Graphing Powers and Roots of Complex Numbers

1993 ◽  
Vol 86 (7) ◽  
pp. 589-597
Author(s):  
Charles Vonder Embse
Keyword(s):  
Complex Plane ◽  
Mathematics Teaching ◽  
Visual Representation ◽  
Complex Numbers ◽  
Graphical Representations ◽  
Numerical Representation ◽  
Algebraic Representation ◽  
Technological Environment ◽  
Evaluation Standards ◽  
Mathematical Connections

Establishing mathematical connections is one of the most important themes that permeates the vision of mathematics teaching as outlined by the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). A graph has the power to help students make connections between the abstract algebraic representation and the visual representation of a concept, relationship, or pattern. With the introduction of sophisticated graphing utilities and graphing calculators, we can also make connections to the numerical representation as well as the algebraic and graphical representations. Even topics as abstract as complex numbers can now easily be visualized with a graph. Using De Moivre's theorem and a parametric graphing utility, we can graph the powers and roots of complex numbers. The “trace” feature of many graphing utilities moves a point along a graph and gives numerical output, helping students make the connection between the visual and numerical representation of the complex numbers. This rich technological environment allows students to conjecture and test many hypotheses on the way to “discovering” the propertieS of complex numbers represemed in the complex plane.

NCTM's Standards in Japanese Elementary Schools

1997 ◽  
Vol 4 (1) ◽  
pp. 20-23
Author(s):  
Daiyo Sawada
Keyword(s):  
Elementary Schools ◽  
Mathematics Education ◽  
Problem Solving ◽  
Mathematics Teaching ◽  
Observational Studies ◽  
Evaluation Standards ◽  
Mathematical Connections ◽  
Standards Documents

In recent years, the NCTM's Standards (1989, 1991) and Asian mathematics education (Becker et al. 1990; Stevenson and Stigler 1992; Stedman 1994; and many others) have, each in its own right, received a great deal of attention. I believe, however, that to look at the connections between the two areas would greatly benefit teaching. In this article, five classroom situations taken from observational studies of mathematics teaching in Japanese elementary schools are described and interpreted from the perspective of the two Standards documents (1989, 1991). More specifically, the classroom situations are examined from the perspective ot the first four standards found in the Curriculum anil Evaluation Standards (1989): Mathematics as Problem Solving. Mathematics as Communication. Mathematics as Reasoning, and Mathematical Connections.

scholarly journals Winding Numbers, Unwinding Numbers, and the Lambert W Function

2021 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Complex Plane ◽  
Complex Number ◽  
Lambert W Function ◽  
Complex Numbers ◽  
Winding Number ◽  
Line Integral ◽  
Complex Functions

AbstractThe unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches $$W_k$$ W k of the Lambert W function as a line integral.

scholarly journals Displaying Labeled Quantitative Data

2019 ◽  
Vol 5 (2) ◽  
pp. 11-20
Author(s):  
Marcin Kozak
Keyword(s):  
Quantitative Data ◽  
Visual Representation ◽  
Graphical Representations ◽  
Data Set ◽  
Quantitative Variable ◽  
Word Cloud

Abstract The information world is full of labeled quantitative data, in which a number of qualitative categories are to be compared based on a quantitative variable. Their graphical representations are various and serve different audiences and purposes. Based on a simple data set and its different visualizations, we will play with the data and their visual representation. We will use well-known charts, such as a regular table, a bar plot, and a word cloud; less-know, such as Cleveland’s dot plot, a fan plot, and a text-table; and new ones, constructed for the very aim of this essay, such as a labeled rectangle plot and a ruler-like graph. Our discussion will not aim to choose the best graph but rather to show the different faces of visualizing labeled quantitative data. I hope to convince the readers that it is always worth spending a minute on pondering how to present their data.

Using Logic Activities to Produce a Class Constitution

1992 ◽  
Vol 40 (3) ◽  
pp. 174-176
Author(s):  
Susan Strand Monchamp
Keyword(s):  
Real World ◽  
Evaluation Standards ◽  
The Real ◽  
Mathematical Connections ◽  
Relationship Of ◽  
The Relationship

One of my major goals in mathematics is to have my student understand the relationship of mathematics to the real world. To this end, we begin the year in my first-gradeclass by doing a series of logic activities that lead to the production of our class constitution. The activities reflect three of the NCTM' curriculum and evaluation standards—Mathematics a Communication, Mathematics as Reasoning, and Mathematical Connections (NCTM 1989).

Activating Assessment Alternatives in Mathematics

1992 ◽  
Vol 39 (6) ◽  
pp. 24-29 ◽  
Author(s):  
David J. Clarke
Keyword(s):  
Mathematics Education ◽  
Mathematics Teaching ◽  
Teaching Practice ◽  
Mathematics Curriculum ◽  
Teaching Program ◽  
Evaluation Standards ◽  
Pencil And Paper ◽  
And Mathematics ◽  
Assessment Strategies

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989, 1, 2) emphasizes the role of evaluation “in gathering information on which teachers can base their subsequent instruction.” This strong sense of assessment's informing instructional practice is also evident in the materials arising from the Australian Mathematics Curriculum and Teaching Program (Clarke 1989: Lovitt and Clarke 1988, 1989). Both projects offer their respective mathematics-education communities a set of goal much broader than those traditionally conceived for mathematics instruction. The adoption of these goals by mathematics teachers and school systems demands the use of new assessment strategies if the restructuring of the mathematics curriculum and mathematics-teaching practice is to be effected. Mathematics education must not restrict itself to those goals that can be assessed only through conventional pencil-and-paper methods.

Illustrating Mathematical Connections: Two Proofs That Only Five Regular Polyhedra Exist

1993 ◽  
Vol 86 (8) ◽  
pp. 657-661
Author(s):  
Peter L. Glidden ◽  
Erin K. Fry
Keyword(s):  
Mathematical Models ◽  
Real World ◽  
Mathematics Curriculum ◽  
Three Dimensional ◽  
Discrete Mathematics ◽  
Evaluation Standards ◽  
Mathematical Connections ◽  
Regular Polyhedra ◽  
Real World Problems

The reforms proposed in the NCTM's Curriculum and Evaluation Standards (1989) call for specific changes in the grades 9-12 mathematics curriculum, as well as for general themes that should be emphasized throughout the curriculum. In particular, the standards document calls for including topics from discrete mathematics and three-dimensional geometry, and it calls for increased emphasis on paragraph-style proofs. Overall, these and other topics should be taught with the ultimate goals of illustrating mathematical connections and constructing mathematical models to solve real-world problems.

Implementing the Standards: Uses of Technology in Prealgebra and Beginning Algebra

1990 ◽  
Vol 83 (3) ◽  
pp. 194-198
Author(s):  
M. Kathleen Heid
Keyword(s):  
Problem Solving ◽  
School Mathematics ◽  
Evaluation Standards ◽  
Mathematics Classrooms ◽  
Beginning Algebra ◽  
Grade Levels ◽  
Mathematical Connections

The NCTM's Curriculum and Evaluation Standards for School Mathematics (Stan dards) (1989) designates four standards that apply to all students at all grade levels: mathematics as problem solving, mathematics as communication, mathematics as reasoning, and mathematical connections. These and NCTM's other standards are embedded in a vision of technologically rich school mathematics classrooms in which students and teachers have constant access to appropriate computing devices and in which students use computers and calculators as tools for the investigation and exploration of problems.

Professional Development: Teachers' Communication and Collaboration Keys to Student Achievement

1995 ◽  
Vol 1 (6) ◽  
pp. 454-458
Author(s):  
Helene J. Sherman ◽  
Thomas Jaeger
Keyword(s):  
Professional Development ◽  
Student Achievement ◽  
Educational Reform ◽  
Mathematics Teaching ◽  
School Mathematics ◽  
Professional Standards ◽  
Teaching Mathematics ◽  
Professional Satisfaction ◽  
Evaluation Standards ◽  
Instructional Changes

The curriculum and evaluation standards for School Mathematics (NCTM 1989) and the Professional Standards for Teaching Mathematics (NCTM 1991) have served as both stimuli for, and responses to, numerous formal and informal programs, conferences, and conversations calling for educational reform and improvement in mathematics teaching. After all the plans are drawn and all the objectives are written, however, reform is most likely to occur and make a lasting difference when teachers are aware of the need for improvement, have a voice in planning it, and derive a real sense of professional satisfaction from implementing the instructional changes.

scholarly journals ON APPROXIMATION OF ANALYTIC FUNCTIONS BY PERIODIC HURWITZ ZETA-FUNCTIONS

2018 ◽  
Vol 24 (1) ◽  
pp. 20-33 ◽  
Author(s):  
Darius Siaučiūnas ◽  
Violeta Franckevič ◽  
Antanas Laurinčikas
Keyword(s):  
Zeta Function ◽  
Complex Plane ◽  
Analytic Functions ◽  
Zeta Functions ◽  
Periodic Sequence ◽  
Hurwitz Zeta Function ◽  
Complex Numbers ◽  
Closed Set

The periodic Hurwitz zeta-function ζ(s, α; a), s = σ +it, with parameter 0 < α ≤ 1 and periodic sequence of complex numbers a = {am } is defined, for σ > 1, by series sum from m=0 to ∞ am / (m+α)s, and can be continued moromorphically to the whole complex plane. It is known that the function ζ(s, α; a) with transcendental orrational α is universal, i.e., its shifts ζ(s + iτ, α; a) approximate all analytic functions defined in the strip D = { s ∈ C : 1/2 σ < 1. In the paper, it is proved that, for all 0 < α ≤ 1 and a, there exists a non-empty closed set Fα,a of analytic functions on D such that every function f ∈ Fα,a can be approximated by shifts ζ(s + iτ, α; a).

Great Circles and Rhumb Lines on the Complex Plane

2017 ◽  
Vol 70 (3) ◽  
pp. 618-627
Author(s):  
Robin G. Stuart
Keyword(s):  
Complex Plane ◽  
Stereographic Projection ◽  
Riemann Sphere ◽  
Complex Numbers ◽  
Spherical Trigonometry ◽  
Numerical Examples ◽  
Computer Evaluation

Mapping points on the Riemann sphere to points on the plane of complex numbers by stereographic projection has been shown to offer a number of advantages when applied to problems in navigation traditionally handled using spherical trigonometry. Here it is shown that the same approach can be used for problems involving great circles and/or rhumb lines and it results in simple, compact expressions suitable for efficient computer evaluation. Worked numerical examples are given and the values obtained are compared to standard references.

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