Thursday, February 2, 2012

Very Small Very Large Numbers

One googologist's quest for the smallest member of his trade
(Written by Sbiis Saibian)

What do you consider to be the smallest Very Large Number? Yes you read that correctly. Today we will be exploring some very small 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 smallest 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?

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 ... :)

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 smallest large number? Perhaps they'd say that's like talking about millikilometers. 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?

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 googolminex is defined by Conway & Guy to be 1 over a googolplex. So if a googolminex 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 is "the smallest number". Negative numbers should not be considered along a "large-small" axis. Instead Negative Numbers should only be considered "less than zero".

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.

What about Complex Numbers? Quaternions? Octonions? Well ask yourself, how do you compare numbers like i , or 3+2i, 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, i , -1 , -i, 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". 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 ... 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 i 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!

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 smallest large number?

Obviously numbers at the forefront of the large number race, like meameamealokkapoowa oompa and Rayo's Number, 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.

But these are the behemoths that resulted from countless hours of mathematical research (meameamealokkapoowa oompa as a result of Bowers time spent defining array notation, and Rayo's Number 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 smallest 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 googolplex and Graham's Number, 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.

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 we are talking about a large number!

Hmm. In that case, we can certainly get a lot smaller while still remaining outside of reality. How about Graham's Number? That's WAY WAY beyond anything known to be real! Is this the smallest large number? Nah, it's still too big. We can do better than that. Let's go even smaller ... a Moser. 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 smallest? 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 very very small very very large number is way beyond reality!

Alright then, how about a googolplex (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 googolplex would be "too small" to be "a large number"!

That's laughable though, because a googolplex IS 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.

Most large number discussions don't even get this far! Surely a googolplex 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.

With that in mind, even a googol, 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 googol 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 googol 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.

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 octillion. Most people probably haven't even heard of an octillion. It's a thousand to the ninth power! Pretty HUGE for a smallish large number. There aren't even an octillion 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 quintillion 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.

What about a trillion. 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 billion 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 billion. 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 billion are really long and hard to say under 2 seconds. Try saying 567,895,234 without skipping a syllable. Say "five sixty seven million eight ninety five thousand two thirty four" 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.

How about a million. Perhaps this is the smallest 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 googol and googolplex, it began a trend of larger and larger numbers in the same vein. A million really is the quintessential large number; the one that started it all. It borders nicely between unfathomable, and just barely obtainable. Is it still staggering 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 million 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.

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 million, billion, and trillion. 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:

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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 thousand? That's still a large number right? Maybe not a "very large" number, but it's still large. Hmm.

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 hundred? 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" IS 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...

If a number is big simply because it's outside our ability to perceive directly (this is known as our number sense) than clearly a lot of numbers in this range are still big. 40,30, 20. How about ten? Maybe THAT's the smallest large number. After all, ten is the base of our decimal system, and therefore the base of numbers like hundred, thousand, million, googol, googolplex, etc. What kind of conversation about large numbers could we have if we couldn't talk about ten?

Not convinced that ten is large? Well how about a couple having ten kids, or worse a mother giving birth to decuplets! That's a lot then. Eating ten pancakes or waffles is likely to make you very full. Try juggling ten balls, or counting ten objects constantly scurrying about! Could you imagine if popular movie series had ten parts! Harry Potter had eight, Star Wars had six. Could you imagine having something like Rocky 10, or Superman 10?! There is now ten 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, ten is large. Just think ... there was a time when there wasn't even a name for this number and the only number words were one, two, and many. Okay. So maybe 10 should be designated the "smallest large number". If not, we are clearly running out of options fast!

If a number is large simply because we can't perceive it all at once, than five would probably be the smallest large number. It's the smallest number that most people can't perceive without counting. Most people can recognize one, two, three, and four objects on sight, but five 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.

After all, it's only seems large because it's just beyond our number sense capabilities. What about birds though. I suppose four might seem large to them? Perhaps three seems large to insects, and can a bateria imagine two? Then there is the fact that two, three, and four 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 two? Remember the Mega and the Moser. A Mega is "two in a pentagon", and a Moser is "two in a megagon". I'm fond of saying that two is the smallest large number. I have often defined a large number, as any number which is larger than others. Well two is large then, because at least it's larger than one. If we restrict ourselves to the counting numbers (as I normally do) then one can't be a large number, because there is no counting number for it to be larger than! On a more intuitive level, one just can't seem large. After all there is one person that we're always in the presense of, ... ourselves. So we can't even escape one if we wanted to. One is so familiar that its status as a number seems somewhat shaky. It makes sense to say that two people form a "group", but it seems silly to call a single person a "group of one". A group is by definition two or more. How about this. Isn't it a bit redundant to speak of "one apple", when it is just an "apple". Seen this way, there is something unique about one. It's the only natural whose every integer power is itself. It's the only number considered to be neither prime nor composite.

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.

This brings me to the whole point of this meandering tour through number-space. If we agree that numbers less than one are small, shouldn't it follow that numbers greater than one are big? Perhaps this seems unnatural. However, any other choice does not seem to be mathematically defensible and seems based purely on human subjectivity.

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.

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 smallest large number, and also no largest 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.

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 "googolminex and one" 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 Graham's Number to get a googolminex and one! It's so absurd, and almost as much fun as coming up with "very large" large numbers :)

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 "superuniary numbers" or just the "superuniaries". 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 superuniaries instead.

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 :)

2 comments:

Jayk Tomaszewski said...

1.1

Unknown said...

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