Look around you. Computers and networks are everywhere, enabling an intricate web of com-
plex human activities: education, commerce, entertainment, research, manufacturing, health
management, human communication, even war. Of the two main technological underpinnings
of this amazing proliferation, one is obvious: the breathtaking pace with which advances in
microelectronics and chip design have been bringing us faster and faster hardware.
This book tells the story of the other intellectual enterprise that is crucially fueling the
It is a fascinating story.
Gather ’round and listen close.
Books and algorithms
Two ideas changed the world. In 1448 in the German city of Mainz a goldsmith named Jo-
hann Gutenberg discovered a way to print books by putting together movable metallic pieces.
Literacy spread, the Dark Ages ended, the human intellect was liberated, science and tech-
nology triumphed, the Industrial Revolution happened. Many historians say we owe all this
to typography. Imagine a world in which only an elite could read these lines! But others insist
that the key development was not typography, but
Today we are so used to writing numbers in decimal, that it is easy to forget that Guten-
berg would write the number 1448 as MCDXLVIII. How do you add two Roman numerals?
What is MCDXLVIII
DCCCXII? (And just try to think about multiplying them.) Even a
clever man like Gutenberg probably only knew how to add and subtract small numbers using
his Fngers; for anything more complicated he had to consult an abacus specialist.
The decimal system, invented in India around AD 600, was a revolution in quantitative
reasoning: using only 10 symbols, even very large numbers could be written down compactly,
and arithmetic could be done efFciently on them by following elementary steps. Nonetheless
these ideas took a long time to spread, hindered by traditional barriers of language, distance,
and ignorance. The most in±uential medium of transmission turned out to be a textbook,
written in Arabic in the ninth century by a man who lived in Baghdad. Al Khwarizmi laid
out the basic methods for adding, multiplying, and dividing numbers—even extracting square
roots and calculating digits of
. These procedures were precise, unambiguous, mechanical,