ERG SES H 07, Sciences, Mathematics and Education
Negative numbers are one of the challenging concepts in mathematics education and many students face with problems regarding this concept (Beswick,2011;Vlassis,2008).The importance of the conceptual understanding of negative numbers is obvious and historical development of negative number concept would help to mathematics educators to prevent the difficulties students have on negative numbers (Gallardo,2001;Heeffer,2011).Therefore, the current study aims to investigate the historical development of negative numbers and indicate some suggestions regarding the teaching of negative numbers based on the historical review.
It is believed that negative numbers firstly occurred in the East, in Chinese texts.In the eighth chapter of ‘Nine Chapters of the Mathematical Art’ (in 250 BCE,one of the earliest mathematical texts in China),negative numbers were used for the solution of linear equations (Gallardo, 2002).These equations were developed to solve daily life problems and negative number concept is constructed on the “gain and loss” and “buy and sell” rationale.Apart from the Chinese texts, negative quantities put also in an appearance in Greek texts.In the book of Diophantus named Arithmetica (3rd century),he came across with an equation which has negative solution but he claimed that the solution is not meaningful and thus, changed the initial data (Gallardo,2002).This kind of thought can be a natural result of the fact that Ancient Greeks’ mathematics was founded on geometrical base and thus,Diophantus might have interpreted the negative solution as absurd (Rogers,2013).Another use of negative numbers could be seen in Indian texts.In the work of Brahmagupta (7th century),negative numbers were used for the idea of debts and moreover, he mentioned some rules to deal with the operations on positive and negative numbers (Schubring,2005).In another Hindu text, Vija-ganita (12nd century),Bhascara used negative quantities for loss and debts similar with Brahmagupta and differentiated negative numbers from the positive numbers by putting a point over the number (Gallardo,2001).In that work, Bhascara gives rules for the operations on positive and negative numbers including the division and multiplication (Gallardo,2002).Bhascara also treated second degree equations but when he encountered with a negative solution he did not accept it as a solution.For instance, in a problem regarding to find the number of monkeys he refused the negative result while he accepted the negative solution for the length of a line segment by interpreting the negativity would change the direction of the line (Gallardo, 2002;Schubring,2005).Therefore, it is possible to say that the negative results were not accepted in these texts unless they could be interpreted in a meaningful way within the problem.In the 9th century, Al Khwarizmi came up with the negative solutions to the linear or quadratic equations but he interpreted such solutions as meaningless since his solution was based on geometrical diagrams (Rogers,2013). However, he accepted negative quantities as debt on the problems regarding the inheritance (Rogers,2013).
Apart from the Diophantus’s Arithmetica, negative quantities does not seem to have a place in European context until 15th century but then, with the effect of ancient Islamic and Byzantine translations, negative numbers took place again in European mathematics (Rogers,2013).In the Viete’s book of Isagoga(1591),the operations of algebra were introduced and thanks to strong beliefs about the correctness of these operations, solutions of algebraic equations were accepted as correct even if they were consist of negative numbers (Heeffer,2011). In the 17th and 18th century many mathematicians were working with negative numbers but the fully meaning of this concept was not formed yet (Schubring,2005).The fully conceptualization of negative numbers occurred in the 19th century: in 1821 Cauchy showed positive numbers with plus sign and negative numbers with minus sign and with Hankel in 1867, negative numbers reached to the status as we know today (Vlassis,2008).
Beswick, K. (2011). Positive experiences with negative numbers: Building on students in and out of school experiences. Australian Mathematics Teachers, 67 (2), 31-40. Galbraith, M. J. (1974). Negative numbers. International Journal of Mathematical Education in Science and Technology, 5 (1), 83-90. Gallardo, A. (2001). Historical-epistemological analysis in mathematics education: Two works in didactics of algebra. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspective on School Algebra (pp. 121-139). Dordrecht, The Netherlands: Kluwer Academic Publishers. Gallardo, A. (2002). The extension of the natural number domain to the integers in the transition from arithmetic to algebra. Educational Studies in Mathematics, 49, 171-192. Heeffer, A. (2011). Historical objections against the number line. Science and Education, 20, 863-880. Rogers, L. (2013). The history of negative numbers. Retrieved March 3, 2013 from http://nrich.maths.org/5961 Schubring, G. (2005). Conflicts between generalization, rigor, and intuition. USA: Springer Science Business Media. Vlassis, J. (2008). The role of mathematical symbols in the development of number conceptualization: The case of the minus sign. Philosophical Psychology, 21 (4), 555-570.
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