Posted 8/25/2005 10:27
PM Putting numbers in their place
Q: Your recent article on socalled Arabic
numerals is all well and good. However, the Indian system of number
symbols is of no real significance to the history of mathematics.
The Indian system of numeration is significant indeed. Please tell
us more [and he elaborates in his email]. (Andy, Cambridge,
Massachusetts)
A: Gladly. Let's start with a quote from
the great 18thcentury mathematician, Pierre Simon Laplace:
"The ingenious method of expressing every
possible number using a set of ten symbols (each symbol having a
place value and an absolute value) emerged in India. The idea seems
so simple nowadays that its significance and profound importance is
no longer appreciated. Its simplicity lies in the way it facilitated
calculation and placed arithmetic foremost amongst useful
inventions."
The Indians devised a placevalue system that
used powers of 10 (base 10) to define the place values. Numbers
stood for different values depending upon their position.
For example, in the dollar amount, $5693, the 5
stands for the highest value since it's the farthest left. It stands
for 5000, which is 5 times 1000 — a power of 10. Thus, we express in
neat shorthand — by position only — the amount $5693 as a sum of
powers of 10. We can write the amount out the long way and express
its meaning fully:
5*1000 (5 thousands of dollars)
6*100 (6 hundreds of dollars)
9*10 (9 tens of dollars)
3*1 (3 single dollars)
= $5693
The long way shows what we have actually done:
counted by thousands and got 5, counted by hundreds and got 6,
etc.
We see the placevalue utility by comparing
this neat system with the former British monetary system:
5 pounds + 6 shillings + 9 pence + 3 halfpence
DOES NOT = 5693 any things.
The British were "stuck" with this enumeration
system until Feb. 15, 1971 (a scant 34 years ago), as Reader Andy (mathematics professor at Harvard
University) points out in his email. "Since 20 shillings made a
pound and 12 pence made a shilling, it was hard to keep accounts in
business."
Or anything else. Consider the problem of
figuring out what two can openers cost. Thirtyfive years ago a Brit
would think: Suppose one can opener costs 2 pounds (£2), 13
shillings (13s), and 8 pennies (8d). Then two can openers
cost...
2* (£2 13s 8d) =
£4 26s 16d =
£4 (20 + 6)s (12+4)d =
£5 6s (12+4)d =
£5 7s 4d
"Incidentally, it is worth noting," says Andy,
"that the United States led the world in switching to the decimal
system of coinage." In 1786, Congress adopted a decimal monetary
system based on the dollar and in 1792 we built the mint. The
French, however, got the idea rolling in the late 1700s and adopted
powersof10 units (the metric system) in 1795.
The Indians invented their placevalue system
at least as early as 594 AD, which is the date of the oldest Indian
document (a legal form) using place values.
They may have gotten the idea from the
Babylonians, who were the first to use place values — 2200 years
ago. The Babylonian system, however, was based on powers of 60 and,
therefore, not so convenient as the Indian system.
We don't know who invented the zero symbol. The
Indians referred to it as sunya, meaning "void" and used it in their
numeration system. In 628, however, the Indian mathematician, Brahmagupta, wrote the first (known) text to treat
zero as a number.
Arab scholars added the concept of decimal
fractions to the Indian system.
The Indian place value system soon spread to
the rest of the world — first to China and Alexandria and then to
the Arab empire by the 700s. The system finally made it to Europe.
By the 1500s, almost all Europeans used it. Although some held out.
In the early 1700s, "the last significant case of an attempt to
abolish the Indian decimal place value system was in Sweden," says
mathematician Ian Pierce of the University of St Andrews,
Scotland.
Further Reading:
WonderQuest: Counting
to 60 by finger joints
University of St Andrews, Scotland: Indian numerals by JJ O'Connor and EF
Robertson
University of St Andrews, Scotland: Decimal numeration and the placevalue system by
Ian Pierce
Wikipedia: Brahmagupta
(Answered Aug. 26, 2005)
April Holladay, science journalist for
USATODAY.com, lives in Albuquerque, New Mexico. A few years ago
Holladay retired early from computer engineering to canoe the
floodswollen Mackenzie, Canada's largest river. Now she writes a
column about nature and science, which appears Fridays at
USATODAY.com. To read April's past WonderQuest columns, please check
out her site. If you have a question for April, visit
this informational page.
