2.3.2 - Imagining Big Numbers

2.3.2

Imagining Big Numbers

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1,2,3, ... Blast Off !

So we've learned the basics about numbers, we've seen how far they go in real life, and we have seen how far the ancients got. Now it's definitely time for us to simply apply what we know so far and see how far we can get with only the aid of our imaginations !

Most people understand that in order to make very large numbers, all you have to do is add alot of zeroes. Obviously 100 is greater than 10, 1000 is greater than 100, and so on. Most people are content to consider numbers like a million, a billion, and a trillion as quite sizable; and make no mistake they are. But we can certainly define much larger ones. Maybe you have cringed once in your life when you momentarily tried to comprehend what 1 followed by a thousand zeroes would mean ! Most people, after having this terrible insight mentally shut down. Why would anyone want to consider anything larger ! Heaven forbid ! Perhaps it's not that different than building the tower of babel.

But this is exactly what we want to do now. With our imaginations we can imagine tremendous numbers, or at least their decimal notations. Imagine starting with a " 1 ". Now imagine adding a zero to the right. Now we have " 10 ". Add another and we have " 100 ", another and we have " 1000 " , or one thousand. A million has six zeroes, which we can seperate with commas as " 1,000,000 ". A billion has 9 zeroes " 1,000,000,000 ", a trillion has 12 zeroes " 1,000,000,000,000 ", ... then what. Most people don't know the next "illion", but I actually mentioned two additional ones in Section I. Next would be a quadrillion, or 1 followed by 15 zeroes ... " 1,000,000,000,000,000 ", and then a quintillion, or 1 followed by 18 zeroes ... " 1,000,000,000,000,000,000 ". Okay now what...

Even if we don't know any more -illions, it doesn't mean we suddenly hit a brick wall. JUST KEEP ADDING ZEROES ! Remember that with each additional zero the number becomes 10 x larger ! Next we could consider 1 followed by 21 zeroes, 24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78, ...

but already we can jump ahead of all this. Why not just imagine 1 followed by 100 zeroes. This number has already been coined a "Googol". But why stop there? Let's keep making even larger numbers and go way past this with ... 1 followed by a thousands zeroes, no a million, a billion, a trillion, a quadrillion, a quintillion ... a ... a googol zeroes !!! hmmm. Now what?

Most people probably stop around here. They say to themselves something like "WOW ! billions of billions of zeroes, that's insane !" and then the light switches off. But for some the journey to infinity has only begun. For some a new thrilling and frightening possiblity is just beginning to dawn on them.

Why not continue in this way. Instead of merely increasing the number of zeroes, just take the largest number we know, ... and then have 1 followed by THAT ! So if we begin with a googol, we can have 1 followed by a googol zeroes. But now "1 followed by a googol zeroes" is the largest number we know so we can now have 1 followed by THAT! That's 1 followed by " 1 followed by a googol zeroes". Woah !? Crazy right? But we can repeat it again and have 1 followed by " 1 followed by " 1 followed by a googol zeroes " ". At this point it usually gets alittle tricky to hold in your head. a googol is 1 followed by a hundred zeroes, 1 followed by a googol zeroes, then have that many zeroes, then have that many zeroes ... hmmm. But wait we can get further than this ...

Just think of 1 followed by googol, as the first step. Then 1 followed by " 1 followed by a googol zeroes" as the second step. Then 1 followed by " 1 followed by " 1 followed by a googol zeroes" as the third step. Now go to the millionth step. WOAH! Mind boggling, but wait, lets go to the billionth, I mean trillionth, no googolth step. Or better yet the 1 followed by a googol zeroes step, or or ... hmm.

Wait let's go to 1 followed by " 1 followed by " 1 followed by a googol zeroes step""... hmm. How can we push further. Wait that is only the third step of our sequence, right? So we are going to the "3rd step" step. So we could in theory go to the "100th step" step, or the "googolth step" step, or the "1 followed by a googol zeroes step" step. But again that's the first step so really that's the same as " " 1st step" step " step. How about the "" ... " googolth step" step " ... " step " step where you say the word step a googol times !! Getting dizzy yet ?! 0_o;

But wait ... I can go further ... let's see, say step "1 followed by a googol zeroes" times. No wait say step 100 times, then that is the number of times you can say step. Now better yet let's Start with the "" .. "" googolth step" step " step " ... step " step, with a googol copies of step. Call that the first super step, next have a number of copies of step equal to the first super step, that's the second super step ! Now we can have a googolth super step, or a " googolth super step" super step... wait a the first super super step would be "googolth super step" ... super step with a googol copies of super step. Then continue the process ... so then the first super super super step would be "googolth super super step" ... " super super step with a googol copies of super super step... then we could have the first super super super ... super super super step with a googol copies of super ...

wait let the number of supers determine the number of supers , then you have levels of that which you again can loop around, then you can do it over and over, and they'll be like some kind of second level of supers, then a third, then a googoloth, then you can have something new to count that and then and then ... more levels of levels and then , then ... um ... my mind went blank -_-;

Mad Number Rants : What just happened ?!

Some of you, especially those of you who have never given big number THAT MUCH THOUGHT, are probably wondering , what the heck was that about? If you were confused by the last paragraph, or even the last few that's fine. If it sounded like utter madness to you, that's also fine. That was the point I WAS trying to make. It get's very confusing and insane. It really is just like building the tower of babel. Just as it seems your about to eclipse infinity every attempt at explaination fails and your left babbling. At the end I just give up trying to make sense of it and just race on ahead. If you have ever thought along these lines you know what I'm talking about. Creating large numbers can become very addictive. You get an adrenaline rush, ideas start racing, ... you feel like your always trying to outrun yourself ... you want to jump ahead ... way ahead of anything you could easily think up ... and yet you can't. You can only imagine what your capable of imagining. Generating large numbers is not a linear process, not even an exponential one, it is just always accelerating ahead of your own expectations. You can never preempt it. How this happens is actually something of a mystery. It's a curious fact that humans CAN imagine things like this even if they can't comprehend what they are talking about. At least they can talk about it, while animals would be utterly mystified by such pure abstraction.

Notice that I used very little real mathematics in my demonstration. Mostly I used words to express my ideas. In fact there was no arithmetic involved. All I really asked you do to was imagine alot of zeroes and then extrapolate that process ... over and over again ... ad infinidum. Although it wasn't really infinite was it. In fact I got my mathematics tied into a nice little knot after a mere 8 paragraphs. I most likely lost quite a few readers because I was extremely vague at the end. This is common for large number enthusiasts. Sometimes it's even hard for us to understand each other ! An idea might seem quite clear in your own mind, but unless you have some way to convey it intelligably others will not understand you. This is what we can call the "mad number rant". These number rants can be fun, but they often leave you in a confusing place mentally. Suddenly you don't know up from down and no sense of scale seems appropriate to the discussion, and none is. Even in my little demonstration I actually went a good way up a few levels of large numbers, ... but not quite as far as you might think. People, and by people I mean amateur mathematicians, have gotten much further than this. But HOW ?! It's already so damn confusing ?!

Well although some of you probably don't want to hear this, we need math to get further. We need a way to properly organize these ideas, give them structure and context. Otherwise the thing builds and builds until suddenly you can't remember the whole process and you go blank. The light bulb switches off and the train that was racing towards infinity suddenly fades out of existence. This is probably why people usually only get so far with these thoughts. Unless your prepared to start writing down some definitions these thoughts can very rapidly spin out of your control. As a matter of fact these are exactly the sorts of recursive thoughts that used to run through my head all the time, but they would always slip out of my grasp and I wouldn't be able to recreate my sudden epiphany.

In Section III we are going to try to tackle this problem again. But this time we will be using the theory of algorithms to help us make sense of it. As you will see, algorithmic methods are VERY powerful, and will allow us to tackle thoughts of even greater complexity than the ones just considered. And better yet, the thoughts we just had will become much more clear and concise.

In the meantime however, we will try to make sense of some of this mess in another fashion. In the next chapter we will be considering the -illion numbers and how to construct vast naming schemes. We will also be introduced to a popular large number enthusiast, Jonathan Bowers, who has created a great deal of -illion numbers himself.

There is a reason why large number ethusiasts like naming large numbers. It's because names can serve as place holders and provide a lose kind of structure. As you will see in the next chapter, we will be able to use the -illions, and alittle bit of simple arithmetic to make more sense of our journey towards the infinite.