Quaternions: The hypercomplex number system

2008 ◽  
Vol 92 (525) ◽  
pp. 431-436 ◽  
Author(s):  
Sandra Pulver
Keyword(s):  
Real Axis ◽  
Imaginary Axis ◽  
Vertical Axis ◽  
Number System ◽  
Real Coefficient ◽  
Complex Numbers ◽  
Square Root ◽  
Real Numbers ◽  
The Real ◽  
Hypercomplex Number

Are there solutions of the equation x2 + 1 = 0 ? Carl Fredrich Gauss (1777–1855) conjectured that there was a solution and that it was the square root of - 1 . But since the squares of all real numbers, positive or negative, are positive, Gauss introduced a fanciful idea. His solution to this equation was , which he named i. He integrated i with the real numbers to form a set known as , the complex numbers, where each element in that set was of the form a + bi, where a, . Gauss illustrated this on a graph, the horizontal axis became the real axis and represented the real coefficient, while the vertical axis became the imaginary axis and represented the imaginary coefficient.

scholarly journals Complex roots of polynomials and their computation with the help of Scientific Calculators

BIBECHANA ◽  
2012 ◽  
Vol 9 ◽  
pp. 18-27
Author(s):  
Mohd Yusuf Yasin
Keyword(s):  
Number System ◽  
Complex Numbers ◽  
Real Numbers ◽  
Engineering Technology ◽  
The Real ◽  
Complex Roots ◽  
Wide Range ◽  
Range Of Functions ◽  
Imaginary Number

Real numbers are something which are associated with the practical life. This number system is one dimensional. Situations arise when the real numbers fail to provide a solution. Perhaps the Italian mathematician Gerolamo Cardano is the first known mathematician who pointed out the necessity of imaginary and complex numbers. Complex numbers are now a vital part of sciences and are used in various branches of engineering, technology, electromagnetism, quantum theory, chaos theory etc. A complex number constitutes a real number along with an imaginary number that lies on the quadrature axis and gives an additional dimension to the number system. Therefore any computation based on complex numbers, is usually complex because both the real and imaginary parts of the number are to be simultaneously dealt with. Modern scientific calculators are capable of performing on a wide range of functions on complex numbers in their COMP and CMPLX modes with an equal ease as with the real numbers. In this work, the use of scientific calculators (Casio brand) for efficient determination of complex roots of various types of equations is discussed. DOI: http://dx.doi.org/10.3126/bibechana.v9i0.7148 BIBECHANA 9 (2013) 18-27

Complex Modular Numbers: Complex Numbers Need not be Complex

1976 ◽  
Vol 69 (1) ◽  
pp. 53-54
Author(s):  
Susan J. Grant ◽  
Ward R. Stewart
Keyword(s):  
Real Number ◽  
Complex Number ◽  
Number System ◽  
Complex Numbers ◽  
Negative Number ◽  
Real Numbers ◽  
The Real ◽  
Equivalent Equation ◽  
Imaginary Number

Most students are faced with the task of solving the equation x2 + 1 = 0 over the real numbers at some time in their algebra classes. After they substitute values for x unsuccessfully, they usually attempt to solve the equivalent equation x2 = -1. They soon realize that it is impossible to square a real number and obtain a negative number. At this point their teacher may define the imaginary number i to be and then proceed to develop the complex number system.

Didactic Features of Studying the Real Numbers in the School Course of Mathematics

2017 ◽  
Vol 6 (1) ◽  
pp. 65-70
Author(s):  
Алексеенко ◽  
A. Alekseenko ◽  
Лихачева ◽  
M. Likhacheva
Keyword(s):  
Real Number ◽  
Complex Numbers ◽  
Algebraic Structures ◽  
Real Numbers ◽  
Irrational Numbers ◽  
The Real ◽  
Clear Idea

The article is devoted to the study of the peculiarities of real numbers in the discipline "Algebra and analysis" in the secondary school. The theme of "Real numbers" is not easy to understand and often causes difficulties for students. However, the study of this topic is now being given enough attention and time. The consequence is a lack of understanding of students and school-leavers, what constitutes the real numbers, irrational numbers. At the same time the notion of a real number is required for further successful study of mathematics. To improve the efficiency of studying the topic and form a clear idea about the different numbers offered to add significantly to the material of modern textbooks, increase the number of hours in the study of real numbers, as well as to include in the school course of algebra topics "Complex numbers" and "Algebraic structures".

scholarly journals Advanced number system – The universe of all numbers where anything is possible

2020 ◽  
Author(s):  
Balram A Shah
Keyword(s):  
Multiple Solutions ◽  
Best Practice ◽  
Number System ◽  
Complex Numbers ◽  
Real Numbers ◽  
The Past ◽  
Indeterminate Form ◽  
The Universe ◽  
Imaginary Number ◽  
Mathematical Constant

This research introduces a new scope in mathematics with new numbers that already exist in everyday mathematics but very difficult to get noticed. These numbers are termed as advanced numbers where entire real numbers, including complex numbers are the subset of this number’s universe. Dividing by zero results in multiple solutions so it is the best practice to not divide by zero, but what if dividing by zero have a unique solution? These numbers carry additional details about every number that it produces unique results for every indeterminate form, it allows us to divide by zero and even allows us to deal with infinite values uniquely. So, related to this number, theories, framework, axioms, theorems and formulas are established and some problems are solved which had no confirmed solutions in the past. Problems solved in this article will help us to understand little more about imaginary number, calculus, infinite summation series, negative factorial, Euler’s number e and mathematical constant π in very new prospective. With these numbers, we also understand that zero and one are very sophisticated numbers than any numbers and can lead to form any number. Advance number system simply opens a new horizon for entire mathematics and holds so much detailed precision about every number that it may require computation intelligence and power in certain situations to evaluate it.

Continuity in Intuitionism

2020 ◽  
pp. 299-327
Author(s):  
Charles McCarty
Keyword(s):  
Real Number ◽  
Number System ◽  
Real Numbers ◽  
Continuity Theorem ◽  
The Real ◽  
Real Number System

The chapter features, first, a critical presentation of Brouwer’s intuitionistic doctrines concerning logic, the real numbers, and continuity in the real number system, including his Principle for Numbers and Continuity Theorem. This is followed by a parallel examination of Hermann Weyl’s quasi-intuitionistic views on logic, continuity, and the real number system, views inspired by (but grossly misrepresenting) ideas of Brouwer. The whole business wraps up with an attempt to place Brouwer’s and Weyl’s efforts within the trajectory of informed thinking, during the late 19th and early 20th centuries, on the subjects of continua, magnitudes, and quantities.

Some James numbers of Stiefel manifolds

1982 ◽  
Vol 92 (1) ◽  
pp. 139-161 ◽  
Author(s):  
Hideaki Ōshima
Keyword(s):  
The Other ◽  
Stiefel Manifold ◽  
Complex Numbers ◽  
Real Numbers ◽  
The Real ◽  
Stiefel Manifolds ◽  
Other Hand ◽  
Usual Norm

The purpose of this note is to determine some unstable James numbers of Stiefel manifolds. We denote the real numbers by R, the complex numbers by C, and the quaternions by H. Let F be one of these fields with the usual norm, and d = dimRF. Let On, k = On, k(F) be the Stiefel manifold of all orthonormal k–frames in Fn, and q: On, k → Sdn−1 the bundle projection which associates with each frame its last vector. Then the James number O{n, k} = OF{n, k} is defined as the index of q* πdn−1(On, k) in πdn−1(Sdn−1). We already know when O{n, k} is 1 (cf. (1), (2), (3), (13), (33)), and also the value of OK{n, k} (cf. (1), (13), (15), (34)). In this note we shall consider the complex and quaternionic cases. For earlier work see (11), (17), (23), (27), (29), (31) and (32). In (27) we defined the stable James number , which was a divisor of O{n, k}. Following James we shall use the notations X{n, k}, Xs{n, k}, W{n, k} and Ws{n, k} instead of OH{n, k}, , Oc{n, k} and respectively. In (27) we noticed that O{n, k} = Os{n, k} if n ≥ 2k– 1, and determined Xs{n, k} for 1 ≤ k ≤ 4, and also Ws{n, k} for 1 ≤ k ≤ 8. On the other hand Sigrist (31) calculated W{n, k} for 1 ≤ k ≤ 4. He informed the author that W{6,4} was not 4 but 8. Since Ws{6,4} = 4 (cf. § 5 below) this yields that the unstable James number does not equal the stable one in general.

scholarly journals Roots, Schottky semigroups, and a proof of Bandt’s conjecture

2016 ◽  
Vol 37 (8) ◽  
pp. 2487-2555 ◽  
Author(s):  
DANNY CALEGARI ◽  
SARAH KOCH ◽  
ALDEN WALKER
Keyword(s):  
Real Axis ◽  
Mandelbrot Set ◽  
Complex Numbers ◽  
Limit Set ◽  
The Real ◽  
Roots Of Polynomials ◽  
Image Position ◽  
Interior Points ◽  
Small Holes

In 1985, Barnsley and Harrington defined a ‘Mandelbrot Set’${\mathcal{M}}$for pairs of similarities: this is the set of complex numbers$z$with$0<|z|<1$for which the limit set of the semigroup generated by the similarities$$\begin{eqnarray}x\mapsto zx\quad \text{and}\quad x\mapsto z(x-1)+1\end{eqnarray}$$is connected. Equivalently,${\mathcal{M}}$is the closure of the set of roots of polynomials with coefficients in$\{-1,0,1\}$. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small ‘holes’ in${\mathcal{M}}$, and conjectured that these holes were genuine. These holes are very interesting, since they are ‘exotic’ components of the space of (2-generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, and he conjectured that the interior points are dense away from the real axis. We introduce the technique oftrapsto construct and certify interior points of${\mathcal{M}}$, and use them to prove Bandt’s conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in${\mathcal{M}}$.

FRACTAL IMAGES OF GENERALIZED JULIA SETS

Fractals ◽  
1996 ◽  
Vol 04 (04) ◽  
pp. 533-541 ◽  
Author(s):  
KIM-KHOON ONG ◽  
AICHYUN SHIAH ◽  
ZDZISLAW E. MUSIELAK
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Julia Sets ◽  
Real Numbers ◽  
Positive Real ◽  
The Real ◽  
Fractal Images ◽  
Zero Values ◽  
Principal Value ◽  
Iteration Function

The iteration function [Formula: see text], where both α and β are positive real numbers, is used to generate families of the generalized Julia sets, [Formula: see text]. The calculations are restricted to the principal value of zα + iβ and the obtained results demonstrate that classical Julia sets, [Formula: see text] are significantly deformed when non-zero values of β are considered. As a result of this deformation, the area of stable regions in the complex plane changes and a process of splitting and shifting takes place along the real axis. It is shown that this process is responsible for the formation of new fractal images of generalized Julia sets.

scholarly journals A New Set Theory for Analysis

2019 ◽  
Author(s):  
Juan Ramirez
Keyword(s):  
Real Number ◽  
Number System ◽  
Number Line ◽  
Physical Models ◽  
Order Structure ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Power Set ◽  
Real Number System

We present the real number system as a natural generalization of the natural numbers. First, we prove the co-finite topology, $Cof(\mathbb N)$, is isomorphic to the natural numbers. Then, we generalize these results to describe the continuum $[0,1]$. Then we prove the power set $2^{\mathbb Z}$ contains a subset isomorphic to the non-negative real numbers, with all its defining structure of operations and order. Finally, we provide two different constructions of the entire real number line. We see that the power set $2^{\mathbb N}$ can be given the defining structure of $\mathbb R$. The constructions here provided give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. The supremum and infimum are explicitly constructed by means of a well defined algorithm that ends in denumerable steps. In section 5 we give evidence our construction of $\mathbb N$ and $\mathbb R$ are canonical; these constructions are as natural as possible. In the same section, we propose a new axiomatic basis for analysis. In the last section we provide a series of graphic representations and physical models that can be used to represent the real number system. We conclude that the system of real numbers is completely defined by the order structure of $\mathbb N$.}

scholarly journals Minimum Rank of Matrices Described by a Graph or Pattern over the Rational, Real and Complex Numbers

10.37236/749 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Avi Berman ◽  
Shmuel Friedland ◽  
Leslie Hogben ◽  
Uriel G. Rothblum ◽  
Bryan Shader
Keyword(s):  
Computational Complexity ◽  
Rational Numbers ◽  
Complex Numbers ◽  
Real Numbers ◽  
Minimum Rank ◽  
The Real ◽  
Sign Patterns ◽  
Symmetric Patterns

We use a technique based on matroids to construct two nonzero patterns $Z_1$ and $Z_2$ such that the minimum rank of matrices described by $Z_1$ is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by $Z_2$ is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture by Arav, Hall, Koyucu, Li and Rao about rational realization of minimum rank of sign patterns. Using $Z_1$ and $Z_2$, we construct symmetric patterns, equivalent to graphs $G_1$ and $G_2$, with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.

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