Simon Stevin Introduces the Arithmetic of Decimal Fractions

In today’s world, it is hard to imagine that there was a time in which people were not using decimal notation and decimal arithmetic. In this post, I will focus on a book published in Dutch by Simon Stevin in 1585, De Thiende, which was quickly translated into Latin, French and English. This is the same man who undertook the test of dropping stones of different weights in Delft, as I wrote in my post Galileo Galilei's and Simon Stevin's Tower Experiments. I will refer to the English version of his text, published in 1608, under the title Dime, The Art of Tenths¹.

Decimal numbers were introduced into Europe by Fibonacci [Leonardo Bonacci] (c.1170 – c.1242), who published in 1202 the Liber Abaci [The Book of the Abacus], and presented the modus Indorum [method of the Indians], which uses positional notation and the zero.

In this method, all numbers are written using the ten digits 0,1,2,3,4,5,6,7,8,9, and successive digits are written next to each other. For example, in the number 1234, that means 1 thousand, 2 hundreds, 3 tens and 4 ones, i.e., 1234 = 1000 + 200 + 30 + 4.

The digit 0 serves a dual purpose. All by itself, it means the value zero, i.e., nothing. However, if it is inserted to the right of a number that already has any of the other digits, that number is multiplied by 10; for example, 12340 is ten times the value of 1234, and 123400 is ten times the value of 12340. If the 0 is inserted in the middle of a number, the part to the left of the 0 is multiplied by 10; for example, 12034 is ten times 1200, plus 34. Should the 0 be placed to the left of a number, it has no influence on the value of the number; for example, 01234 is the same as 1234. Therefore, except for the number 0 itself, one does not normally begin a number with the digit 0.

This new—for Europeans—technique was picked up by the Florentine merchants, and made their calculations a lot more efficient, giving them an edge over those Europeans who continued to use the old, Roman number system. The new system had two clear advantages. First, there was no limit to the size of the numbers that could be expressed. Second, the algorithms for doing the basic calculations of addition, subtraction, multiplication and division were greatly simplified.

The question remained how to handle fractions. It is quite possible that people were secretly using some kind of positional notation for their own purposes. However, for public consumption, it was Stevin’s publication of De Thiende that provided a means for handling decimal fractions by only slightly adjusting the algorithms already used for ordinary decimal numbers.

Stevin allowed the insertion of circled numbers, called signs, such as ⓪,①,②,③, inside decimal numbers, in order to designate fractions. For example, the number 1234⓪5①6②7③, would mean 1234 wholes, 5 tenths, 6 hundredths and 7 thousandths, i.e., 1234⓪5①6②7③ = 1234 + 5/10 + 6/100 + 7/1000. In today’s notation, 1234⓪5①6②7③ = 1234 + 0.5 + 0.06 + 0.007 = 1234.567.

The names Stevin gave to the fractional signs were commencements, primes, seconds, thirds, etc. Hence 1234⓪5①6②7③ would be read as 1234 commencements, 5 primes, 6 seconds and 7 thirds. We still use similar terms for the sexagesimal notation of degrees of angle and their fractions. For example, 93º12'24''23''' is read as 93 degrees, 12 minutes, 24 seconds and 23 thirds, where a third is one sixtieth of a second.

For a small fraction, not all successive circled numbers need appear. Hence 6②7③ would mean 6/100 + 7/1000 = 67/1000. Similarly, 5①7③ would mean 5/10 + 7/1000 = 507/1000. As for 1234⓪7③, it would mean 1234 + 7/1000.

Stevin then presents how to adjust the existing algorithms for the four basic operations. We give some examples from his text.

Addition is undertaken by vertically aligning the digits of the same decimal fraction. Here is the addition 8.56 +5.07 = 13.63:

If the two numbers do not end with the same sign, then the number with the smaller last sign is padded with the appropriate number of 0s.

Subtraction takes place in the same manner as for addition. Here is the addition 237.578 - 59.739 = 177.839:

For multiplication, once again the digits are aligned vertically, and the multiplication takes place as if the signs were not there. At the end, the last signs of the two numbers are added together to produce the last sign of the result. Here ②+② = ④, and so 32.57 × 89.46 = 2913.7122:

For division, normal division is undertaken, as if the signs were not there, then the result consists of subtracting the last sign of the divisor from the last sign of the dividend to get the last sign of the quotient. Here ⑤-③=②, so 3.44342 ÷ .96 = 3.587:

Should the last sign of the dividend be smaller than the last sign of the divisor, then the dividend needs to be padded with 0s.

Once the above ideas are presented, Stevin examines different examples, ranging from measurements of different kinds (land, the human body, liquor vessels, etc.), to astronomical calculations and computations of money-masters and merchants. Clearly he was responding to the needs of the different trades and the growing merchant class in the Netherlands: his decimal fractions made everyone’s life easier.

The move to a full positional notation for decimal fractions would take place when John Napier (1550-1617) introduced the decimal point to separate the whole part from the fractional part of a decimal number, thereby removing the need for Stevin’s signs.

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1. Simon Stevin. Dime, The Art of Tenths, or Decimal Arithmetic, Teaching how to perform all computations whatsoever by whole numbers without fractions, by the four principles of common arithmetic, namely: addition, subtraction, multiplication, and division. Published in English with some additions by Robert Norton, Gentleman. In The Principal Works. Volume II: Mathematics. Edited by D.J. Struik. Amsterdam: C.V. Swets & Zeitlinger, pp.371-475.