scholarly journals 486. The Definition of a Complex Number

1916 ◽  
Vol 8 (124) ◽  
pp. 305
Author(s):  
G. W. Palmer
Keyword(s):  
Complex Number ◽  
Definition Of

Finding Complex Roots: Can You Trust Your Calculator?

2006 ◽  
Vol 99 (5) ◽  
pp. 366-371
Author(s):  
John W. Watson ◽  
Barbara A. Ciesia
Keyword(s):  
Complex Number ◽  
Specific Instance ◽  
Complex Roots ◽  
Complex Power ◽  
Arbitrary Complex ◽  
The Difference ◽  
Definition Of

This article investigates a specific instance when the textbook answer for finding a root of a complex number differed with the answer given by the TI-83. After explaining the reason for the difference the article then expands the definition of the integral root of a complex number to an arbitrary complex power of a complex number. Read now to see where false assumptions might be made based on the results of a calculator and see explanations of how to overcome those assumptions with logic and proof.

scholarly journals Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 206
Author(s):  
Ji-Eun Kim
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Complex Number ◽  
Complex Function ◽  
Taylor Series Expansion ◽  
Complex Numbers ◽  
Definition Of ◽  
Generalized Quaternion ◽  
Complex Valued ◽  
Complex Basis

The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.

Fields and Near-Fields of Ordered Pairs of Reals

1966 ◽  
Vol 59 (4) ◽  
pp. 335-341
Author(s):  
Janet Dorman ◽  
Jeffrey L. Tollefson ◽  
F. Max Stein
Keyword(s):  
Complex Number ◽  
Rational Number ◽  
Binary Operations ◽  
Definition Of ◽  
Ordered Pairs

A Complex number may be written as a + ib or as an ordered pair of numbers (a, b), with a and b real. Also a rational number may be written as c/d or as the ordered pair (c, d), d≠0, with c and d integers. Then in each case, after a definition of equality is made, the binary operations of addition and multiplication can be considered.

scholarly journals 439. The Definition of a Complex Number

1915 ◽  
Vol 8 (116) ◽  
pp. 48
Author(s):  
G. H. Hardy
Keyword(s):  
Complex Number ◽  
Definition Of

scholarly journals Exponentials and Logarithms Properties in an Extended Complex Number Field

2021 ◽  
Author(s):  
Daniel Tischhauser
Keyword(s):  
Number Field ◽  
Complex Number ◽  
Complex Numbers ◽  
Algebraic Properties ◽  
Extended Complex ◽  
Definition Of ◽  
Complex Number Field

In this study we demonstrate the complex logarithm and exponential multivalued results and identity failures are not induced by the exponentiation and logarithm operations, but are solely induced by the definition of complex numbers and exponentiation as in C. We propose a new definition of the complex number set, in which the issues related to the identity failures and the multivalued results resolve. Furthermore the exponentiation is no longer defined by the logarithm, instead the complex logarithm formula can be deduced from the exponentiation. There is a cost as some algebraic properties of the addition and substraction will be diminished, though remaining valid to a certain extent. Finally we attempt a geometric and algebraic formalization of the new complex numbers set. It will appear clearly the new complex numbers system is a natural and harmonious complement to the C field.

scholarly journals Exponentials and Logarithms Properties in an Extended Complex Number Field

2021 ◽  
Author(s):  
Daniel Tischhauser
Keyword(s):  
Number Field ◽  
Complex Number ◽  
Complex Numbers ◽  
Algebraic Properties ◽  
Extended Complex ◽  
Definition Of ◽  
Complex Number Field

In this study we demonstrate the complex logarithm and exponential multivalued results and identity failures are not induced by the exponentiation and logarithm operations, but are solely induced by the definition of complex numbers and exponentiation as in C. We propose a new definition of the complex number set, in which the issues related to the identity failures and the multivalued results resolve. Furthermore the exponentiation is no longer defined by the logarithm, instead the complex logarithm formula can be deduced from the exponentiation. There is a cost as some algebraic properties of the addition and substraction will be diminished, though remaining valid to a certain extent. Finally we attempt a geometric and algebraic formalization of the new complex numbers set. It will appear clearly the new complex numbers system is a natural and harmonious complement to the C field.

scholarly journals Conceptual Understanding of Complex Analysis Number using Flipped Learning

2021 ◽  
Vol 4 (1) ◽  
pp. 56
Author(s):  
Fariz Setyawan ◽  
Siti Nur Rohmah
Keyword(s):  
Conceptual Understanding ◽  
Complex Number ◽  
Teaching And Learning ◽  
Complex Analysis ◽  
Mathematics Classroom ◽  
Learning Activity ◽  
Flipped Learning ◽  
Education Department ◽  
The Subject ◽  
Definition Of

Flipped Learning is one of the alternatives of teaching and learning approach in mathematics classroom. The objective of this study is exploring students’ conceptual understanding about complex number using flipped learning with handout. The subject of the study are the students in 5th semester students of mathematics education department in 2019/2020. The study used qualitative approach to describe the implementation of flipped learning.There are 31,6% of 19 respondents give score very satisfied. This result then observed by using the test with all the students understand with the definition of complex numbers. Besides they can adapt their learning activity using flipped learning with complex analysis handout. As legibility aspect of the handout, there are 52,6% of the respondents gives score satisfied and 26,3% of the respondents are very satisfied. The score indicates that the flipped learning with handout helps students to understand about the complex number concepts.

scholarly journals Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 668
Author(s):  
Ji Eun Kim
Keyword(s):  
Real Function ◽  
Complex Number ◽  
Elementary Function ◽  
Complex Function ◽  
Elementary Functions ◽  
Analytical Studies ◽  
Definition Of ◽  
Derivatives Of

We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i, which is used to derive the step derivative of a real function, we intend to derive the complex function using the base direction of the quaternion. Because many analytical studies on quaternions have been conducted, various examples can be presented using the expression of the elementary function of a quaternion. In a previous study, the base direction of the quaternion was regarded as the base separate from the basis of the complex number. However, considering the properties of the quaternion, we propose two types of step derivatives in this study. The step derivative is first defined in the j direction, which includes a quaternion. Furthermore, the step derivative in the j+k2 direction is determined using the rule between bases i, j, and k defined in the quaternion. We present examples in which the definition of the j-step derivative and (j,k)-step derivative are applied to elementary functions ez, sinz, and cosz.

Introductory paper

1966 ◽  
Vol 24 ◽  
pp. 3-5
Author(s):  
W. W. Morgan
Keyword(s):  
Line Intensity ◽  
Classification Scheme ◽  
Two Dimensional ◽  
Spectral Classification ◽  
Dimensional Array ◽  
Rr Lyrae ◽  
Metallic Line ◽  
Introductory Paper ◽  
Parameter Varying ◽  
Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

2.2. Working Group 2: Conceptual Definitions of Reference Coordinate Systems for Earth Dynamics

1975 ◽  
Vol 26 ◽  
pp. 21-26
Keyword(s):  
Coordinate System ◽  
Working Group ◽  
General Character ◽  
Coordinate Systems ◽  
Reference Coordinate System ◽  
Group 2 ◽  
Definition Of ◽  
Earth Dynamics

An ideal definition of a reference coordinate system should meet the following general requirements:1. It should be as conceptually simple as possible, so its philosophy is well understood by the users.2. It should imply as few physical assumptions as possible. Wherever they are necessary, such assumptions should be of a very general character and, in particular, they should not be dependent upon astronomical and geophysical detailed theories.3. It should suggest a materialization that is dynamically stable and is accessible to observations with the required accuracy.

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