1.10. Review I
REVIEW & SUMMARY FOR SECTION I
We have now finished section 1 of this web-book, where we've gone over a variety of topics. First we examined numbers in the English language, leading up to the -illions, how they came to be, and how they're used today. Then we went over the variety of large numbers that can occur in the physical world and in probability, as well as some famous large numbers around that range, namely the googol family and Skewes' numbers. After that we went over schemes people have devised to extend the -illions to outrageous heights, finishing this section off with Knuth's -yllions.
So are we ready to begin and explore the wonders that await in section 2? Not entirely: there are a few points I'd like to make before we are ready for section 2.
First off, think about what we've been using to generate all these large numbers—what tools have we been using to make large numbers? We've mostly made use of exponents to generate large numbers, not much else. With those we got to the largest number we envisioned in section 1, the confusing latinlatinmyllionalatinmyllionlatinmylliononeayllionalatinmyllionlatinmyllionalatinmyllionlatinmylliononeaylliononeayllion.
If you're new to large numbers, you may be wondering: are these the largest kinds of numbers humans can devise?! Nope, not even close! But why is that, aren't these numbers ridiculously mind-bogglingly huge? They may seem huge right now, but with the right tools those numbers will be totally left in the dust!
Now, what kinds of tools am I referring to? To make some REALLY big numbers you'll need to invent tools that are more powerful than those exponents and factorials! In fact, there are already many such tools. Several mathematicians have invented simple but powerful tools for generating large numbers mainly to give people a taste of what infinity is like. Some people however (Jonathan Bowers for instance), took such tools way to the extreme, generating large numbers way way beyond the tools mathematicians devised! This is the basis of googology, generating numbers as big as you can, for their own sake!
But the most important thing is that without tools for making large numbers, you would NOT EVEN COME CLOSE to making numbers googologists would consider mildly impressive. Hell, you wouldn't even be able to scratch the surface of Graham's number, which is itself humble among the truly enormous numbers googologists have studied!!!
Believe it or not, making tools to generate large numbers is easy if you know how. How exactly can you do that? The key to making large numbers is recursion. That is exactly what we'll go over next.
So we will start off section 2 with an article on what exactly recursion is, and then we'll start going over some of the tools people have devised to make large numbers, as well as some large numbers defined with those tools. Section 2 in particular is devoted primarily to the more well-known and simpler tools used to generate large numbers. Read on to continue and begin section 2.