Today, big numbers are an old source of
fascination. 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 81 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.
(Theme music)
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.
The Engines of Our Ingenuity is
Copyright © 1988-1999 by John H.
Lienhard.
