Really big numbers
In the book Innumeracy, John Allen Paulos describes our problems in dealing with big numbers. I agree that we can, and should, learn to distinguish easily between a million, a billion, and a trillion. However, some numbers are just too large to grasp. We can’t really comprehend a number as large as a googol , or 10^100. And I don’t really know of any useful analogies that would allow me to compare the quantity represented by a googolplex, 10^(10^100), to that of a googol. These quantities, and the difference between them, are just so vast that we can’t really wrap our minds around them. There are only 10^78 or so atoms in the universe, and even that is a number which we cannot really fathom. But large numbers do pop up even when you ask relatively simple questions in mathematics.
For instance, in how many ways can you arrange a full deck of cards? You can easily show that the answer is 52 x 51 x … x 1, or 52! We can use Stirling’s approximation to estimate this number as sqrt(2*pi*52)(52/e)^52 which is approximately 8 x 10^67. This number is getting close to the number of atoms in the universe. Indeed, whenever you shuffle a deck of cards well, you can be nearly certain that nobody in the history of the world has played with the same arrangement before.
Large numbers appear frequently whenever we count the number of possible ways of arranging different objects. You have probably heard that if you put a number of monkeys in a room and let them pound away on typewriters, they will eventually produce all the works of Shakespeare. This may be true, but the amount of time it would take is not really comprehensible.
Let us look at a related question: In his story “The Library of Babel“, the Argentinian writer Luis Borges described a strange universe that consists of hexagonal cells lined by shelves full of books. Each book is unique and consists of different permutations of 25 letters and symbols. The narrator reveals that any possible arrangement of these 25 characters is contained somewhere in the library.
Borges writes that therefore:
Everything: the minutely detailed history of the future, the archangels’ autobiographies, the faithful catalogues of the Library, thousands and thousands of false catalogues, the demonstration of the fallacy of those catalogues, the demonstration of the fallacy of the true catalogue, the Gnostic gospel of Basilides, the commentary on that gospel, the commentary on the commentary on that gospel, the true story of your death, the translation of every book in all languages, the interpolations of every book in all books.
Taking into account Borges’ description of each book we can compute exactly how many of them are contained in the library: Each book has 410 pages, each page 40 lines, and line 80 characters. Therefore there are 410 x 40 x 80 = 1,312,000 characters in each book. The Library of Babel therefore consists of 25^(410x40x80) = 25^(1,312,000) books. (You can also check here for a more detailed explanation, or go here to download Borges’ number). I was trying to come up with a way to illustrate how large the Library or Babel. But I don’t think there is a useful analogy – the best we can do with numbers this big is to write them down. Perhaps my imagination is lacking, but I don’t see any way to illustrate their vastness other than essentially saying “This number is really, really big” in a different way.
This is pretty weird stuff. But these numbers are not really all that big. You can easily come up with much bigger numbers yourself. For example, you can order operations in an ascending order as follows: Repeated summation leads to multiplication, and repeated multiplication leads to exponentiation. Next repeated exponentiation give you towers of powers. You can keep on going, and soon you’ll reach operations that quickly produce immense integers (check out Ackerman’s notation to get a handle on how to write such numbers).
Still we have not come even close to infinity. Here is one way to see how far we are: We know that irrational numbers have a series of non-repeating digits. But there are different types of irrational numbers. For instance, you can just arrange 1s and 2s in a non-repeating pattern. Or you can take .1010010001000001000001… Since this expansion is not eventually periodic, this is an irrational number. However, the arrangement of digits in this number seems non-random. When we think of an arbitrary irrational numbers, we have a notion that the appearance of any digit, and group of digits, should be equally likely. Irrational numbers that have this property are called normal (I will actually consider numbers that are normal in any base. These are sometimes called absolutely normal numbers). More precisely, in the decimal expansion of a normal irrational number, each digit, each pair of digits, each triplet and so on appear with equal frequency. Interestingly, while we do know that almost all irrational numbers are normal, we don’t know whether specific irrational numbers are normal. Nobody knows whether pi or sqrt(2) are normal.
As a consequence, any and all sequences of digits appear in the expansions of normal numbers: Suppose that you have a normal number in base 25. We can map each of the numbers in this base to one of 25 characters that fill the books in the Library of Babel. That means that any normal number contains the text of all books in the Library of Babel. To see this, expand the number in base 25, convert the numbers to the corresponding characters, and break the expansion into blocks of 1,312,000 letters. Each such block corresponds to a book. Indeed, each book from the Library, and the entire Library itself appear in the expansion infinitely many times.
Let me finish with a last point that I owe to Lawrence Krauss. I recommend his book entitled A Universe from Nothing. Krauss talks a lot about quantum fluctuations which are present even in a vacuum. They allow particles to pop in and out existence all the time. Indeed, Krauss proposes that our entire universe may have simply popped into existence out of nothingness. Even more bizarrely, it is possible that any arrangements of particles may pop into existence out of nothingness, and disappear in the next instant. If you allow for infinitely many possibilities of this type, it means that we, along with all our memories, could be nothing more than a quantum fluctuation that exists only for a moment and disappears. Indeed, this may be a much more likely possibility than the alternative that we truly exist continuously in time.