The Birth Of Decimal Fractions

Salisbury University

Our system of numeration is called a decimal system of numeration, and it is a positional system with so called place-value positions.  Mathematicians and scientists routinely use decimal numerals and decimal representations of fractions. Students in elementary school learn that ¼ and .25 are equivalent quantities in an early grade.  The use of hand-held calculators has made it necessary to understand these equivalences.  But how many people know the origins or history of our decimal system of numeration and the use of decimal representations of fractions?

In the last few years, there has been an emphasis on using history to teach mathematics (Katz, 2000; Wilson et. al., 2000).  There are many benefits from this approach, ranging from connections to society to a deeper understanding of mathematics.  Traditionally, although much of the history of mathematics seems to have been focused on the contributions of Western mathematicians, recently there have been efforts to be more inclusive in highlighting the contributions from other cultures.

The contributions of Persian mathematicians have been significant, particularly of those mathematicians who wrote in the Islamic era (about 6th century to 17th century).  Even the origin of our word "algebra" came from this period ( Michalowicz, 1993).

It is worth pointing out that in Western cultures, the Middle-Eastern mathematicians during the Islamic era are often referred to as "Arab" mathematicians.  Indeed many of them were Arab mathematicians, but not all.  It is a type of Venn-diagram argument.  A large percentage of Arabs are Islamic people, but also a large percentage of Islamic people are not Arabs, as is the case with the Persian people.  So these Persian mathematicians who wrote during the Islamic era were Islamic mathematicians that were non-Arab mathematicians.  There are two such Persian mathematicians associated with the decimal system of numeration and the use of decimal fractions.  They are Kharazmi and Kashi. Kharazmi's work has been associated with the Hindu system of numeration, and Khasi's work is significant in the use of decimal fractions.

The book Al-Jama’-val-Tafrigh, that was authored by “Kharazmi,” a Persian Mathematician between 780-850, is the earliest available arithmetic book that uses the Hindu numerals (Karpinski, 1910-11; Encyclopedia de L’Islam, p.1163). In writing this book Kharazmi combined his thorough knowledge of Persian mathematics with his knowledge of Indian mathematics, particularly the Hindu system of writing numbers.

The original copy of this book is not available.  According to several Latin translations of this textbook made in Europe in the twelfth century, Kharazmi first introduced (to the rest of the world) in this book nine characters to represent the first nine positive integers and a circle to represent zero (Boncompagni,1857; Katz, 1993). Laying a foundation for the present place-value system, he showed how to express any number using the characters that he had introduced.  Kharazmi also designed algorithms that demonstrated how to use these numbers and perform the first four fundamental operations (addition, subtraction, division and multiplication), halving, doubling and extraction of square roots. He explained and demonstrated all of his algorithms by providing real-number examples.  The decimal fractions are one of the most important parts of the place-value system, but Kharazmi did not discuss this subject; he focused on the sum of common fractions instead.

Research in the history of decimal fractions reveals their usage in Al-fusool-fel-Hesab-al-Hendi, a book that was authored by a mathematician “Ahmad Ughlidusi” in 962 (Ghurbani, 1971).  The use of decimal fractions can also be found in the work of another mathematician “al-Samaul", also known as "al-Maghrebi,” who lived in the time period of 1125-1180 (Ghurbani, 1986). Both Ughlidusi and Maghrebi used the decimal fractions in their works, but they were seemingly unaware of the importance of the decimal fractions.  They used decimal fractions in the same manner as the rational fractions and those of sexagesimal (base sixty) system.  They worked with them without much insight-- like persons who had learned something from some prior conditioning that had provided little meaning to what they were doing.  Ughlidusi and Maghrebi treated decimal fractions as an ad-hoc device beside the rational fractions.  Because they themselves were unaware of the significance of their work with decimal fractions, nobody became aware of their work and of the importance of the decimal fractions. As a result, the decimal fractions were missing from the number system from the time of Kharazmi until the fourteenth century, a period of about six hundred years.  It was then that Kashi, a Persian mathematician of the fourteenth and fifteenth centuries, reintroduced them and completed the place-value system in that he used the place positions to the right of the decimal point (Ghurbani, 1971; Luckey, 1951).

Kashi was a great Persian mathematician and a very well-known astronomer.  He discovered the importance of the decimal fractions, used them, and recommended their use to others. In this way the use of the decimal fractions became popular.

Kashi calculated the value of 2π as 6.2831853071795865, a value that remained unchanged for more than 150 years (Ghurbani, 1986; Ghurbani, 1954). Additionally, he calculated the sine of a one-degree angle by solving a third degree equation using an iterative method.  Kashi obtained the result that

sin 1° = 0. 0174524064372835103712

an approximation that agrees with the actual value for seventeen decimal places (Ghurbani, 1986).  Because we are so familiar with the decimals fractions, it is difficult for us to imagine these results without using decimal fractions for our representations.

Although some other mathematicians before Kashi had offered other methods for evaluating the nth root of a number, Kashi believed that he was the first person to give a complete and general solution to the problem (Ghurbani, 1971; Luckey, 1948).  He stated that the methods employed by others before him were excessively complicated, especially when the nth root of a particularly large number is desired.

Kashi gave the nth root of a number as follows:

Example of Kashi’s formula

The square number 25 can be expressed as:

25 = 42 + 9

Let x = 4, r = 9, and n = 2; then:

If one were asked to approximate the fifth root of a rather large number, say 44240899506197 using Kashi's algorithm, he/she would be perplexed until it was discovered that this large number is equal to

5365 + 21. If we let x = 536 and r =21, then it follows from Kashi's approximation that

An interesting exercise for the reader is to discover how good this approximation is! That is, for how many decimal places does this result agree with the actual value?  It is difficult to imagine thinking of the accuracy of an approximation without relying on decimal fractions!

In conclusion, the introduction of decimal fractions into our decimal system of numeration was a significant contribution. Kharazmi and Kashi were two key players in the history of the use of decimal fractions in mathematics; however they were but two of the Persian mathematicians who, in the Islamic era with their writings, made mathematics available to a wider audience.  The following  table time-line summarizes the contributions of the preceeding mathematicians to the history of decimal numbers:

# Contributors to the Development of the Decimal Numbers

 Name Time Period Contributions Kharazmi: a Persian mathematician known as al-Kharazmi 780-850 Introduction of nine characters to represent the first nine positive integers and a circle to represent zero.  Plus algorithms for writing all numbers using those characters and working with them. Ahmad:  an Islamic mathematician known as Ughlidusi 962 He used decimal fractions besides the rational fractions and those of sexagesimal (base sixty) system.  (He showed 12.35 in the form of 12ˉ35 and he treated the decimals  as separate parts in multiplication. al-Samaul: an Islamic mathematician known as "al-Maghrebi,” 1125-1180[ The use of decimal fractions can also be found in his works, but he was seemingly unaware of the importance of the decimal fractions. Jamsheed Kashani: a Persian mathematician known as Jhias-al-din al-Kashi 1375-1429 He discovered the importance of the decimal fractions, used them the way they are used today, and recommended their use to others. In this way the use of the decimal fractions became popular.

References

Boncompagni, Baldassare (1857).  Loannis Hispalensis Liber algorismi de prctica arismetriceTrattati d’ arithmetica II.

Ghurbani, A. (1954).  A History of πSukhan Vol. 5.

Ghurbani, A.  (1986).  Islamic Mathematics from 9th to 17th Centuries.

Ghurbani, A.  (1971).  Kashani-nameh.  Tehran University.

[No author provided].  (1960-1983).  Encyclopedie de L’Islam Vol. III, Nouvelle edition.

Karpinski, L.C.  (1910-1911).  Robert of Chester’s Translation of the ‘Algebra of al-Khowarizmi’ Bibliotheca Mathematica, Vol. 11, p. 125-131.

Katz, Victor (2000).  Using History to Teach Mathematics, MAA Notes.

Katz, Victor (1993).  A History of Mathematics.  P. 228.

Luckey, Paul  (1948).  Die Ausziehung der n-ten Wurzel und der binomische Lehrsatz in der islamischen Mathematik, Mathematics Annalen, Vol. 120, pp. 217-274.

Luckey, Paul.  (1951).  Die Rechenkunst bei Gamsid b. Mas ud al-kasi, mit Ruckbliken auf altere Geschichte des Rechnens, Abhandlungen fur die Kunde des Morgenlandas, Vol. XXXI.

Michalowicz, Karen Dee.  (1993). A Bit of History:  The Origin of the Word AlgebraMathematical Connections.

Chauvot, Jennifer B., and Wilson, Patricia.  (2000).  Who? How? What?Mathematics Teacher, Vol. 93.