic;">Graham's</span> someone will inevitably have to mention <span style="font-style: italic;">gongulus</span> just to be a smart ass :)<br /><br /><span style="font-style: italic;">Gongulus</span> unlike <span style="font-style: italic;">googolplex</span> and <span style="font-style: italic;">Graham's</span> however is not so easy to explain. It requires notation and concepts that require some real imagination to understand. Even here I've only given a cursory explanation for how they work. They are both surprisingly simple to explain, and surprisingly difficult to define rigorously. For this reason it may never be as popular as <span style="font-style: italic;">googolplex</span> and <span style="font-style: italic;">Graham's Number</span>. But I really don't think that's the reason it is not recognized. The <span style="font-style: italic;">gongulus</span> just needs to proper exposure and recognition (that's what this article is here for!). It really is a legitimate number using legitimate mathematics.<br /><br />Professional mathematicians simply regard the art of creating large numbers for their own sake as trivial. They might appreciate it as an impressive computable function, but to them anything computable is trivial because the Busy Beaver function "grows faster than any computable function". This maybe so, but this oft repeated phrase overshadows the fact that the more complex the computable function, the more states are required to surpass it. Bowers has certainly increased the complexity of known computable functions, and taken them to a new frontier.<br /><br />I've heard it said that before you would reach BB(100) the resourcefulness of the human imagination would be overwhelmed. Is this so? A <span style="font-style: italic;">gongulus</span> already seems pretty complex. Just how close is it to something like BB(100)? The world may never know, but at least once mathematicians need a handy way to express BB(10) or BB(20), Bowers notations will already be on hand! That's got to count for something.<br /><br />In any case, even if the <span style="font-style: italic;">gongulus</span> is not officially recognized, and even if it is complicated, I still think people would like to hear about it. Numbers are the domain of everyone who is interested in them, so who cares what Guinness says, <span style="font-style: italic;">gongulus</span> IS one of the WORLDS LARGEST NUMBERS period.<br /><br />It's so large in fact, that it knocks Guinness "world champion" right out of the list of contenders! In fact it gets completely crowded out because Bowers goes even further than this! He starts talking about Super dimensional arrays, Trimensional arrays, tetrational arrays , pentational arrays , etc.<br /><br />Some of these numbers are so large that Bowers can't fully explain them. Even he doesn't fully understand them, and he comments " how do these arrays work ? only God knows ...".<br /><br />The largest number Bowers has coined was finally given a definition on March 13 , 2008 . The number is called the "<span style="font-style: italic;">meameamealokkapoowa oompa</span>" and it could very well be the largest number in the world. You really can't overstate how huge this number is. Unfortunately it literally defies explanation, and so it would be hard to convince a professional mathematician that such a number exists (Technically speaking its not the number that doesn't exist, but rather the name fails to signify a number if the function is not well defined).<br /><br />For this reason I think <span style="font-style: italic;">gongulus</span> has a better chance making it in the Guinness Book of World Records, as the largest number defined through the use of an algorithm. Then every time someone blogs about Large Numbers we'll hear about <span style="font-style: italic;">gongulus</span> in addition to <span style="font-style: italic;">googolplex</span> and <span style="font-style: italic;">Graham's Number</span>. Of the three of these, the largest would be the only one coined by an amateur mathematician :)<br /><br />Then again, it doesn't need to be in Guinness to become well known. We live in the information age where memes can replicate themselves through blogs and chat rooms. Next time someone ends a large number conversation with <span style="font-style: italic;">Graham's Number</span>, you can always top them and try to explain <span style="font-style: italic;">gongulus</span>. If it gets brought up often enough it might eventually become officially recognized. Then you'll find it in the dictionary a little before <span style="font-style: italic;">googolplex</span> under G. Websters will probably have to settle for the abbreviated definition " An extremely large number defined by Jonathan Bowers", rather than an exact definition since that would require a couple pages of mathematics ! :)<br /><br />If you find this stuff interesting and want to learn more you can visit Jonathan Bowers new website ...<br /><br /></a,b></a><a style="font-family: trebuchet ms;" href="http://www.polytope.net/hedrondude/home.htm">http://www.polytope.net/hedrondude/home.htm</a><br /><br /><span style="font-family:trebuchet ms;">Be warned though, that Bowers does not sufficiently explain his notation, so one can only speculate about what he means at times. Thankfully Bowers work is being deciphered by others. Sam Hughes also has studied array notation and explains how dimensional arrays work. He has a series of "everything 2" articles where he explains various levels of array notation up to and including arbitrary dimensional arrays. ( You can also find a compact form of <span style="font-style: italic;">gongulus</span> at the end of the articles ).</span><br /><br /><a style="font-family: trebuchet ms;" href="http://everything2.com/e2node/Linear%2520array%2520notation">http://everything2.com/e2node/Linear%2520array%2520notation</a><br /><br /><span style="font-family:trebuchet ms;">Chain arrows arrived on the public scene in 1996, in the well known "The Book of Numbers". In a way you could consider John Conway, the inventor of Chain arrows, as the "Number Champ" at the close of the 20th Century. If this is so then Jonathan Bowers and his arrays represent the dawn of the 21st Century. Interestingly he is a testament to the power of the information age. An amateur mathematician who would otherwise be unknown is able to present his ideas to the world through the web. These are the Extremely Large Numbers of the 21st Century ! ... and the game has only just begun ...</span><br /><br /><span style="font-family:trebuchet ms;">-- Sbiis Saibian</span>Sbiis Saibianhttp://www.blogger.com/profile/12289340153316642062noreply@blogger.com1