Ordered Field

Ordered fieldFrom Wikipedia, the free encyclopedia

Contents

1 Complex number 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.4 History in brief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Elementary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Addition and subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.3 Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.4 Square root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Absolute value and argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.2 Multiplication and division in polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.1 Eulers formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.2 Natural logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.3 Integer and fractional exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6.1 Field structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6.2 Solutions of polynomial equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6.3 Algebraic characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6.4 Characterization as a topological field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.7 Formal construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7.1 Formal development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7.2 Matrix representation of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.8 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8.1 Complex exponential and related functions . . . . . . . . . . . . . . . . . . . . . . . . . . 171.8.2 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

i

ii CONTENTS

1.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9.1 Control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9.2 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9.3 Fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.9.4 Dynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.9.5 Electromagnetism and electrical engineering . . . . . . . . . . . . . . . . . . . . . . . . . 191.9.6 Signal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.9.7 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.8 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.9 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.10 Algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.11 Analytic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.10 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.11 Generalizations and related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.12 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.13 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.14.1 Mathematical references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.14.2 Historical references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.15 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.16 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Hypercomplex number 272.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Two-dimensional real algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 Higher-dimensional examples (more than one non-real axis) . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2 CayleyDickson construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.3 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.4 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Ordered field 323.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.1 Total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.2 Positive cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.3 Equivalence of the two definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1.4 Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Properties of ordered fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 Vector spaces over an ordered field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

CONTENTS iii

3.3 Examples of ordered fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Which fields can be ordered? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Topology induced by the order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6 Harrison topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.7 Superordered fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Surreal number 374.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.2 Numeric forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.3 Equivalence classes of numeric forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.4 Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.5 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3.1 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3.2 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3.4 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.5 Arithmetic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.4.1 Contents of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5 Transfinite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.6 Powers of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.7 Surcomplex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.8 Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.9 Application to combinatorial game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.10 Alternative realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.10.1 Sign expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.10.2 Axiomatic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.10.3 Hahn series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.11 Relation to hyperreals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.12 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.14 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.15 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.16 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.16.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.16.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

iv CONTENTS

4.16.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapter 1

Complex number

Im

Re

ba+bi

a0A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram,representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the imaginary unit which satisfies i2 = 1.

1
https://en.wikipedia.org/wiki/Argand_diagramhttps://en.wikipedia.org/wiki/Complex_planehttps://en.wikipedia.org/wiki/Imaginary_unit

2 CHAPTER 1. COMPLEX NUMBER

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i isthe imaginary unit, that satisfies the equation i2 = 1.[1] In this expression, a is the real part and b is the imaginarypart of the complex number and i is the positive square root of 1.Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane(also called Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part.The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whosereal part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a realnumber. In this way, the complex numbers contain the ordinary real numbers while extending them in order to solveproblems that cannot be solved with real numbers alone.As well as their use within mathematics, complex numbers have practical applications in many fields, includingphysics, chemistry, biology, economics, electrical engineering, and statistics. The Italian mathematician GerolamoCardano is the first known to have introduced complex numbers. He called them fictitious during his attempts tofind solutions to cubic equations in the 16th century.[2]

1.1 Overview

Complex numbers allow for solutions to certain equations that have no solutions in real numbers. For example, theequation

(x+ 1)2 = 9

has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution tothis problem. The idea is to extend the real numbers with the imaginary unit i where i2 = 1, so that solutions toequations like the preceding one can be found. In this case the solutions are 1 + 3i and 1 3i, as can be verifiedusing the fact that i2 = 1:

((1 + 3i) + 1)2 = (3i)2 = (32)(i2) = 9(1) = 9,

((1 3i) + 1)2 = (3i)2 = (3)2(i2) = 9(1) = 9.

According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in asingle variable have a solution in complex numbers.

1.1.1 Definition

A complex number is a number of the form a + bi, where a and b are real numbers and i is an imaginary unit, satisfyingi2 = 1. For example, 3.5 + 2i is a complex number.The real number a is called the real part of the complex number a + bi; the real number b is called the imaginary partof a + bi. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginarypart.[3][4] The real part of a complex number z is denoted by Re(z) or (z); the imaginary part of a complex numberz is denoted by Im(z) or (z). For example,

Re(3.5 + 2i) = 3.5Im(3.5 + 2i) = 2.

Hence, in terms of its real and imaginary parts, a complex number z is equal to Re(z) + Im(z) i . This expressionis sometimes known as the Cartesian form of z.A real number a can be regarded as a complex number a + 0i whose imaginary part is 0. A purely imaginary numberbi is a complex number 0 + bi whose real part is zero. It is common to write a for a + 0i and bi for 0 + bi. Moreover,when the imaginary part is negative, it is common to write a bi with b > 0 instead of a + (b)i, for example 3 4iinstead of 3 + (4)i.The set of all complex numbers is denoted by , C or C .
https://en.wikipedia.org/wiki/Numberhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Number_linehttps://en.wikipedia.org/wiki/Imaginary_numberhttps://en.wikipedia.org/wiki/Field_extensionhttps://en.wikipedia.org/wiki/Physicshttps://en.wikipedia.org/wiki/Chemistryhttps://en.wikipedia.org/wiki/Biologyhttps://en.wikipedia.org/wiki/Economicshttps://en.wikipedia.org/wiki/Electrical_engineeringhttps://en.wikipedia.org/wiki/Statisticshttps://en.wikipedia.org/wiki/Gerolamo_Cardanohttps://en.wikipedia.org/wiki/Gerolamo_Cardanohttps://en.wikipedia.org/wiki/Cubic_equationshttps://en.wikipedia.org/wiki/Real_numbershttps://en.wikipedia.org/wiki/Field_extensionhttps://en.wikipedia.org/wiki/Imaginary_unithttps://en.wikipedia.org/wiki/The_fundamental_theorem_of_algebrahttps://en.wikipedia.org/wiki/Polynomial_equationhttps://en.wikipedia.org/wiki/Imaginary_numberhttps://en.wikipedia.org/wiki/Set_(mathematics)

1.1. OVERVIEW 3

Im

Re

y

y

0 x

r

r

z=x+iy

z=xiyAn illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y.

1.1.2 Notation

Some authors[5] write a + ib instead of a + bi. In some disciplines, in particular electromagnetism and electricalengineering, j is used instead of i,[6] since i is frequently used for electric current. In these cases complex numbers
https://en.wikipedia.org/wiki/Complex_planehttps://en.wikipedia.org/wiki/Electromagnetismhttps://en.wikipedia.org/wiki/Electrical_engineeringhttps://en.wikipedia.org/wiki/Electrical_engineeringhttps://en.wikipedia.org/wiki/Electric_current


are written as a + bj or a + jb.

1.1.3 Complex plane

Main article: Complex planeA complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system

Figure 1: A complex number plotted as a point (red) and position vector (blue) on an Argand diagram; a + bi is the rectangularexpression of the point.

called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001), named after Jean-RobertArgand. The numbers are conventionally plotted using the real part as the horizontal component, and imaginary partas vertical (see Figure 1). These two values used to identify a given complex number are therefore called its Cartesian,rectangular, or algebraic form.A position vector may also be defined in terms of its magnitude and direction relative to the origin. These areemphasized in a complex numbers polar form. Using the polar form of the complex number in calculations may leadto a more intuitive interpretation of mathematical results. Notably, the operations of addition and multiplication takeon a very natural geometric character when complex numbers are viewed as position vectors: addition correspondsto vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e.the angles they make with the x axis). Viewed in this way the multiplication of a complex number by i corresponds torotating the position vector counterclockwise by a quarter turn (90) about the origin: (a+bi)i = ai+bi2 = b+ai.

1.1.4 History in brief

Main section: History

The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots ofnegative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the
https://en.wikipedia.org/wiki/Complex_planehttps://en.wikipedia.org/wiki/Vector_(geometric)https://en.wikipedia.org/wiki/Cartesian_coordinate_systemhttps://en.wikipedia.org/wiki/Argand_diagramhttps://en.wikipedia.org/wiki/Complex_number#CITEREFPedoe1988https://en.wikipedia.org/wiki/Complex_number#CITEREFSolomentsev2001https://en.wikipedia.org/wiki/Jean-Robert_Argandhttps://en.wikipedia.org/wiki/Jean-Robert_Argandhttps://en.wikipedia.org/wiki/Complex_number#Polar_formhttps://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtractionhttps://en.wikipedia.org/wiki/Orientation_(geometry)https://en.wikipedia.org/wiki/Turn_(geometry)https://en.wikipedia.org/wiki/Right_anglehttps://en.wikipedia.org/wiki/Complex_number#History

1.2. RELATIONS 5

rational root test if the cubic is irreducible (the so-called casus irreducibilis). This conundrum led Italian mathemati-cian Gerolamo Cardano to conceive of complex numbers in around 1545, though his understanding was rudimentary.Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows thatwith complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbersthus form an algebraically closed field, where any polynomial equation has a root.Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction,multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.[7] Amore abstract formalism for the complex numbers was further developed by the Irish mathematician William RowanHamilton, who extended this abstraction to the theory of quaternions.

1.2 Relations

1.2.1 Equality

Two complex numbers are equal if and only if both their real and imaginary parts are equal. In symbols:

z1 = z2 (Re(z1) = Re(z2) Im(z1) = Im(z2)).

1.2.2 Ordering

Because complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linearordering on the set of complex numbers.[8]

There is no linear ordering on the complex numbers that is compatible with addition and multiplication. Formally, wesay that the complex numbers cannot have the structure of an ordered field. This is because any square in an orderedfield is at least 0, but i2 = 1.

1.3 Elementary operations

1.3.1 Conjugation

Main article: Complex conjugateThe complex conjugate of the complex number z = x + yi is defined to be x yi. It is denoted z or z*.Formally, for any complex number z:

z = Re(z) Im(z) i.

Geometrically, z is the reflection of z about the real axis. Conjugating twice gives the original complex number:z = z .The real and imaginary parts of a complex number z can be extracted using the conjugate:

Re (z) = 12 (z + z),

Im (z) = 12i (z z).Moreover, a complex number is real if and only if it equals its conjugate.Conjugation distributes over the standard arithmetic operations:

z + w = z + w,
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Im

Re

y

y

0 x

r

r

z=x+iy

z=xiyGeometric representation of z and its conjugate z in the complex plane

z w = z w,

zw = zw,

(z/w) = z/w.

1.3. ELEMENTARY OPERATIONS 7

The reciprocal of a nonzero complex number z = x + yi is given by

1

z=

z

zz=

z

x2 + y2.

This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangularcoordinates. Inversive geometry, a branch of geometry studying reflections more general than ones about a line, canalso be expressed in terms of complex numbers. In the network analysis of electrical circuits, the complex conjugateis used in finding the equivalent impedance when the maximum power transfer theorem is used.

1.3.2 Addition and subtraction

b

b

aaa+

b

Addition of two complex numbers can be done geometrically by constructing a parallelogram.

Complex numbers are added by adding the real and imaginary parts of the summands. That is to say:

(a+ bi) + (c+ di) = (a+ c) + (b+ d)i.

Similarly, subtraction is defined by
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(a+ bi) (c+ di) = (a c) + (b d)i.

Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpre-tation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtainedby building a parallelogram three of whose vertices are O, A and B. Equivalently, X is the point such that the triangleswith vertices O, A, B, and X, B, A, are congruent.

1.3.3 Multiplication and division

The multiplication of two complex numbers is defined by the following formula:

(a+ bi)(c+ di) = (ac bd) + (bc+ ad)i.

In particular, the square of the imaginary unit is 1:

i2 = i i = 1.

The preceding definition of multiplication of general complex numbers follows naturally from this fundamental prop-erty of the imaginary unit. Indeed, if i is treated as a number so that di means d times i, the above multiplication ruleis identical to the usual rule for multiplying two sums of two terms.

(a+ bi)(c+ di) = ac+ bci+ adi+ bidi (distributive law)

= ac+bidi+bci+adi (commutative law of additionthe order of the summandscan be changed)= ac+ bdi2 + (bc+ ad)i (commutative and distributive laws)= (ac bd) + (bc+ ad)i (fundamental property of the imaginary unit).

The division of two complex numbers is defined in terms of complex multiplication, which is described above, andreal division. When at least one of c and d is non-zero, we have

a+ bi

c+ di=

(ac+ bd

c2 + d2

)+

(bc adc2 + d2

)i.

Division can be defined in this way because of the following observation:

a+ bi

c+ di=

(a+ bi) (c di)(c+ di) (c di)

=

(ac+ bd

c2 + d2

)+

(bc adc2 + d2

)i.

As shown earlier, c di is the complex conjugate of the denominator c + di. At least one of the real part c and theimaginary part d of the denominator must be nonzero for division to be defined. This is called "rationalization" ofthe denominator (although the denominator in the final expression might be an irrational real number).

1.3.4 Square root

See also: Square roots of negative and complex numbers

The square roots of a + bi (with b 0) are ( + i) , where

=

a+

a2 + b2

2
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1.4. POLAR FORM 9

and

= sgn(b)

a+

a2 + b2

2,

where sgn is the signum function. This can be seen by squaring ( + i) to obtain a + bi.[9][10] Herea2 + b2 is

called the modulus of a + bi, and the square root with non-negative real part is called the principal square root; alsoa2 + b2 =

zz , where z = a+ bi .[11]

1.4 Polar form

Main article: Polar coordinate system

Im

Re0

r

x

y

Figure 2: The argument and modulus r locate a point on an Argand diagram; r(cos+ i sin) or rei are polar expressions ofthe point.
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1.4.1 Absolute value and argument

An alternative way of defining a point P in the complex plane, other than using the x- and y-coordinates, is to use thedistance of the point from O, the point whose coordinates are (0, 0) (the origin), together with the angle subtendedbetween the positive real axis and the line segment OP in a counterclockwise direction. This idea leads to the polarform of complex numbers.The absolute value (or modulus or magnitude) of a complex number z = x + yi is

r = |z| =

x2 + y2.

If z is a real number (i.e., y = 0), then r = | x |. In general, by Pythagoras theorem, r is the distance of the point Prepresenting the complex number z to the origin. The square of the absolute value is

|z|2 = zz = x2 + y2.

where z is the complex conjugate of z .The argument of z (in many applications referred to as the phase) is the angle of the radius OP with the positivereal axis, and is written as arg(z) . As with the modulus, the argument can be found from the rectangular form x+yi:[12]

= arg(z) =

arctan( yx ) if x > 0arctan( yx ) + if x < 0 and y 0arctan( yx ) if x < 0 and y < 02 if x = 0 and y > 02 if x = 0 and y < 0indeterminate if x = 0 and y = 0.

The value of is expressed in radians in this article. It can increase by any integer multiple of 2 and still give thesame angle. Hence, the arg function is sometimes considered as multivalued. Normally, as given above, the principalvalue in the interval (,] is chosen. Values in the range [0,2) are obtained by adding 2 if the value is negative.The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the angle 0 is common.The value of equals the result of atan2: = atan2(imaginary, real) .Together, r and give another way of representing complex numbers, the polar form, as the combination of modulusand argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates fromthe polar form is done by the formula called trigonometric form

z = r(cos+ i sin).

Using Eulers formula this can be written as

z = rei.

Using the cis function, this is sometimes abbreviated to

z = r cis.

In angle notation, often used in electronics to represent a phasor with amplitude r and phase , it is written as[13]

z = r.
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1.4. POLAR FORM 11

Multiplication of 2 + i (blue triangle) and 3 + i (red triangle). The red triangle is rotated to match the vertex of the blue one andstretched by 5, the length of the hypotenuse of the blue triangle.

1.4.2 Multiplication and division in polar form

Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulasin Cartesian coordinates. Given two complex numbers z1 = r1(cos 1 + i sin 1) and z2 = r2(cos 2 + i sin 2),because of the well-known trigonometric identities

cos(a) cos(b) sin(a) sin(b) = cos(a+ b)

cos(a) sin(b) + sin(a) cos(b) = sin(a+ b)we may derive

z1z2 = r1r2(cos(1 + 2) + i sin(1 + 2)).

In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product.For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i2 = 1. The pictureat the right illustrates the multiplication of

(2 + i)(3 + i) = 5 + 5i.
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Since the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or /4 (in radian).On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) andarctan(1/2), respectively. Thus, the formula

4= arctan 1

2+ arctan 1

3

holds. As the arctan function can be approximated highly efficiently, formulas like thisknown as Machin-likeformulasare used for high-precision approximations of .Similarly, division is given by

z1z2

=r1r2

(cos(1 2) + i sin(1 2)) .

1.5 Exponentiation

1.5.1 Eulers formula

Eulers formula states that, for any real number x,

eix = cosx+ i sinx

where e is the base of the natural logarithm. This can be proved through induction by observing that

i0 = 1, i1 = i, i2 = 1, i3 = i,i4 = 1, i5 = i, i6 = 1, i7 = i,

and so on, and by considering the Taylor series expansions of eix, cos(x) and sin(x):

eix = 1 + ix+(ix)2

2!+

(ix)3

3!+

(ix)4

4!+

(ix)5

5!+

(ix)6

6!+

(ix)7

7!+

(ix)8

8!+

= 1 + ix x2

2! ix

3

3!+

x4

4!+

ix5

5! x

6

6! ix

7

7!+

x8

8!+

=

(1 x

2

2!+

x4

4! x

6

6!+

x8

8!

)+ i

(x x

3

3!+

x5

5! x

7

7!+

)= cosx+ i sinx .

The rearrangement of terms is justified because each series is absolutely convergent.

1.5.2 Natural logarithm

Eulers formula allows us to observe that, for any complex number

z = r(cos+ i sin).

where r is a non-negative real number, one possible value for z's natural logarithm is

ln(z) = ln(r) + i
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1.6. PROPERTIES 13

Because cos and sin are periodic functions, the natural logarithm may be considered a multi-valued function, with:

ln(z) = {ln(r) + (+ 2k)i | k Z}

1.5.3 Integer and fractional exponents

We may use the identity

ln(ab) = b ln(a)

to define complex exponentiation, which is likewise multi-valued:

ln(zn) = ln((r(cos+ i sin))n)

= n ln(r(cos+ i sin))= {n(ln(r) + (+ k2)i)|k Z}= {n ln(r) + ni+ nk2i|k Z}.When n is an integer, this simplifies to de Moivres formula:

zn = (r(cos+ i sin))n = rn (cosn+ i sinn).

The nth roots of z are given by

nz = n

r

(cos

(+ 2k

n

)+ i sin

(+ 2k

n

))for any integer k satisfying 0 k n 1. Here nr is the usual (positive) nth root of the positive real number r.While the nth root of a positive real number r is chosen to be the positive real number c satisfying cn = x there isno natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root of zis considered as a multivalued function (in z), as opposed to a usual function f, for which f(z) is a uniquely definednumber. Formulas such as

nzn = z

(which holds for positive real numbers), do in general not hold for complex numbers.

1.6 Properties

1.6.1 Field structure

The set C of complex numbers is a field. Briefly, this means that the following facts hold: first, any two complexnumbers can be added and multiplied to yield another complex number. Second, for any complex number z, itsadditive inverse z is also a complex number; and third, every nonzero complex number has a reciprocal complexnumber. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition andmultiplication for any two complex numbers z1 and z2:

z1 + z2 = z2 + z1,
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z1z2 = z2z1.

These two laws and the other requirements on a field can be proven by the formulas given above, using the fact thatthe real numbers themselves form a field.Unlike the reals, C is not an ordered field, that is to say, it is not possible to define a relation z1 < z2 that is compatiblewith the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, soi2 = 1 precludes the existence of an ordering on C.When the underlying field for a mathematical topic or construct is the field of complex numbers, the topics nameis usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, andcomplex Lie algebra.

1.6.2 Solutions of polynomial equations

Given any complex numbers (called coefficients) a0, , an, the equation

anzn + + a1z + a0 = 0

has at least one complex solution z, provided that at least one of the higher coefficients a1, , an is nonzero. This isthe statement of the fundamental theorem of algebra. Because of this fact, C is called an algebraically closed field.This property does not hold for the field of rational numbers Q (the polynomial x2 2 does not have a rational root,since 2 is not a rational number) nor the real numbers R (the polynomial x2 + a does not have a real root for a > 0,since the square of x is positive for any real number x).There are various proofs of this theorem, either by analytic methods such as Liouvilles theorem, or topological onessuch as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degreehas at least one real root.Because of this fact, theorems that hold for any algebraically closed field, apply to C. For example, any non-emptycomplex square matrix has at least one (complex) eigenvalue.

1.6.3 Algebraic characterization

The field C has the following three properties: first, it has characteristic 0. This means that 1 + 1 + + 1 0 forany number of summands (all of which equal one). Second, its transcendence degree over Q, the prime field of C, isthe cardinality of the continuum. Third, it is algebraically closed (see above). It can be shown that any field havingthese properties is isomorphic (as a field) to C. For example, the algebraic closure of Qp also satisfies these threeproperties, so these two fields are isomorphic. Also, C is isomorphic to the field of complex Puiseux series. However,specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization isthat C contains many proper subfields that are isomorphic to C.

1.6.4 Characterization as a topological field

The preceding characterization ofC describes only the algebraic aspects ofC. That is to say, the properties of nearnessand continuity, which matter in areas such as analysis and topology, are not dealt with. The following descriptionof C as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence)does take into account the topological properties. C contains a subset P (namely the set of positive real numbers) ofnonzero elements satisfying the following three conditions:

P is closed under addition, multiplication and taking inverses.

If x and y are distinct elements of P, then either x y or y x is in P.

If S is any nonempty subset of P, then S + P = x + P for some x in C.

Moreover, C has a nontrivial involutive automorphism x x* (namely the complex conjugation), such that x x* is inP for any nonzero x in C.
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1.7. FORMAL CONSTRUCTION 15

Any field F with these properties can be endowed with a topology by taking the sets B(x, p) = { y | p (y x)(y x)* P } as a base, where x ranges over the field and p ranges over P. With this topology F is isomorphic as a topologicalfield to C.The only connected locally compact topological fields are R and C. This gives another characterization of C as atopological field, since C can be distinguished from R because the nonzero complex numbers are connected, whilethe nonzero real numbers are not.

1.7 Formal construction

1.7.1 Formal development

Above, complex numbers have been defined by introducing i, the imaginary unit, as a symbol. More rigorously, theset C of complex numbers can be defined as the set R2 of ordered pairs (a, b) of real numbers. In this notation, theabove formulas for addition and multiplication read

(a, b) + (c, d) = (a+ c, b+ d)

(a, b) (c, d) = (ac bd, bc+ ad).

It is then just a matter of notation to express (a, b) as a + bi.Though this low-level construction does accurately describe the structure of the complex numbers, the followingequivalent definition reveals the algebraic nature ofCmore immediately. This characterization relies on the notion offields and polynomials. A field is a set endowed with an addition, subtraction, multiplication and division operationsthat behave as is familiar from, say, rational numbers. For example, the distributive law

(x+ y)z = xz + yz

must hold for any three elements x, y and z of a field. The set R of real numbers does form a field. A polynomialp(X) with real coefficients is an expression of the form

anXn + + a1X + a0

where the a0, ..., an are real numbers. The usual addition and multiplication of polynomials endows the set R[X] ofall such polynomials with a ring structure. This ring is called polynomial ring.The quotient ring R[X]/(X 2 + 1) can be shown to be a field. This extension field contains two square roots of 1,namely (the cosets of) X and X, respectively. (The cosets of) 1 and X form a basis of R[X]/(X 2 + 1) as a realvector space, which means that each element of the extension field can be uniquely written as a linear combinationin these two elements. Equivalently, elements of the extension field can be written as ordered pairs (a, b) of realnumbers. Moreover, the above formulas for addition etc. correspond to the ones yielded by this abstract algebraicapproachthe two definitions of the field C are said to be isomorphic (as fields). Together with the above-mentionedfact that C is algebraically closed, this also shows that C is an algebraic closure of R.

1.7.2 Matrix representation of complex numbers

Complex numbers a + bi can also be represented by 2 2 matrices that have the following form:

(a bb a

).

Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, andthe sum and product of complex numbers corresponds to the sum and product of such matrices. The geometricdescription of the multiplication of complex numbers can also be phrased in terms of rotation matrices by using
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this correspondence between complex numbers and such matrices. Moreover, the square of the absolute value of acomplex number expressed as a matrix is equal to the determinant of that matrix:

|z|2 =a bb a

= (a2) ((b)(b)) = a2 + b2.The conjugate z corresponds to the transpose of the matrix.Though this representation of complex numbers with matrices is the most common, many other representations arisefrom matrices other than

(0 11 0

)that square to the negative of the identity matrix. See the article on 2 2 real

matrices for other representations of complex numbers.

1.8 Complex analysis

Color wheel graph of sin(1/ z). Black parts inside refer to numbers having large absolute values.

Main article: Complex analysis

The study of functions of a complex variable is known as complex analysis and has enormous practical use in appliedmathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis
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1.8. COMPLEX ANALYSIS 17

or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlikereal functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensionalgraphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or byanimating the complex functions dynamic transformation of the complex plane.

1.8.1 Complex exponential and related functions

The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis.A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalentto the (, )-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers.From a more abstract point of view, C, endowed with the metric

d(z1, z2) = |z1 z2|

is a complete metric space, which notably includes the triangle inequality

|z1 + z2| |z1|+ |z2|

for any two complex numbers z1 and z2.Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponentialfunction exp(z), also written ez, is defined as the infinite series

exp(z) := 1 + z + z2

2 1+

z3

3 2 1+ =

n=0

zn

n!.

and the series defining the real trigonometric functions sine and cosine, as well as hyperbolic functions such as sinhalso carry over to complex arguments without change. Eulers identity states:

exp(i) = cos() + i sin()

for any real number , in particular

exp(i) = 1

Unlike in the situation of real numbers, there is an infinitude of complex solutions z of the equation

exp(z) = w

for any complex number w 0. It can be shown that any such solution zcalled complex logarithm of asatisfies

log(x+ iy) = ln |w|+ i arg(w),

where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, uniqueonly up to a multiple of 2, log is also multivalued. The principal value of log is often taken by restricting theimaginary part to the interval (,].Complex exponentiation z is defined as

z = exp( log z).
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Consequently, they are in general multi-valued. For = 1 / n, for some natural number n, this recovers the non-uniqueness of nth roots mentioned above.Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities,particularly when navely treated as single-valued functions; see failure of power and logarithm identities. For examplethey do not satisfy

abc = (ab)c.

Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values onthe left are a subset of those on the right.

1.8.2 Holomorphic functions

A function f : C C is called holomorphic if it satisfies the CauchyRiemann equations. For example, any R-linearmap C C can be written in the form

f(z) = az + bz

with complex coefficients a and b. This map is holomorphic if and only if b = 0. The second summand bz is real-differentiable, but does not satisfy the CauchyRiemann equations.Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functionsf and g that agree on an arbitrarily small open subset of C necessarily agree everywhere. Meromorphic functions,functions that can locally be written as f(z)/(z z0)n with a holomorphic function f, still share some of the featuresof holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0.

1.9 Applications

Complex numbers have essential concrete applications in a variety of scientific and related areas such as signal pro-cessing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis.Some applications of complex numbers are:

1.9.1 Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplacetransform. The systems poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, andNichols plot techniques all make use of the complex plane.In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e.have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are

in the right half plane, it will be unstable,

all in the left half plane, it will be stable,

on the imaginary axis, it will have marginal stability.

If a system has zeros in the right half plane, it is a nonminimum phase system.

1.9.2 Improper integrals

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means ofcomplex-valued functions. Several methods exist to do this; see methods of contour integration.
https://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identitieshttps://en.wikipedia.org/wiki/Holomorphic_functionhttps://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equationshttps://en.wikipedia.org/wiki/Linear_transformation#Definition_and_first_consequenceshttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equationshttps://en.wikipedia.org/wiki/Open_subsethttps://en.wikipedia.org/wiki/Meromorphic_functionhttps://en.wikipedia.org/wiki/Essential_singularityhttps://en.wikipedia.org/wiki/Signal_processinghttps://en.wikipedia.org/wiki/Signal_processinghttps://en.wikipedia.org/wiki/Control_theoryhttps://en.wikipedia.org/wiki/Electromagnetismhttps://en.wikipedia.org/wiki/Fluid_dynamicshttps://en.wikipedia.org/wiki/Quantum_mechanicshttps://en.wikipedia.org/wiki/Cartographyhttps://en.wikipedia.org/wiki/Vibration#Vibration_analysishttps://en.wikipedia.org/wiki/Control_theoryhttps://en.wikipedia.org/wiki/Time_domainhttps://en.wikipedia.org/wiki/Frequency_domainhttps://en.wikipedia.org/wiki/Laplace_transformhttps://en.wikipedia.org/wiki/Laplace_transformhttps://en.wikipedia.org/wiki/Pole_(complex_analysis)https://en.wikipedia.org/wiki/Zero_(complex_analysis)https://en.wikipedia.org/wiki/Root_locushttps://en.wikipedia.org/wiki/Nyquist_plothttps://en.wikipedia.org/wiki/Nichols_plothttps://en.wikipedia.org/wiki/Pole_(complex_analysis)https://en.wikipedia.org/wiki/Zero_(complex_analysis)https://en.wikipedia.org/wiki/Unstablehttps://en.wikipedia.org/wiki/BIBO_stabilityhttps://en.wikipedia.org/wiki/Marginal_stabilityhttps://en.wikipedia.org/wiki/Nonminimum_phasehttps://en.wikipedia.org/wiki/Improper_integralhttps://en.wikipedia.org/wiki/Methods_of_contour_integration

1.9. APPLICATIONS 19

1.9.3 Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in two dimensions.

1.9.4 Dynamic equations

In differential equations, it is common to first find all complex roots r of the characteristic equation of a lineardifferential equation or equation system and then attempt to solve the system in terms of base functions of the formf(t) = ert . Likewise, in difference equations, the complex roots r of the characteristic equation of the differenceequation system are used, to attempt to solve the system in terms of base functions of the form f(t) = rt .

1.9.5 Electromagnetism and electrical engineering

Main article: Alternating current

In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment ofresistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistancesfor the latter two and combining all three in a single complex number called the impedance. This approach is calledphasor calculus.In electrical engineering, the imaginary unit is denoted by j, to avoid confusion with I, which is generally in use todenote electric current, or, more particularly, i, which is generally in use to denote instantaneous electric current.Since the voltage in an AC circuit is oscillating, it can be represented as

V (t) = V0ejt = V0 (cost+ j sint) ,

To obtain the measurable quantity, the real part is taken:

v(t) = Re(V ) = Re[V0e

jt]= V0 cost.

The complex-valued signal V (t) is called the analytic representation of the real-valued, measurable signal v(t) . [14]

1.9.6 Signal analysis

Complex numbers are used in signal analysis and other fields for a convenient description for periodically varyingsignals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corre-sponding complex functions are considered of which the real parts are the original quantities. For a sine wave of agiven frequency, the absolute value | z | of the corresponding z is the amplitude and the argument arg(z) is the phase.If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodicfunctions are often written as complex valued functions of the form

x(t) = Re{X(t)}

and

X(t) = Aeit = aeieit = aei(t+)

where represents the angular frequency and the complex number A encodes the phase and amplitude as explainedabove.This use is also extended into digital signal processing and digital image processing, which utilize digital versions ofFourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, stillimages, and video signals.
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Another example, relevant to the two side bands of amplitude modulation of AM radio, is:

cos(( + )t) + cos (( )t) = Re(ei(+)t + ei()t

)= Re

((eit + eit) eit

)= Re

(2 cos(t) eit

)= 2 cos(t) Re

(eit

)= 2 cos(t) cos (t) .

1.9.7 Quantum mechanics

The complex number field is intrinsic to themathematical formulations of quantummechanics, where complexHilbertspaces provide the context for one such formulation that is convenient and perhaps most standard. The originalfoundation formulas of quantum mechanicsthe Schrdinger equation and Heisenbergs matrix mechanicsmakeuse of complex numbers.

1.9.8 Relativity

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the timecomponent of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity,but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are ageneralization of the tensors used in relativity.

1.9.9 Geometry

Fractals

Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set and Julia sets.

Triangles

Every triangle has a unique Steiner inellipsean ellipse inside the triangle and tangent to the midpoints of the threesides of the triangle. The foci of a triangles Steiner inellipse can be found as follows, according to Mardens theo-rem:[15][16] Denote the triangles vertices in the complex plane as a = xA + yAi, b = xB + yBi, and c = xC + yCi. Writethe cubic equation (xa)(xb)(xc)=0 , take its derivative, and equate the (quadratic) derivative to zero. MardensTheorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci ofthe Steiner inellipse.

1.9.10 Algebraic number theory

As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in C. A fortiori,the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers they are a principal object of study in algebraic number theory. Compared to Q, the algebraic closure of Q, whichalso contains all algebraic numbers, C has the advantage of being easily understandable in geometric terms. In thisway, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, morespecifically applying the machinery of field theory to the number field containing roots of unity, it can be shown thatit is not possible to construct a regular nonagon using only compass and straightedge a purely geometric problem.Another example are Gaussian integers, that is, numbers of the form x + iy, where x and y are integers, which can beused to classify sums of squares.
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1.10. HISTORY 21

Construction of a regular pentagon using straightedge and compass.

1.9.11 Analytic number theory

Main article: Analytic number theory

Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can beregarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoreticinformation in complex-valued functions. For example, the Riemann zeta function (s) is related to the distributionof prime numbers.

1.10 History

The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of theGreek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considers, apparentlyin error, the volume of an impossible frustum of a pyramid to arrive at the term 81144=3i7 in his calculations,although negative quantities were not conceived of in Hellenistic mathematics and Heron merely replaced it by itspositive ( 14481=37 ).[17]
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The impetus to study complex numbers proper first arose in the 16th century when algebraic solutions for the roots ofcubic and quartic polynomials were discovered by Italian mathematicians (see Niccol Fontana Tartaglia, GerolamoCardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimesrequired the manipulation of square roots of negative numbers. As an example, Tartaglias formula for a cubicequation of the form x3=px+q [18] gives the solution to the equation x3 = x as

13

((1)1/3 + 1

(1)1/3

).

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equationz3 = i has solutions i, 32 + 12 i and

3

2 +12 i . Substituting these in turn for

11/3 in Tartaglias cubic formula

and simplifying, one gets 0, 1 and 1 as the solutions of x3 x = 0. Of course this particular equation can be solvedat sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, aslater mathematicians showed rigorously, the use of complex numbers is unavoidable. Rafael Bombelli was the firstto explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complexarithmetic trying to resolve these issues.The term imaginary for these quantities was coined by Ren Descartes in 1637, although he was at pains to stresstheir imaginary nature[19]

[...] sometimes only imaginary, that is one can imagine as many as I said in each equation, butsometimes there exists no quantity that matches that which we imagine.

([...] quelquefois seulement imaginaires cest--dire que lon peut toujours en imaginer autant quej'ai dit en chaque quation, mais quil ny a quelquefois aucune quantit qui corresponde celle quonimagine.)

A further source of confusion was that the equation 12=11=1 seemed to be capriciously inconsistent withthe algebraic identity ab=ab , which is valid for non-negative real numbers a and b, and which was also usedin complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity(and the related identity 1

a=

1a ) in the case when both a and b are negative even bedeviled Euler. This difficulty

eventually led to the convention of using the special symbol i in place of 1 to guard against this mistake. Even soEuler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementaryalgebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a naturalway throughout.In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complexexpressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abrahamde Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle topowers of trigonometric functions of that angle could be simply re-expressed by the following well-known formulawhich bears his name, de Moivres formula:

(cos + i sin )n = cosn + i sinn.

In 1748 Leonhard Euler went further and obtained Eulers formula of complex analysis:

cos + i sin = ei

by formally manipulating complex power series and observed that this formula could be used to reduce any trigono-metric identity to much simpler exponential identities.The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799,although it had been anticipated as early as 1685 in Walliss De Algebra tractatus.Wessels memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of thefundamental theorem of algebra. Gauss had earlier published an essentially topological proof of the theorem in1797 but expressed his doubts at the time about the true metaphysics of the square root of 1. It was not until
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1.11. GENERALIZATIONS AND RELATED NOTIONS 23

1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largelyestablishing modern notation and terminology. The English mathematician G. H. Hardy remarked that Gauss was thefirst mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians suchas Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his1831 treatise.[20] Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complexanalysis to a high state of completion, commencing around 1825 in Cauchys case.The common terms used in the theory are chiefly due to the founders. Argand called cos+i sin the direction factor,and r=a2+b2 themodulus; Cauchy (1828) called cos+i sin the reduced form (l'expression rduite) and apparentlyintroduced the term argument; Gauss used i for 1 , introduced the term complex number for a + bi, and called a2 +b2 the norm. The expression direction coefficient, often used for cos+ i sin , is due to Hankel (1867), and absolutevalue, for modulus, is due to Weierstrass.Later classical writers on the general theory include Richard Dedekind, Otto Hlder, Felix Klein, Henri Poincar,Hermann Schwarz, Karl Weierstrass and many others.

1.11 Generalizations and related notions

The process of extending the field R of reals to C is known as CayleyDickson construction. It can be carried furtherto higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4and 8, respectively.However, just as applying the construction to reals loses the property of ordering, more properties familiar from realand complex numbers vanish with increasing dimension. The quaternions are only a skew field, i.e. for some x,y: xy yx for two quaternions, the multiplication of octonions fails (in addition to not being commutative) to beassociative: for some x, y, z: (xy)z x(yz).Reals, complex numbers, quaternions and octonions are all normed division algebras over R. However, by Hurwitzstheorem they are the only ones. The next step in the CayleyDickson construction, the sedenions, in fact fails to havethis structure.The CayleyDickson construction is closely related to the regular representation of C, thought of as an R-algebra (anR-vector space with a multiplication), with respect to the basis (1, i). This means the following: the R-linear map

C C, z 7 wz

for some fixed complex number w can be represented by a 2 2 matrix (once a basis has been chosen). With respectto the basis (1, i), this matrix is

(Re(w) Im(w)Im(w) Re(w)

)i.e., the one mentioned in the section on matrix representation of complex numbers above. While this is a linearrepresentation of C in the 2 2 real matrices, it is not the only one. Any matrix

J =

(p qr p

), p2 + qr + 1 = 0

has the property that its square is the negative of the identity matrix: J2 = I. Then

{z = aI + bJ : a, b R}

is also isomorphic to the field C, and gives an alternative complex structure on R2. This is generalized by the notionof a linear complex structure.Hypercomplex numbers also generalizeR,C,H, andO. For example this notion contains the split-complex numbers,which are elements of the ring R[x]/(x2 1) (as opposed to R[x]/(x2 + 1)). In this ring, the equation a2 = 1 has foursolutions.
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The field R is the completion of Q, the field of rational numbers, with respect to the usual absolute value metric.Other choices of metrics on Q lead to the fields Qp of p-adic numbers (for any prime number p), which are therebyanalogous to R. There are no other nontrivial ways of completing Q than R and Qp, by Ostrowskis theorem. Thealgebraic closure Qp of Qp still carry a norm, but (unlike C) are not complete with respect to it. The completion Cpof Qp turns out to be algebraically closed. This field is called p-adic complex numbers by analogy.The fields R and Qp and their finite field extensions, including C, are local fields.

1.12 See also

Algebraic surface

Circular motion using complex numbers

Complex base systems

Complex geometry

Complex square root

Domain coloring

Eisenstein integer

Eulers identity

Gaussian integer

Mandelbrot set

Quaternion

Riemann sphere (extended complex plane)

Root of unity

Unit complex number

1.13 Notes[1] Charles P. McKeague (2011), Elementary Algebra, Brooks/Cole, p. 524, ISBN 978-0-8400-6421-9

[2] Burton (1995, p. 294)

[3] Complex Variables (2nd Edition), M.R. Spiegel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaums Outline Series, McGraw Hill (USA), ISBN 978-0-07-161569-3

[4] Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007), Chapter P, College Algebra and Trigonometry (6ed.), Cengage Learning, p. 66, ISBN 0-618-82515-0

[5] For example Ahlfors (1979).

[6] Brown, James Ward; Churchill, Ruel V. (1996), Complex variables and applications (6th ed.), New York: McGraw-Hill,p. 2, ISBN 0-07-912147-0, In electrical engineering, the letter j is used instead of i.

[7] Katz (2004, 9.1.4)

[8] http://mathworld.wolfram.com/ComplexNumber.html

[9] Abramowitz, Milton; Stegun, Irene A. (1964), Handbook of mathematical functions with formulas, graphs, and mathemat-ical tables, Courier Dover Publications, p. 17, ISBN 0-486-61272-4, Section 3.7.26, p. 17

[10] Cooke, Roger (2008), Classical algebra: its nature, origins, and uses, John Wiley and Sons, p. 59, ISBN 0-470-25952-3,Extract: page 59
https://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Absolute_valuehttps://en.wikipedia.org/wiki/Metric_(mathematics)https://en.wikipedia.org/wiki/Metric_(mathematics)https://en.wikipedia.org/wiki/P-adic_numberhttps://en.wikipedia.org/wiki/Prime_numberhttps://en.wikipedia.org/wiki/Ostrowski%2527s_theoremhttps://en.wikipedia.org/wiki/Local_fieldhttps://en.wikipedia.org/wiki/Algebraic_surfacehttps://en.wikipedia.org/wiki/Circular_motion#Using_complex_numbershttps://en.wikipedia.org/wiki/Complex_base_systemshttps://en.wikipedia.org/wiki/Complex_geometryhttps://en.wikipedia.org/wiki/Complex_square_roothttps://en.wikipedia.org/wiki/Domain_coloringhttps://en.wikipedia.org/wiki/Eisenstein_integerhttps://en.wikipedia.org/wiki/Euler%2527s_identityhttps://en.wikipedia.org/wiki/Gaussian_integerhttps://en.wikipedia.org/wiki/Mandelbrot_sethttps://en.wikipedia.org/wiki/Quaternionhttps://en.wikipedia.org/wiki/Riemann_spherehttps://en.wikipedia.org/wiki/Root_of_unityhttps://en.wikipedia.org/wiki/Unit_complex_numberhttps://books.google.com/?id=etTbP0rItQ4C&pg=PA524https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-8400-6421-9https://en.wikipedia.org/wiki/Complex_number#CITEREFBurton1995https://en.wikipedia.org/wiki/Special:BookSources/9780071615693http://books.google.com/?id=g5j-cT-vg_wC&pg=PA66https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-618-82515-0https://en.wikipedia.org/wiki/Complex_number#CITEREFAhlfors1979https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-07-912147-0https://en.wikipedia.org/wiki/Complex_number#CITEREFKatz2004http://mathworld.wolfram.com/ComplexNumber.htmlhttp://books.google.com/books?id=MtU8uP7XMvoChttp://books.google.com/books?id=MtU8uP7XMvoChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-486-61272-4http://www.math.sfu.ca/~cbm/aands/page_17.htmhttp://books.google.com/books?id=lUcTsYopfhkChttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-470-25952-3http://books.google.com/books?id=lUcTsYopfhkC&pg=PA59

1.14. REFERENCES 25

[11] Ahlfors (1979, p. 3)

[12] Kasana, H.S. (2005), Chapter 1, Complex Variables: Theory And Applications (2nd ed.), PHI Learning Pvt. Ltd, p. 14,ISBN 81-203-2641-5

[13] Nilsson, James William; Riedel, Susan A. (2008), Chapter 9, Electric circuits (8th ed.), Prentice Hall, p. 338, ISBN0-13-198925-1

[14] Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN 0-471-92712-0

[15] Kalman, Dan (2008a), An Elementary Proof of Mardens Theorem, The American Mathematical Monthly 115: 33038,ISSN 0002-9890

[16] Kalman, Dan (2008b), TheMostMarvelous Theorem inMathematics, Journal of OnlineMathematics and its Applications

[17] Nahin, Paul J. (2007), An Imaginary Tale: The Story of 1, Princeton University Press, ISBN 978-0-691-12798-9,retrieved 20 April 2011

[18] In modern notation, Tartaglias solution is based on expanding the cube of the sum of two cube roots: ( 3u+ 3v)3=3 3

uv( 3u+ 3

v)+u+v With x= 3u+ 3v , p=3 3uv , q=u+v , u and v can be expressed in terms of p and q as u=q/2+

(q/2)2(p/3)3 and v=q/2(q/2)2(p/3)3 , respectively. Therefore, x= 3

q/2+

(q/2)2(p/3)3+ 3

q/2

(q/2)2(p/3)3

. When (q/2)2(p/3)3 is negative (casus irreducibilis), the second cube root should be regarded as the complex conjugateof the first one.

[19] Descartes, Ren (1954) [1637], La Gomtrie | The Geometry of Ren Descartes with a facsimile of the first edition, DoverPublications, ISBN 0-486-60068-8, retrieved 20 April 2011

[20] Hardy, G. H.; Wright, E. M. (2000) [1938], An Introduction to the Theory of Numbers, OUP Oxford, p. 189 (fourthedition), ISBN 0-19-921986-9

1.14 References

1.14.1 Mathematical references

Ahlfors, Lars (1979), Complex analysis (3rd ed.), McGraw-Hill, ISBN 978-0-07-000657-7

Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN 0-387-90328-3

Joshi, Kapil D. (1989), Foundations of Discrete Mathematics, New York: John Wiley & Sons, ISBN 978-0-470-21152-6

Pedoe, Dan (1988), Geometry: A comprehensive course, Dover, ISBN 0-486-65812-0

Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), Section 5.5 Complex Arithmetic, Nu-merical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN978-0-521-88068-8

Solomentsev, E.D. (2001), Complex number, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

1.14.2 Historical references

Burton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-009465-9

Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2

Nahin, Paul J. (1998), An Imaginary Tale: The Story of 1 (hardcover edition ed.), Princeton UniversityPress, ISBN 0-691-02795-1

A gentle introduction to the history of complex numbers and the beginnings of complex analysis.

H.D. Ebbinghaus; H. Hermes; F. Hirzebruch; M. Koecher; K. Mainzer; J. Neukirch; A. Prestel; R. Remmert(1991), Numbers (hardcover edition ed.), Springer, ISBN 0-387-97497-0

An advanced perspective on the historical development of the concept of number.
https://en.wikipedia.org/wiki/Complex_number#CITEREFAhlfors1979http://books.google.com/?id=rFhiJqkrALIC&pg=PA14https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/81-203-2641-5http://books.google.com/?id=sxmM8RFL99wC&pg=PA338https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-13-198925-1https://en.wikipedia.org/wiki/Special:BookSources/0471927120http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1https://en.wikipedia.org/wiki/American_Mathematical_Monthlyhttps://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttps://www.worldcat.org/issn/0002-9890http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1663http://mathdl.maa.org/mathDL/4/http://mathforum.org/kb/thread.jspa?forumID=149&threadID=383188&messageID=1181284https://en.wikipedia.org/wiki/Princeton_University_Presshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-691-12798-9https://en.wikipedia.org/wiki/Ren%C3%A9_Descarteshttp://www.gutenberg.org/ebooks/26400https://en.wikipedia.org/wiki/Dover_Publicationshttps://en.wikipedia.org/wiki/Dover_Publicationshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-486-60068-8https://en.wikipedia.org/wiki/Oxford_University_Presshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-19-921986-9https://en.wikipedia.org/wiki/Lars_Ahlforshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-07-000657-7https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-387-90328-3https://en.wikipedia.org/wiki/John_Wiley_&_Sonshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-470-21152-6https://en.wikipedia.org/wiki/Special:BookSources/978-0-470-21152-6https://en.wikipedia.org/wiki/Dan_Pedoehttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-486-65812-0http://apps.nrbook.com/empanel/index.html?pg=225https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-521-88068-8http://www.encyclopediaofmath.org/index.php?title=c/c024140https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/McGraw-Hillhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-07-009465-9https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-009465-9https://en.wikipedia.org/wiki/Addison-Wesleyhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0-321-16193-2https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-691-02795-1https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-387-97497-0


1.15 Further reading The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf,2005; ISBN 0-679-45443-8. Chapters 47 in particular deal extensively (and enthusiastically) with complexnumbers.

Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations andthe structures of modern algebra.

Visual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN 0-19-853447-7 (hardcover, 1997).History of complex numbers and complex analysi

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