MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000

Introduction

Readers may wonder why the historical thread of the problem solving algebra begins with Simon Stevin. Stevin is often referred to as the inventor of the decimal fraction and thus he was important because he brought to common usage our method of adding, subtracting, multiplying, and dividing numbers that contain decimals. Stevin published La Disme in 1585 and with it he hoped to standardize number symbolism for everyday use different from methods developed by scholars in earlier times. Stevin’s standardization of arithmetic procedures allowed for common usage and understanding in commerce, banking, science, and engineering. (Swetz 411)

Stevin was a Flemish mathematician who was born in 1548 and died in 1620. He came from a wealthy family living in what is now Belgium and he was educated at Leiden, Holland. He was an engineer until 1604 when he became quartermaster-general of the Dutch Army while also serving as the Prince Maurice of Orange’s mathematics tutor and advisor on navigation and military matters. (Calinger 273)

Inspired by the then recent discovery of major scientific and mathematical ancient writings, especially Archimedes and Diophantus, Stevin included practical applications of his research by publishing a book of interest tables in 1582. Such tables previously were only available and understood by big banking houses. Following his pamphlet De Thiende (Tenth) in 1585 and the French translation, La Disme, as well as the book L’arthimetique, Stevin extended his interest by writing a major book on statics and hydrostatics engineering in 1586. ( http://www.newadvent.org ) In 1586 Stevin also published his report of a gravity experiment where two unequal weighted lead spheres were dropped. His work received scant credibility compared to Galileo’s report of a similar experiment three years later. (Encyclopedia Britannica)

Stevin’s De Thiende introduced to the world, in a short 29-page pamphlet, a simplified method of subtraction, addition, multiplication, and division. While his idea was certainly neither original nor the first used, the broad distribution and Stevin’s practical application of the method to commerce, science, and engineering brought some notoriety. (Swetz 407)

Stevin's Notation

Stevin may have borrowed from a decimal value notation of Bombelli, who used a half circle with a number to denote a geometric regression of ten from each place value to the right of an integer. Stevin instead used a circle with a number inside to represent the place value and also introduced i as the commencement, j as the prime, k as the second, and so on.

As an example, the decimal number we know as 125.739 would have been written as 125i 7j 3k 9l. Eventually Stevin reduced the decimal notation to just

125 739l. Although this system seems cumbersome, after further simplification by subsequent mathematicians, it reflects the way we conduct mathematics operations today including those that are used to solve algebraic equations.

Stevin’s decimal fractions eliminated the complicated fractions referred by him as vulgar fractions. Division was greatly simplified using Stevin’s decimal notation. From his De Thiende text he divided the number 3.44352 by .96. In his notation this was represented by:

Dividend 3i 4j 4k 3l 5m 2n

Divisor 9j 6k

By the vulgar division of whole numbers the value would be 3587, but where does the decimal point go? Stevin’s method subtracted the latter sign of the divisor k, from the latter sign of the dividend n. The result is the latter sign of the quotient or

3i 5j 8k 7l (Calinger 279)

In special cases of repeating decimals Stevin reasoned that the dropping of the most minute part of that which is being measured (such as .333…) is much more convenient than achieving perfection. (Swetz 416)

Historical Inspiration

Stevin’s publication L’arithmetique included arguments that opposed ideas from the Greeks. Inspired by the translation of Diophantus, he proposed that, one or unity, is a number and divisible and he rejected the idea of zero as a number but he used it freely in his definitions. (Struik 460) Countering Greek ideas of numbers, Stevin touted the idea that numbers are not distinct from magnitude and that number is a continuous quantity or in his words, "To a continuous magnitude corresponds the continuous number to which it is attributed." (Struik 501)

How did Stevin arrive at his use of decimal fractions? The Chinese apparently were using decimal place-value notation as early as 1400 B.C. The decimal system was a natural one with man’s anatomically suited digits. Other number based systems such as the Babylonian’s sexagesimal or base 60 system persisted through history. Our divisions of time, division of angles, and positioning methods for air, land, and sea navigation of the world are the best examples that remain of this system. Even the sexigesimal system still had portions that were based on a decimal system. Prior to the decimal fractions that Stevin popularized, mankind existed with cumbersome fractions that used any denominator or sexagesimal fractions. (Swetz 408)

Stevin’s contribution of the decimal fraction allowed computations with integers without fractions. Before decimal fractions the process of computation was convoluted, low-level and mechanical limiting the process to a few selected individuals. Algebra equations worked with decimal fractions became less thought provoking and allowed further progress in calculus.

Prior to Stevin’s attempt to popularize use of decimal fractions, mathematicians had used the Babylonian notation of numbers where a vertical wedge stood for 1 while the characters < and a vertical wedge combined with > signified 10 and 100 respectively. In addition, the Babylonian system discovered on a tablet dated to between 2300 and 1600 B.C., relied on a sexagesimal scale. The problem with using such a scale was an ambiguity in reading and computing with numbers that arose with a system that had no symbol for zero. A vertical wedge could stand for 1, or 60¹, or 60² or 60^-1. From one cuneiform tablet showing a mathematical table of computations it is relatively easy to understand the value of the numbers, but these numbers taken out of context of the table may have several meanings. As the symbols were written there is no way to discern or separate the fractional form from the integral parts of a number since each place value was a multiple of 60 from the next. Sexagesimal fractions were in use by the Babylonians from evidence of the same mathematical tablets. The process of dealing with such fractions must have been laborious through history and so Stevin’s easier approach to working with fractions appealed to many and particular in applications of algebra problems. (Cajori 2-10)

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