Abstract Algebra Maths 113 Lecture 1

1. Re(3 + 5i) = 3 and Im(3 + 5i) = 5.

1998

Abstract In this workshop we will provide our motivation for an approach to developing a better understanding of structural notions in the number system as one of the routes to understanding algebraic structures. Firstly we analyse the notion of a structural view in both numerical and algebraic contexts. We will then present an overview of the approach as well as tasks from the teaching sequence to illustrate this.

For the Learning of Mathematics, 2010

The attainment and retention of later algebra skills in high school has been identified as a factor significantly impacting the postsecondary success of students majoring in STEM fields. Researchers maintain that learners develop meaning for algebraic procedures by forming connections to the basic number system properties. The present study investigated the connections participants formed between algebraic procedures and basic number properties in the context of rational expressions. An assessment, given to 107 undergraduate students in precalculus, contained three pairs of closely matched algebraic and numeric rational expressions with the operations of addition, subtraction, and division. The researcher quantitatively analyzed the distribution of scores in the numeric and algebraic context. Qualitative methods were used to analyze the strategies and errors that occurred in the participants‟ written work. Finally, task-based interviews were conducted with eight participants to reveal their mathematical thinking related to numeric and algebraic rational expressions. Statistical analysis using McNemar‟s test indicated that the undergraduate participants' abilities related to algebraic rational expressions and rational numbers were significantly different, although serious deficiencies were noted in both cases. A small intercorrelation was found in only one of the three pairs of problems, suggesting that the participants had not formed connections between algebraic procedures and basic number properties. The analysis of the participants' written work revealed that the percent of participants who consistently applied the same procedure in the numeric and algebraic items of Problem Sets A, B, and C were 56%, 47%, and 37%, respectively. Correct strategies led to fewer correct solutions in the algebraic context because of a diverse collection of errors. These errors exposed a lack of understanding for the distributive and multiplicative identity properties, as well as the mathematical ideas of equivalence and combining monomials. These fundamental mathematical ideas need to be better developed in primary and secondary education. At the post-secondary level, these ideas should serve as the foundation for interventions that are designed to support underprepared students. The results of the interviews were consistent with the quantitative analyses and the qualitative examination of the strategies used by the participants. The findings in all three areas of the study point to a disconnect between numeric and algebraic contexts in the participants‟ thinking.

Journal of the Association For Computing Machinery, 2007

We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0 -1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting system?

GRF is an ALGEBRA course, and specifically a course about algebraic structures. This introductory section revisits ideas met in the early part of Analysis I and in Linear Algebra I, to set the scene and provide motivation.

1990

While computer algebra systems have dealt with polynomials and rational functions with integer coefficients for many years, dealing with more general constructs from commutative algebra is a more recent problem. In this paper we explain how one system solves this problem, what types and operators it is necessary to introduce and, in short, how one can construct a computational theory of commutative algebra. Of necessity, such a theory is rather different from the conventional, non-constructive, theory. It is also somewhat different from the theories of Seidenberg [1974] and his school, who are not particularly concerned with practical questions of efficiency.