Today, big numbers find new meaning. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.
Large numbers are an old source of fascination. However, they're raising interesting new issues these days. Try this question: How many numbers can your 32-bit PC deal with? It's a very powerful machine. Yet it can manipulate only a number equal to two raised to the power of 32. That gives us only the numbers one to about four billion - a four followed by nine zeros.
Compare that with the universe, whose diameter in miles is roughly a two followed by twenty-three zeros. In other words, if you tried to use your computer to list locations along a line through the universe, the points would have to be placed four trillion miles apart. Chances are you'd miss most our solar system completely, even if it lay on the line. If you were to graph the three-dimensional universe, labeling points one foot apart, you'd need numbers all the way up to a one followed by 53 zeros.
But we don't have to go to the cosmos to find big numbers. The number of air molecules in your office is like a one followed by thirty zeros. Now think about this: when physicists set out to predict the behavior of those molecules, the first step is to count the number of ways their speeds and locations can be rearranged. And here we reach one of the 20th-century frontiers of big numbers.
Think about arranging books on a shelf. If we have only one book, of course there's only one way to arrange it. If we have four books, there are 24 ways. Increase the number of books to only ten, and we can find almost four million ways to distribute those books on the shelf. When we come to arrangements of air molecules around us, that number would be so long we couldn't begin to write it down if we filled every piece of paper in the world.
So we find ways of approximating, or of focusing on, parts of big collections of things or of long calculations. Mathematicians have created all kinds of means for going far beyond our ability to count, and they constantly look for more.
Now the 21st-century frontier for huge numbers will lie in questions about connectivity. Think how words like web and network infuse our lives these days. We can find a trillion ways to connect two million telephones with one another. And if each phone can teleconference with four others, that number goes through the roof.
And so we reach the most important network of all. A trillion axons within the human brain offer vastly more than a trillion ways to look at things. For all practical purposes, that number is uncountable. The saving irony in all this is that our capacity for finding new ways to deal with inconceivably large numbers is itself inconceivably large.
So our capacities are much greater than we realize. As the meaning of large-number complexity becomes clearer, I'm betting that we'll ultimately find ways to harness our own brains in ways we haven't yet imagined were there to harness.
I'm John Lienhard, at the University of Houston, where we're interested in the way inventive minds work.
I am grateful to N. Shamsundar, Mechanical Engineering Department, University of Houston, for steering me toward the topic and for his counsel.
The statistical description of the air in your office is developed in any text on statistical thermodynamics. See, e.g., Tien, C-l. and Lienhard, J.H., Statistical Thermodynamics. New York: Hemisphere Pub. Corp., 1979 (See especially Chapters 2 through 5.)
About putting a number of books equal to N on a shelf: There are N ways to place the first book in the row. Then there are only N-1 ways to place the second book, N-2 ways to place the third, and so forth. Thus we can place the first two books in N(N-1) ways, the first three books in N(N-1)(N-2) ways, and all N books in N! = N(N-1)(N-2) Ö (2)(1) ways. N! is the symbol for this product; it's called "N factorial." If we evaluate N!, we get:
1! = 1
2! = 2
3! = 6
4! = 24
10! = 3,628,800
20! = 2,432,902,008,176,640,000, etc.