Complex Numbers and Quaternions for Calc III
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Math 651 Homework 1 - Algebras and Groups Due 2/22/2013
Math 651 Homework 1 - Algebras and Groups Due 2/22/2013 1) Consider the Lie Group SU(2), the group of 2 × 2 complex matrices A T with A A = I and det(A) = 1. The underlying set is z −w jzj2 + jwj2 = 1 (1) w z with the standard S3 topology. The usual basis for su(2) is 0 i 0 −1 i 0 X = Y = Z = (2) i 0 1 0 0 −i (which are each i times the Pauli matrices: X = iσx, etc.). a) Show this is the algebra of purely imaginary quaternions, under the commutator bracket. b) Extend X to a left-invariant field and Y to a right-invariant field, and show by computation that the Lie bracket between them is zero. c) Extending X, Y , Z to global left-invariant vector fields, give SU(2) the metric g(X; X) = g(Y; Y ) = g(Z; Z) = 1 and all other inner products zero. Show this is a bi-invariant metric. d) Pick > 0 and set g(X; X) = 2, leaving g(Y; Y ) = g(Z; Z) = 1. Show this is left-invariant but not bi-invariant. p 2) The realification of an n × n complex matrix A + −1B is its assignment it to the 2n × 2n matrix A −B (3) BA Any n × n quaternionic matrix can be written A + Bk where A and B are complex matrices. Its complexification is the 2n × 2n complex matrix A −B (4) B A a) Show that the realification of complex matrices and complexifica- tion of quaternionic matrices are algebra homomorphisms. -
The Enigmatic Number E: a History in Verse and Its Uses in the Mathematics Classroom
To appear in MAA Loci: Convergence The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom Sarah Glaz Department of Mathematics University of Connecticut Storrs, CT 06269 [email protected] Introduction In this article we present a history of e in verse—an annotated poem: The Enigmatic Number e . The annotation consists of hyperlinks leading to biographies of the mathematicians appearing in the poem, and to explanations of the mathematical notions and ideas presented in the poem. The intention is to celebrate the history of this venerable number in verse, and to put the mathematical ideas connected with it in historical and artistic context. The poem may also be used by educators in any mathematics course in which the number e appears, and those are as varied as e's multifaceted history. The sections following the poem provide suggestions and resources for the use of the poem as a pedagogical tool in a variety of mathematics courses. They also place these suggestions in the context of other efforts made by educators in this direction by briefly outlining the uses of historical mathematical poems for teaching mathematics at high-school and college level. Historical Background The number e is a newcomer to the mathematical pantheon of numbers denoted by letters: it made several indirect appearances in the 17 th and 18 th centuries, and acquired its letter designation only in 1731. Our history of e starts with John Napier (1550-1617) who defined logarithms through a process called dynamical analogy [1]. Napier aimed to simplify multiplication (and in the same time also simplify division and exponentiation), by finding a model which transforms multiplication into addition. -
Irrational Numbers Unit 4 Lesson 6 IRRATIONAL NUMBERS
Irrational Numbers Unit 4 Lesson 6 IRRATIONAL NUMBERS Students will be able to: Understand the meanings of Irrational Numbers Key Vocabulary: • Irrational Numbers • Examples of Rational Numbers and Irrational Numbers • Decimal expansion of Irrational Numbers • Steps for representing Irrational Numbers on number line IRRATIONAL NUMBERS A rational number is a number that can be expressed as a ratio or we can say that written as a fraction. Every whole number is a rational number, because any whole number can be written as a fraction. Numbers that are not rational are called irrational numbers. An Irrational Number is a real number that cannot be written as a simple fraction or we can say cannot be written as a ratio of two integers. The set of real numbers consists of the union of the rational and irrational numbers. If a whole number is not a perfect square, then its square root is irrational. For example, 2 is not a perfect square, and √2 is irrational. EXAMPLES OF RATIONAL NUMBERS AND IRRATIONAL NUMBERS Examples of Rational Number The number 7 is a rational number because it can be written as the 7 fraction . 1 The number 0.1111111….(1 is repeating) is also rational number 1 because it can be written as fraction . 9 EXAMPLES OF RATIONAL NUMBERS AND IRRATIONAL NUMBERS Examples of Irrational Numbers The square root of 2 is an irrational number because it cannot be written as a fraction √2 = 1.4142135…… Pi(휋) is also an irrational number. π = 3.1415926535897932384626433832795 (and more...) 22 The approx. value of = 3.1428571428571.. -
Arxiv:1001.0240V1 [Math.RA]
Fundamental representations and algebraic properties of biquaternions or complexified quaternions Stephen J. Sangwine∗ School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom. Email: [email protected] Todd A. Ell† 5620 Oak View Court, Savage, MN 55378-4695, USA. Email: [email protected] Nicolas Le Bihan GIPSA-Lab D´epartement Images et Signal 961 Rue de la Houille Blanche, Domaine Universitaire BP 46, 38402 Saint Martin d’H`eres cedex, France. Email: [email protected] October 22, 2018 Abstract The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically. 1 Introduction It is typical of quaternion formulae that, though they be difficult to find, once found they are immediately verifiable. J. L. Synge (1972) [43, p34] arXiv:1001.0240v1 [math.RA] 1 Jan 2010 The quaternions are relatively well-known but the quaternions with complex components (complexified quaternions, or biquaternions1) are less so. This paper aims to set out the fundamental definitions of biquaternions and some elementary results, which, although elementary, are often not trivial. The emphasis in this paper is on the biquaternions as an applied algebra – that is, a tool for the manipulation ∗This paper was started in 2005 at the Laboratoire des Images et des Signaux (now part of the GIPSA-Lab), Grenoble, France with financial support from the Royal Academy of Engineering of the United Kingdom and the Centre National de la Recherche Scientifique (CNRS). -
Hypercomplex Algebras and Their Application to the Mathematical
Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory Torsten Hertig I1, Philip H¨ohmann II2, Ralf Otte I3 I tecData AG Bahnhofsstrasse 114, CH-9240 Uzwil, Schweiz 1 [email protected] 3 [email protected] II info-key GmbH & Co. KG Heinz-Fangman-Straße 2, DE-42287 Wuppertal, Deutschland 2 [email protected] March 31, 2014 Abstract Quantum theory (QT) which is one of the basic theories of physics, namely in terms of ERWIN SCHRODINGER¨ ’s 1926 wave functions in general requires the field C of the complex numbers to be formulated. However, even the complex-valued description soon turned out to be insufficient. Incorporating EINSTEIN’s theory of Special Relativity (SR) (SCHRODINGER¨ , OSKAR KLEIN, WALTER GORDON, 1926, PAUL DIRAC 1928) leads to an equation which requires some coefficients which can neither be real nor complex but rather must be hypercomplex. It is conventional to write down the DIRAC equation using pairwise anti-commuting matrices. However, a unitary ring of square matrices is a hypercomplex algebra by definition, namely an associative one. However, it is the algebraic properties of the elements and their relations to one another, rather than their precise form as matrices which is important. This encourages us to replace the matrix formulation by a more symbolic one of the single elements as linear combinations of some basis elements. In the case of the DIRAC equation, these elements are called biquaternions, also known as quaternions over the complex numbers. As an algebra over R, the biquaternions are eight-dimensional; as subalgebras, this algebra contains the division ring H of the quaternions at one hand and the algebra C ⊗ C of the bicomplex numbers at the other, the latter being commutative in contrast to H. -
Hypercomplex Numbers
Hypercomplex numbers Johanna R¨am¨o Queen Mary, University of London [email protected] We have gradually expanded the set of numbers we use: first from finger counting to the whole set of positive integers, then to positive rationals, ir- rational reals, negatives and finally to complex numbers. It has not always been easy to accept new numbers. Negative numbers were rejected for cen- turies, and complex numbers, the square roots of negative numbers, were considered impossible. Complex numbers behave like ordinary numbers. You can add, subtract, multiply and divide them, and on top of that, do some nice things which you cannot do with real numbers. Complex numbers are now accepted, and have many important applications in mathematics and physics. Scientists could not live without complex numbers. What if we take the next step? What comes after the complex numbers? Is there a bigger set of numbers that has the same nice properties as the real numbers and the complex numbers? The answer is yes. In fact, there are two (and only two) bigger number systems that resemble real and complex numbers, and their discovery has been almost as dramatic as the discovery of complex numbers was. 1 Complex numbers Complex numbers where discovered in the 15th century when Italian math- ematicians tried to find a general solution to the cubic equation x3 + ax2 + bx + c = 0: At that time, mathematicians did not publish their results but kept them secret. They made their living by challenging each other to public contests of 1 problem solving in which the winner got money and fame. -
Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space
Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space Murat Babaarslan and Yusuf Yayli Abstract. A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternions and spacelike constant slope surfaces in Minkowski 3-space. We show that spacelike constant slope surfaces can be reparametrized by using rotation matrices corresponding to unit timelike quaternions with the spacelike vector parts and homothetic motions. Subsequently we give some examples to illustrate our main results. Mathematics Subject Classification (2010). Primary 53A05; Secondary 53A17, 53A35. Key words: Spacelike constant slope surface, split quaternion, homothetic motion. 1. Introduction Quaternions were discovered by Sir William Rowan Hamilton as an extension to the complex number in 1843. The most important property of quaternions is that every unit quaternion represents a rotation and this plays a special role in the study of rotations in three- dimensional spaces. Also quaternions are an efficient way understanding many aspects of physics and kinematics. Many physical laws in classical, relativistic and quantum mechanics can be written nicely using them. Today they are used especially in the area of computer vision, computer graphics, animations, aerospace applications, flight simulators, navigation systems and to solve optimization problems involving the estimation of rigid body transformations. Ozdemir and Ergin [9] showed that a unit timelike quaternion represents a rotation in Minkowski 3-space. -
0.999… = 1 an Infinitesimal Explanation Bryan Dawson
0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder . -
Measuring Fractals by Infinite and Infinitesimal Numbers
MEASURING FRACTALS BY INFINITE AND INFINITESIMAL NUMBERS Yaroslav D. Sergeyev DEIS, University of Calabria, Via P. Bucci, Cubo 42-C, 87036 Rende (CS), Italy, N.I. Lobachevsky State University, Nizhni Novgorod, Russia, and Institute of High Performance Computing and Networking of the National Research Council of Italy http://wwwinfo.deis.unical.it/∼yaro e-mail: [email protected] Abstract. Traditional mathematical tools used for analysis of fractals allow one to distinguish results of self-similarity processes after a finite number of iterations. For example, the result of procedure of construction of Cantor’s set after two steps is different from that obtained after three steps. However, we are not able to make such a distinction at infinity. It is shown in this paper that infinite and infinitesimal numbers proposed recently allow one to measure results of fractal processes at different iterations at infinity too. First, the new technique is used to measure at infinity sets being results of Cantor’s proce- dure. Second, it is applied to calculate the lengths of polygonal geometric spirals at different points of infinity. 1. INTRODUCTION During last decades fractals have been intensively studied and applied in various fields (see, for instance, [4, 11, 5, 7, 12, 20]). However, their mathematical analysis (except, of course, a very well developed theory of fractal dimensions) very often continues to have mainly a qualitative character and there are no many tools for a quantitative analysis of their behavior after execution of infinitely many steps of a self-similarity process of construction. Usually, we can measure fractals in a way and can give certain numerical answers to questions regarding fractals only if a finite number of steps in the procedure of their construction has been executed. -
Quaternion Algebra and Calculus
Quaternion Algebra and Calculus David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Created: March 2, 1999 Last Modified: August 18, 2010 Contents 1 Quaternion Algebra 2 2 Relationship of Quaternions to Rotations3 3 Quaternion Calculus 5 4 Spherical Linear Interpolation6 5 Spherical Cubic Interpolation7 6 Spline Interpolation of Quaternions8 1 This document provides a mathematical summary of quaternion algebra and calculus and how they relate to rotations and interpolation of rotations. The ideas are based on the article [1]. 1 Quaternion Algebra A quaternion is given by q = w + xi + yj + zk where w, x, y, and z are real numbers. Define qn = wn + xni + ynj + znk (n = 0; 1). Addition and subtraction of quaternions is defined by q0 ± q1 = (w0 + x0i + y0j + z0k) ± (w1 + x1i + y1j + z1k) (1) = (w0 ± w1) + (x0 ± x1)i + (y0 ± y1)j + (z0 ± z1)k: Multiplication for the primitive elements i, j, and k is defined by i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, and ki = −ik = j. Multiplication of quaternions is defined by q0q1 = (w0 + x0i + y0j + z0k)(w1 + x1i + y1j + z1k) = (w0w1 − x0x1 − y0y1 − z0z1)+ (w0x1 + x0w1 + y0z1 − z0y1)i+ (2) (w0y1 − x0z1 + y0w1 + z0x1)j+ (w0z1 + x0y1 − y0x1 + z0w1)k: Multiplication is not commutative in that the products q0q1 and q1q0 are not necessarily equal. -
Complex Numbers Sigma-Complex3-2009-1 in This Unit We Describe Formally What Is Meant by a Complex Number
Complex numbers sigma-complex3-2009-1 In this unit we describe formally what is meant by a complex number. First let us revisit the solution of a quadratic equation. Example Use the formula for solving a quadratic equation to solve x2 10x +29=0. − Solution Using the formula b √b2 4ac x = − ± − 2a with a =1, b = 10 and c = 29, we find − 10 p( 10)2 4(1)(29) x = ± − − 2 10 √100 116 x = ± − 2 10 √ 16 x = ± − 2 Now using i we can find the square root of 16 as 4i, and then write down the two solutions of the equation. − 10 4 i x = ± =5 2 i 2 ± The solutions are x = 5+2i and x =5-2i. Real and imaginary parts We have found that the solutions of the equation x2 10x +29=0 are x =5 2i. The solutions are known as complex numbers. A complex number− such as 5+2i is made up± of two parts, a real part 5, and an imaginary part 2. The imaginary part is the multiple of i. It is common practice to use the letter z to stand for a complex number and write z = a + b i where a is the real part and b is the imaginary part. Key Point If z is a complex number then we write z = a + b i where i = √ 1 − where a is the real part and b is the imaginary part. www.mathcentre.ac.uk 1 c mathcentre 2009 Example State the real and imaginary parts of 3+4i. -
1.1 the Real Number System
1.1 The Real Number System Types of Numbers: The following diagram shows the types of numbers that form the set of real numbers. Definitions 1. The natural numbers are the numbers used for counting. 1, 2, 3, 4, 5, . A natural number is a prime number if it is greater than 1 and its only factors are 1 and itself. A natural number is a composite number if it is greater than 1 and it is not prime. Example: 5, 7, 13,29, 31 are prime numbers. 8, 24, 33 are composite numbers. 2. The whole numbers are the natural numbers and zero. 0, 1, 2, 3, 4, 5, . 3. The integers are all the whole numbers and their additive inverses. No fractions or decimals. , -3, -2, -1, 0, 1, 2, 3, . An integer is even if it can be written in the form 2n , where n is an integer (if 2 is a factor). An integer is odd if it can be written in the form 2n −1, where n is an integer (if 2 is not a factor). Example: 2, 0, 8, -24 are even integers and 1, 57, -13 are odd integers. 4. The rational numbers are the numbers that can be written as the ratio of two integers. All rational numbers when written in their equivalent decimal form will have terminating or repeating decimals. 1 2 , 3.25, 0.8125252525 …, 0.6 , 2 ( = ) 5 1 1 5. The irrational numbers are any real numbers that can not be represented as the ratio of two integers.