Today, we seek nothing. 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.
In his book, The Nothing
That Is, Robert Kaplan tells the history of
zero. It's tricky because we have to simultaneously
see zero as a symbol, a mathematical concept, and a
metaphor. How to capture all those contradictory
roles?
Even today, it might seem confusing that the symbol
for nothing is what we use to make numbers
very large. "I'll give you two hundred dollars for
your horse," says farmer A. "Add a zero," says
farmer B, "then we can talk business."
So Kaplan deals with very-large numbers as well as
with the absence of number. When Archimedes wanted
to show how large numbers might get, he began with
a known Greek word, myriad. Myriad meant ten
thousand: a number that no one yet had any way to
write out.
Archimedes said, imagine a myriad grains of sand
making up a pile the size of a seed. Now imagine a
pile of seeds the size of your finger. A myriad
finger-widths is a tenth of mile. How many of those
would make up the diameter of Earth; how many
Earths, a universe? Archimedes arrives at a number
of grains of sand equal to a one followed by
sixty-three zeros. But, without a zero, he can only
recite this tortuous assembly of grains of sand.
For a zero to take its place among numbers, it had to
sneak up on people. By six hundred AD, the
mathematician Brahmagupta in India was clearly
asserting that any number subtracted from itself
was zero. He also struggled with adding,
multiplying — even dividing by zero. At the same
time, Mayas in the western hemisphere had many
symbols for zero. They even had a god of zero — a
god of death whom they used in some their more
ghastly rituals.
India took up the use of a dot to signify zero. The
concept eventually seeped out of India. It seems
to've traveled the Silk Road to the Arab world, in
the tenth century. There it mutated from a dot into
a circle. And the Arabs learned how to make it a
functional part of arithmetic and algebra.
But the transfer of ideas between Europe and the
Middle East was about to founder on the Crusades.
And, by the twelfth century, William of Malmsbury
was still writing about algebra as dangerous
Saracen magic. Zero wouldn't get full use in
Europe until the fifteenth century. Even then it
kept setting traps for us.
Divide zero by itself and you can get any number.
Multiply it by anything and it grows no larger.
Kaplan offers a tongue-in-cheek riddle: If a total
of four people occupy a room and seven people
leave, how many must enter before the room is
empty? (The answer, of course, is three.)
He ends with octopuses: If he says all octopuses
have nine tentacles, we need produce only
one with eight tentacles to disprove him.
But if he says every octopus in his room
has nine tentacles we can't disprove him, since the
number of octopuses in his room is zero.
Zero does indeed seem to toy with us. Yet without
zero, it is mathematics itself that would be
nothing.
I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
work.
(Theme music)
R. Kaplan, The Nothing That Is: A Natural History
of Zero. New York: Oxford University Press,
1999.
I am grateful to Lewis Wheeler, UH Mechanical
Engineering Dept., for lending me the Kaplan book
and for his counsel.
The Engines of Our Ingenuity is
Copyright © 1988-2003 by John H.
Lienhard.
