An Early Appearance of Nondecimal Notation in Secondary Education
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Jemma Lorenat, Editor
An Early Appearance of Nondecimal Notation in Secondary Education FRANKA MIRIAM BRUECKLER VLADIMIR STILINOVIC´
AND
Years Ago features essays by historians and mathematicians that take us back in time. Whether addressing special topics or general trends, individual mathematicians or ‘‘schools’’ (as in schools of fish), the idea is always the same: to shed new light on the mathematics of the past. Submissions are welcome.
â Submissions should be uploaded to http://tmin.edmgr.com or sent directly to Jemma Lorenat, e-mail: [email protected].
ositional number systems have a long history. Although decimal notation has been the most widespread number system since the Middle Ages, the oldest known positional number system was that of the Babylonians, who developed a combination of decimal and sexagesimal (base 60) notation during the reign of the third dynasty of Ur (end of the third millennium B.C.E.) [Chrisomalis 2003, Ifrah et al. 2000, Medic´ 2011]. The sexagesimal number system has persisted to the present day in time and angle measurement, but its aspect as an alternative positional number system is usually not emphasized in school textbooks. Today’s secondary school students usually learn about the binary and hexadecimal number systems because of their application in computer science, but such exposure was far from usual before the mid-twentieth century. However, elements of the binary system can be found in ancient Egypt and other ancient systems of weights and measures. Some of these early number systems, in particular the Babylonian and Mesoamerican, were positional systems with a base different from 10, but lacking 0 and not using our digit symbols 1, 2, … (Chrisomalis 2003). The first nondecimal positional system in modern notation (using B digits 0 to B - 1 if the base is B) was introduced in 1703 by Gottfried Wilhelm Leibniz in his Explication de l’Arithme´tique Binaire (Leibniz 1703) (he had developed the idea some years earlier), even if he did attribute the discovery of this system to the ancient Chinese (I Ching) [Glaser 1981, Lande 2014, Strickland 2007]. Leibniz’s work later became important for the development of Boolean algebra and further applications in computer science [Biggs 1979, Glaser 1981, Lande 2014]. Although numerous scholarly works on the binary number system (and others) have been published since the eighteenth century, it was not until the beginning of the twentieth century that such number systems appeared in secondary education textbooks (Glaser 1981). In our ongoing research into mathematics textbooks that were used in central Europe during the eighteenth and nineteenth centuries, we came upon an eighteenth-century secondary-level textbook that is exceptional in that it appeared only seventy years after Leibniz first published his description of the binary number system. In contrast, although decimal fractions were introduced in Europe by Simon Stevin in 1585, they rarely appeared in arithmetic books of the seventeenth and eighteenth centuries, for at that
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