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This paper presents an analysis of a hitherto barely known conceptual problem in the foundations of arithmetic. An unknown type of number, so-called complex numbers, first emerged in 18th century France and became connected with the claim of non-commutativity of their multiplication. These first French practitioners are here analysed, along with examination of the impact of the introduction of the metric decimal system. International dissemination of the notion is then analysed, in particular in the case of Brazil. The paper concludes by looking at the mathematical foundations of multiplying (physical) quantities. (c) 2020 Elsevier Inc. All rights reserved.

This chapter is concerned with the representation and exploitation of complex numbers, which have wide application in the mathematics of the physical sciences. A complex number is a kind of two-dimensional vector whose components are the so-called real part and imaginary part. The real part usually corresponds to physical quantities and the imaginary part is a purely mathematical construction. The biased treatment that states “positive scalars have positive and negative square roots while negative scalars seem to have no square roots at all” was only resolved with the invention of complex numbers.

Complex radix (−1 + j) allows the arithmetic operations of complex numbers to be done without treating the divide and conquer rules, which offers the significant speed improvement of complex numbers computation circuitry. Design and hardware implementation of complex radix (−1 + j) converter has been introduced in this paper. Extensive simulation results have been incorporated and an application of this converter towards the implementation of discrete Fourier transformation (DFT) processor has been presented. The functionality of the DFT processor have been verified in Xilinx ISE design suite version 14.7 and performance parameters like propagation delay and dynamic switching power consumption have been calculated by Virtuoso platform in Cadence. The proposed DFT processor has been implemented through conversion, multiplication and addition. The performance parameter matrix in terms of delay and power consumption offered a significant improvement over other traditional implementation of DFT processor.

In this chapter, selected advanced mathematical concepts and related built-in functions in the MATLAB® software are introduced. The first section in this chapter first explores fitting simple polynomial curves to data. Other topics include complex numbers and a brief introduction to differentiation and integration in calculus. Some symbolic math functions, all of which are in Symbolic Math Toolbox™ in MATLAB, are also introduced. Solutions to sets of linear algebraic equations are important in many applications. To solve systems of equations using MATLAB, there are basically two methods, both of which will be covered in this chapter: using a matrix representation and using the solve function (which is part of Symbolic Math Toolbox™).

Publisher Summary This chapter presents a discusion of linear systems in a frequency domain. Linear systems can be represented by differential equations and transfer functions that are combinations of four basic elements. If signals are restricted to steady-state sinusoids, then phasor techniques can be used to represent these elements by algebraic equations that are functions of only frequency. Phasor techniques represent steady-state sinusoids as a single complex number. With this representation, the calculus operation of differentiation can be implemented in algebra simply as multiplication by jω, where j = √(−1) ω is frequency in radians. If all the elements in a system can be represented by algebraic operations, they can be combined into a single equation termed the transfer function. The transfer function relates the output of a system to the input through a single equation. The transfer function can only deal with signals that are sinusoidal or can be decomposed into sinusoids. The transfer function not only offers a succinct representation of the system, it also gives a direct link to the frequency spectrum of the system. The concepts of analog and system representations of linear processes are explained in the chapter. It explains linearity, time invariance, causality, and superposition. The response of system elements to sinusoidal inputs is also discussed in the chapter.

The extension to Python, NumPy, that enables fast calculations using matrices, vectors, and similar containers is detailed. We discuss the creation of objects in NumPy, manipulating them, and applying mathematical functions to them. We also demonstrate operator overloading, complex arithmetic, indexing, and iterating using NumPy objects. All of these concepts are illustrated using worked examples. To make scientific figures and plots the module Matplotlib is introduced. This toolkit has similar syntax to the plotting tools of Matlab. Several examples of plots and their customization is shown using Python codes.

Trigonometry is an essential part of engineering mathematics. For example, in robotics, trigonometry can be useful in calculating the positions of robotic arms, rotations as well as other quantities. In addition, trigonometrical functions are also intrinsically related to complex numbers. This chapter introduces the fundamentals of trigonometrical functions.

Complex numbers can be useful in solving many engineering problems such as linear circuits, mechanical vibrations, signal processing and image processing. This chapter introduces the fundamentals of complex numbers and complex functions.

Research in the cognitive sciences has yielded unique and critical insights into how the human mind represents, processes, and learns mathematical concepts and procedures. The chapters throughout this volume illustrate these insights for complex arithmetic, rational numbers, algebra, geometry, and trigonometry. In this chapter, we review and discuss several basic themes, such as the importance of prior knowledge for learning that cut across most or all of the other chapters, and highlight several challenges to translating these findings into instructional approaches and policies.

This chapter provides an overview of equations by constructing models to solve problems with one variable, solving equations with two variables, with quadratic, rational, radical, and absolute value. The chapter shows how to solve inequality equations and explains intervals and complex numbers.

Help for those who are not so familiar with basic mathematical tools such as vectors, scalars, and complex numbers. Electrical circuit equivalents with tissue electrical properties and their respective equations are listed. Also Caspar Wessel and the complex plane.

Generalizations of bent functions with respect to their algebraic, combinatorial, and cryptographic properties are becoming more numerous and more widely studied from year to year. It is quite difficult not only to determine connections between generalizations, but also to collect information about all of them and briefly review the progress in this area. In this chapter and the next two chapters we provide a systematic survey of the existing generalizations of bent functions and try whenever possible to establish relations between various generalizations. In this chapter the following algebraic generalizations are considered: q -valued bent functions, p -ary bent functions, bent functions over a finite field, generalized Boolean bent functions of Schmidt, bent functions from a finite Abelian group into the set of complex numbers on the unit circle, bent functions from a finite Abelian group into a finite Abelian group, non-Abelian bent functions, vectorial G -bent functions, and multidimensional bent functions on a finite Abelian group.

Relying on the ideas of Stampfli [14] and Magajna [12] we introduce, for operators S and T on a separable complex Hilbert space, a new notion called the numerical range of S relative to T at r is an element of sigma(vertical bar T vertical bar). Some properties of these numerical ranges are proved. In particular, it is shown that the relative numerical ranges are non-empty convex subsets of the closure of the ordinary numerical range of S. We show that the position of zero with respect to the relative numerical range of S relative to T at parallel to T parallel to gives an information about the distance between the involved operators. This result has many interesting corollaries. For instance, one can characterize those complex numbers which are in the closure of the numerical range of S but are not in the spectrum of S. info:eu-repo/semantics/acceptedVersion

Part I of this book was concerned entirely with real numbers. That is, all quantities were part of the number system RR in which any number can be represented as a point on a number line. In this chapter we broaden our number system to the set of complex numbers, CC. As we shall see, complex numbers are a much broader number system, indeed RR is a subset of CC. While at first sight complex numbers may appear to have limited use, they are in fact a very powerful mathematical tool. This chapter is intended to give a solid understanding of the properties of complex numbers and a glimpse of their power in applied mathematics.

There are hundreds of PTC Mathcad functions. The following is a partial list of the different categories that have functions: Bessel Functions, Complex Numbers, Curve Fitting, Data Analysis, Differential Equations, Finance, Fourier Transforms, Graphing, Hyperbolic Functions, Image Processing, Interpolation, Logs, Number Theory, Probability, Solving, Sorting, Statistics, Trigonometry, Vectors, and Waves.

This chapter introduces complex numbers and functions, and discusses their representation in Cartesian and polar forms and their plotting on an Argand diagram (the complex plane). The exponential function of imaginary argument is defined and related to the sine and cosine. Hyperbolic functions of complex argument and Multiple-valued functions are defined and described. The illustrations include fractional powers, roots, and the logarithm.

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values known as variables. This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Most quantitative results in science and mathematics are expressed as algebraic equations.

In this chapter we present matrix differential equations governing transient problems for a variety of physical situations. A finite element discretization in the space dimension is used and a semi-discretization process followed (as introducled in Chapters 3 and 5). For structural problems the result is a set of equations involving a mass, damping and stiffness matrix. The second order equation in time may be solved formally using analytical methods in which time is represented by exponential functions. For undamped systems this leads to a classical eigen-problem in which the natural frequencies and mode shapes for a system may be identified. Including damping makes the solution feasible in complex arithmetic. General transient solutions are obtained from modal representations. A second form is forced periodic response in which applied force is represented in periodic form.

We discuss the secant method: x i + 1 = x i - f i x i - x i - 1 f i - f i - 1 ( i = 1 , 2 , 3 , … ) where ( x 0 , f 0 ) , ( x 1 , f 1 ) are initial guesses. In the Regula Falsi variation we start with initial guesses ( a , f ( a ) ) and ( b , f ( b ) ) such that f ( a ) f ( b ) 0 ; after an iteration similar to the above we replace either a or b by the new value x i + 1 depending on which of f ( a ) or f ( b ) has the same sign as f ( x i + 1 ) . Often one of the points gets “stuck,” and several variants such as the Illinois or Pegasus methods and variations are used to “unstick” it. We discuss convergence and efficiency of most of the methods considered. We treat methods involving quadratic of higher order interpolation and rational approximation. We also discuss the bisection method where again f ( a ) f ( b ) 0 and we set c = a + b 2 . We replace a or b by c according to the sign of f ( c ) as in the Regula Falsi method. Various generalizations are described, including some for complex roots. Finally we consider hybrid methods involving two or more of the previously described methods.

First we consider the Jenkins–Traub 3-stage algorithm. In stage 1 we define H ( 0 ) ( z ) = P ′ ( z ) ; H ( k + 1 ) ( z ) = 1 z H ( k ) ( z ) - H ( k ) ( 0 ) P ( 0 ) P ( z ) ( k = 0 , 1 , 2 , … , M ) In the second stage the factor 1 z is replaced by 1 z - s for fixed s , and in the third stage by 1 z - s k where s k is re-computed at each iteration. Then s k → a root. A slightly different algorithm is given for real polynomials. Another class of methods uses minimization, i.e. we try to find z ∗ such that V = R 2 + J 2 is a minimum, where f ( z ) = f ( x + iy ) = R ( x , y ) + iJ ( x , y ) . At this minimum we must have R = J = 0 , i.e. f ( z ∗ ) = 0 . Several authors search along the coordinate axes or at various angles with them, while others move along the negative gradient, which is probably more efficient. Some use a hybrid of Newton and minimization. Finally we come to Lin and Bairstow’s methods, which divide the polynomial by a quadratic and iteratively reduce the remainder to 0. This enables us to find pairs of complex roots using only real arithmetic.

Topics covered include: fundamental operations of algebra, powers, roots and logarithms, the quadratic formula, the imaginary unit i , complex numbers, permutations and combinations, complex numbers, the exponential base e , the binomial theorem, exponential growth and decay.

This chapter consists of two short appendices that provide methodological details. The first provides a geometrical derivation of Lagrange multipliers, a technique used in the book to implement minimization of a function subject to equality constraints. The second discusses how the formulas of probability theory and inverse theory can be generalized to handle complex numbers. While the most typical applications of inverse theory arise in cases where the data are purely real, complex data are occasionally encountered, especially in spectral analysis.

Publisher Summary This chapter introduces outline the importance of complex numbers and exponentials which are an integral part of digital communications and digital signal processing (DSP). The reason of delving deep into this area leads the readers to two-dimensional number plane which help to understand DSP. The operators such as addition, subtraction, multiplication take people through the complex side of numbers which are explained in two dimensions. The use of complex conjugate and complex exponential—in the number plane—allows readers understand the relationship of basic characteristics of numbers and its treatment. The chapter further looks into measuring angles in radians which leads to the use of pi and the equivalence of the degrees and radians in measuring angles. DSP therefore uses this concept in later stages to understand the clockwise and anticlockwise movement around the circle.

Digital signal processing (DSP) deals with the way numbers are processed. Most texts on DSP add an appendix or two to review complex numbers. This is unfortunate, since the key algorithms in DSP are virtually incomprehensible without a strong foundation in the basic numerical concepts. This chapter explores the applied mathematics as a study of functions and the way they behave. It is often handy to have a couple of different ways to express a function. For some applications, one expression may make one's work simpler than another. The study states that polynomials are the workhorse of applied mathematics. One another topic to explore is limits. Limits play a key role in many modern mathematical concepts. They are particularly important in studying integrals and derivatives. The basic mathematical concept of a limit closely parallels what most people think of as a limit in the physical world. Many concepts in DSP have geometrical interpretations. One example is the geometrical interpretation of the process of integration. Integration is effectively a matter of finding the area under the curve f ( x ).

Reflection in a line is represented by a sandwiching construction involving the geometric product. Though that may have seemed like a curiosity in the previous chapter, it is crucial to the representation of operators in geometric algebra. Geometrically, all orthogonal transformations can be considered as multiple reflections. Algebraically, this leads to their representation as a geometric product of unit vectors. An even number of reflection gives a rotation, represented as a rotor—the geometric product of an even number of unit vectors. This chapter shows that rotors encompass and extend complex numbers and quaternions, and present a real 3-D visualization of the quaternion product. Rotors transcend quaternions in that they can be applied to elements of any grade, in a space of any dimension. The distinction between subspaces and operators fades when it is realized that any subspace generates a reflection operator, which can act on any element. The concept of a versor (a product of vectors to be used as an operator in a sandwiching product) combines all these representations of orthogonal transformations. This chapter shows that versors preserve the structure of geometric constructions and can be universally applied to any geometrical element. This is a unique feature of geometric algebra and it can simplify code considerably. The chapter ends with a discussion of the difference between geometric algebra and Clifford algebra, and a preliminary consideration of issues in efficient implementation to convince users of the practical usability of the versor techniques in writing efficient code for geometry.

Publisher Summary This chapter aims to provide a comprehensive review of the principles and techniques for implementing discrete time Fourier analysis. This is valuable because Fourier analysis is used to calculate the frequency content of deterministic signals and is fundamental to spectral analysis. Although aperiodic signals are much more prevalent than periodic ones, a review of Fourier series is warranted. This is because the concepts of frequency content and the treatment of complex numbers that are used in all aspects of frequency analysis are easier to understand in the series form. The formal development of Fourier analysis dates back to the beginning of the eighteenth century. Fourier analysis will indicate what other frequency components, in addition to the perturbation frequency, are present. For different types of signals, there are different versions of Fourier analysis. The chapter reviews the Fourier series and explores various properties related to it. Finally, the chapter concludes with selected applications of the discrete Fourier transform (DFT) to show its utility over a broad range of engineering fields. A review of the definition of continuous time Fourier transformation is provided in the appendix of this chapter for the reader's convenience.

Publisher Summary This chapter focuses on the vector spaces. A nonempty set F of scalars (real or complex numbers) on which the operations of addition and multiplication are defined, so that to every pair of scalars α, β ∊ F there corresponds scalars α+β and αβ in F is called a field. A vector space V over the field F is a set of vectors on which the operations of addition and multiplication by a scalar are defined, that is, for every pair of vectors X, Y ∊ V, there corresponds a vector X + Y ∊ V and also α X ∊ V, where α is a scalar. If F is the field R of real numbers, V is called a real vector space and if F is the field C of complex numbers, V is a complex vector space. A nonempty subset U of a vector space V is called a subspace of V or a linear manifold if it is closed under the operation of addition and scalar multiplication. A subspace U of a vector space V is itself a vector space and as all vector spaces, it always contains the null vector 0. The whole space V and the set consisting of the null vector alone are two special examples of subspaces.

Publisher Summary This chapter presents the common methods of obtaining phases. Phase is the only additional information the crystallographer needs in order to obtain an image of the unit cell contents. The chapter emphasizes the fact that each reflection has its own phase, so the phase problems must be solved for every one of the thousands of reflections used to construct the Fourier sum. It aims to illuminate both the phase problem and its solution and represents the structure factors as vectors on a two-dimensional plane of complex numbers. This allows one to show geometrically how to compute phases. The discussion begins by introducing complex numbers and their representation as points having coordinates (a,b) on the complex plane. Lastly, the vector representation of structure factors is used to explain a few common methods of obtaining phases. The other topics covered in the chapter include isomorphous replacement, anomalous scattering, and iterative improvement of phases.

The algebraic structure of central molecular chirality can be achieved starting from the geometrical representation of bonds of tetrahedral molecules, as complex numbers in polar form, and the empirical Fischer projections used in organic chemistry. A general orthogonal O(4) algebra is derived from which we obtain a chirality index related to the classification of a molecule as achiral, diastereoisomer or enantiomer. Consequently, the chiral features of tetrahedral chains can be predicted by means of a molecular Aufbau. Moreover, a consistent Schroedinger equation is developed, whose solutions are the bonds of tetrahedral molecules in complex number representation. Starting from this result, the O(4) algebra can be considered as a quantum chiral algebra. It is shown that the operators of such an algebra preserve the parity of the whole system.

This chapter reviews complex numbers that are crucial for dealing with alternating current (AC) signals and circuits. The varied and complex nature of signals and electronic systems require a thorough understanding of the mathematical description of signals and the circuits that process them. Complex numbers play a major role in AC circuit analysis through the use of the phasor concept and associated analysis. The chapter introduces complex numbers and the different ways of representing them. It describes the elementary algebraic operations for these types of numbers. This also includes the polar representation of complex numbers and the exponential representation that is the phasor representation. The chapter concludes with the calculation of powers and roots of complex numbers.

This chapter introduces some elementary facts from set theory, topology of the real numbers, and the theory of real functions. The concepts of real analysis rest on the system of real numbers and on notions such as sets, relations, and functions. A set is a well-defined collection of objects. The objects in the collection are called the elements of the set. For example, A is equal to {x, y, z} is a set with three elements x, y, and z. The notation x Є A denotes that x belongs to A or, equivalently, x is in A . The set of natural numbers is denoted by N , the set of integers by Z , the set of rational numbers by Q , and the set of complex numbers by C . For two sets A and B , A is called a subset of B , denoted by A I B , if every element of A is also an element of B . The intersection of two sets A and B is the set of all elements that are in both A and B , denoted by ACB .

Publisher Summary This chapter describes the numerical, algebraic, and analytical results for series and calculus. Bernoulli numbers enter into many summations, and their use can often lead to the form of the general term in a series expansion of a function that may be unobtainable by other means. Alternative definitions of Bernoulli and Euler numbers are in use, in which the indexing and choice of sign of the numbers differ from the conventions adopted. When reference is made to other sources using Bernoulli and Euler numbers, it is essential to determine which definition is in use. The most commonly used alternative definitions lead to the sequences of Bernoulli numbers B*n, and Euler numbers £*n, where an asterisk has been added to distinguish them from the other numbers. A related property of determinants is that the sum of the products of the elements in any row, and the cofactors of the corresponding elements in any other row is zero.

This chapter deals with the case study regarding the design of a pipelined multiplier accumulator (MAC) for a stream of complex numbers. Many digital signal–processing algorithms, such as digital demodulation, filtering, and equalization, make use of MACs. A complex MAC operates on two sequences of complex numbers, {x i } and {y i }. The MAC multiplies corresponding elements of the sequences and accumulates the sum of the products. MAC calculates the result by taking successive pairs of complex numbers, one each from the two input sequences, forming their complex product and adding it to an accumulator register. The accumulator is initially cleared to zero and is reset after each pair of sequences gets processed. Since the operations must be performed in this order, the time taken to complete the processing of one pair of inputs is the sum of the delays for the three steps. One can avoid the delay by pipelining the MAC, which is, organizing it like an assembly line. Initializing and restarting the pipeline is needed to accumulate sums of products of a number of input sequences one after another. To represent the data, a 16-bit, two's-complements and a fixed-point binary representation is used. The first implementation of the MAC is a behavioral model, which allows focusing on the algorithm without being distracted by other details at the early stage of the design. Register–transfer–level model is based on the pipeline diagram.

This chapter provides an introduction to pythagorean-hodograph (PH) Curves, which offer many advantages, such as rational offsets, superior shape properties, and real-time interpolators. Apart from these practical advantages, the investigation of PH curves is of considerable intellectual appeal for the wealth of mathematical ideas it entails, including the geometry of complex numbers, inaugurated by Wessel, Argand, and Gauss; Hamilton's quaternions and the geometrical algebras of Clifford; medial axis transforms and the Minkowski metric of special relativity; projective geometry and dual representations; the cyclographic map and Laguerre geometry; and connections with classical geometrical optics. The distinguishing feature of a polynomial PH curve is that the components of its hodograph satisfy a Pythagorean condition—that is, the sum of their squares is equal to the square of a polynomial. By virtue of their special algebraic structure, PH curves provide exact solutions to a number of basic geometrical problems in computer-aided design and manufacturing.

The chapter reviews the use of imaginary numbers, complex numbers, and j -operators to solve RLC circuits, which otherwise are difficult to solve using other methods. The total impedance of a circuit can be converted to its polar form to yield the magnitude of the impedance and the phase angle. The chapter concludes that one can perform the addition, subtraction, multiplication, and division of complex numbers in rectangular or polar form. These basic operations simplify complex impedances of AC circuits, and can be applied to phasor algebra techniques to solve voltage current and impedance values in a series RLC circuit. The results of the calculations are then used to draw the voltage phasor diagram. Analyzing and understanding the phasor diagram makes one realize that the circuit solutions are similar to results that are obtained with much simpler circuits.