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Putting numbers in their place

Q: Your recent article on so-called 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 18^th-century 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."

Old coins, some using the place value system and others not. [Courtesy Corel Corporation]

The Indians devised a place-value 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 place-value 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. Thirty-five 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 powers- of-10 units (the metric system) in 1795.

The Indians invented their place-value 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 place-value system by Ian Pierce

Wikipedia: Brahmagupta

(Answered Aug. 26, 2005)