• Journal for History of Mathematics
• /
• v.16 no.1
• /
• pp.25-44
• /
• 2003

In this paper, we deal with the ‘dearithmetization’ of algebra. Historically, the origin of algebra comes from arithmetic. Also school algebra is related to arithmetic in general. However, we have many difficulties in teaching school algebra, and there is many problems for students to learn algebra from elementary arithmetic knowledges. This paper supposed that the solution of these problem may be founded in the ‘dearithmetization’ of algebra. And we supposed that the ‘dearithmetization’ of algebra may be developed by three historical achievements - the completion of symbolic algebra, the principle of permanence of form, and the expansion of number concepts. In order to justify these supposition, we investigate Peacock's ideas, i.e. ‘symbolic algebra’, ‘the principle of permanence of form’, and consider how the integer is introduced in modern mathematics. And we analyze various textbooks, and investigate the ‘dearithmetization’ of school algebra which has been progressed in the three fields - symbolic algebra, the principle of permanence of form, and the expansion of number concepts.

• Journal for History of Mathematics
• /
• v.21 no.4
• /
• pp.61-78
• /
• 2008

In this paper, we discuss about De Morgan's view on the development of algebra according to following distinctions: arithmetic, universal arithmetic, symbolic algebra, significant algebra. De Morgan thought that the differences between arithmetic and universal arithmetic lie in the usage of letters and the immediate performance of computation. In his viewpoint, universal arithmetic is a transitional phase, in which absurd phenomena occur, from arithmetic to algebra and these absurd phenomena call for algebra. The feature of De Morgan's view on the development of algebra is that symbolic calculus which consist of symbol system without symbol's meaning is acquired, then as extended meanings are furnished to symbols, symbolic calculus become logical so significant calculus is developed. For example, Single algebra is developed, as an extended meaning is furnished to a symbol -1, and double algebra is developed, as an extended meaning is furnished to a symbol $\sqrt{-1}$. According to De Morgan, a symbol system is derived from the incompleteness of a prior symbol system.

• Bulletin of the Society of Naval Architects of Korea
• /
• v.32 no.6
• /
• pp.39-45
• /
• 1995

• Journal of the Korean Institute of Telematics and Electronics
• /
• v.20 no.5
• /
• pp.1-7
• /
• 1983

Generally, multiple-valued logic algebra is based on the number system of modulo-M. In this paper, characters a, b, c‥… each of them represents the independent state, are regarded as the elements of the symbolic multiple-valued logic. By using the set theory, the symbolic multiple - valued logic and their functions are defined. And Varation for the symbolic logic function due to the variation of a variable and their properties are suggested and analized. With these variations, the MacLaurin's and Taylor's Series expansions of the symbolic logic functions are proposed and proved.

• Journal for History of Mathematics
• /
• v.20 no.3
• /
• pp.167-180
• /
• 2007

In this study, we explored the role of analysis in the algebra development. For this, we classified ancient geometric analysis into an analysis by reduction and a Pappusian problematic analysis. this shows that both analyses have the function of reduction. Pappus' analysis consists of four steps; transformation, resolution, construction, demonstration. The transformation, by which conditions of given problem is transformed into other conditions which suggest a problem-solving, seems to be a kind of reduction. Mathematicians created new problems as a result of the reductional function of analysis, and became to see mathematics in the different view. An analytical thinking was a background at the birth of symbolic algebra, the reductional function of analysis played an important role in the development of symbolic algebra.

• Journal for History of Mathematics
• /
• v.25 no.4
• /
• pp.69-82
• /
• 2012

The objective of this paper is to study De Morgan's perspective on teaching and learning complex numbers. De Morgan's didactical approaches reflect the process of development of his thoughts about algebra from universal arithmetic, symbolic algebra to meaning algebra. De Morgan develop his perspective on algebra by justifying and explaining complex numbers. This implies that teaching and learning complex numbers is a catalyst for mathematical development of De Morgan.

• Journal for History of Mathematics
• /
• v.12 no.2
• /
• pp.15-23
• /
• 1999

This paper deals with the origin of the concept of lattices in mathematics and its development until 1930's. Although it is purely mathematical, its formation is due to the development of symbolic logic Further, logicians were mostly concerned about how to imitate the methods and duplicate the problems of algebra but not the application to mathematics. The first purely mathematical approach was given by Dedekind and his results were neglected and then reappeared in 1930's.

• Kyungpook Mathematical Journal
• /
• v.56 no.4
• /
• pp.1141-1160
• /
• 2016

In this paper, a symbolic algorithm for solving a regular initial value problem (IVP) for a system of linear differential-algebraic equations (DAEs) with constant coeffcients has been presented. Algebra of integro-differential operators is employed to express the given system of DAEs. We compute a canonical form of the given system which produces another simple equivalent system. Algorithm includes computing the matrix Green's operator and the vector Green's function of a given IVP. Implementation of the proposed algorithm in Maple is also presented with sample computations.

• 제어로봇시스템학회:학술대회논문집
• /
• 1987.10b
• /
• pp.19-22
• /
• 1987

A program is developed for generations the dynamic equations for robotic manipulators using the symbolic language muSIMP/MATH. The muSIMP/MATH is a LISP-based computer algebra package, devoted to the manipulation of algebraic expressions including number, variables, functions, and matrix. The muSIMP/MATH can operate on IBM-PC compatibles with MS-DOS. The program is developed, on the e formalism. This is program is applicable to the manipulators of any number of degrees of freedom, maximum six degree of freedom in this program. To control robotic manipulators by using dynamic equation is required a symbolic equations. The generated dynamic equation can be applied directly to the robotic manipulators, for the generated dynamic equation is a reduced form of symbolic expression.

• Journal for History of Mathematics
• /
• v.22 no.4
• /
• pp.129-144
• /
• 2009

The purpose of this study is what exactly De Morgan contributed to abstract algebra and mathematical logic. He recognised the purely symbolic nature of algebra and was aware of the existence of algebras other than ordinary algebra. He madealgebra as a science by introducing the ordered field and made the base for abstract algebra. He was one of the reformer of classical mathematical logic. Looking into De Morgan's works, we made it clear that the developments of algebra and mathematical logic in 19C.