Complex number

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Illustration of the complex plane. The real numbers are found in the horizontal coordinate axis and the imaginaries in the vertical axis.

Them complex numbersdesignated with notation ${displaystyle scriptstyle mathbb {C} }$ , are an extension of the real numbers $scriptstyle {mathbb {R}}$ and form an algebraicly closed body. Between both sets of numbers it is fulfilled that ${displaystyle scriptstyle mathbb {R} subset mathbb {C} }$ I mean, $scriptstyle {mathbb {R}}$ is strictly contained in $scriptstyle {mathbb {C}}$ . Complex numbers include all the roots of polynomials, unlike the real ones. Everything complex number can be represented as the sum of a real number and an imaginary number (which is a real multiple of the imaginary unit, which is indicated with the letter ior in polar form.

Complex numbers are the working tool of algebra, analysis, as well as branches of pure and applied mathematics such as complex variable, differential equations, facilitates the calculation of integrals, in aerodynamics, hydrodynamics and electromagnetism among others of great importance. Furthermore, complex numbers are used everywhere in mathematics, in many fields of physics (notably quantum mechanics) and in engineering, especially electronics and telecommunications, for their utility in representing electromagnetic waves and electric current.

In mathematics, these numbers constitute a field and are generally considered to be points on the plane: the complex plane. This body contains the real numbers and the pure imaginary ones.

History

The general formula for solving the roots (without using trigonometric functions) of a quadratic equation contains the square roots of a negative number when all three roots are real numbers, a situation that cannot be rectified by factoring with the help of of rational root theorem if the cubic polynomial is irreducible (the so-called casus irreducibilis). This enigma led the Italian mathematician Gerolamo Cardano to conceive of complex numbers around 1545, although his understanding was rudimentary.

Work on the problem of general polynomials eventually led to the Fundamental Theorem of Algebra, which shows that in the domain of complex numbers, there exists a solution to every polynomial equation of degree one or higher. The complex numbers form an algebraically closed field, where any polynomial equation has a root.

Numerous mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and taking roots of complex numbers were developed by the Italian mathematician Rafael Bombelli, and it was the Irish mathematician William Rowan Hamilton who developed a more abstract formalism for complex numbers, extending this abstraction to the quaternion theory.

Perhaps it can be said that the earliest fleeting reference to square root of negative number appears in the work of the Greek mathematician of the centuryI Heron of Alexandria. In his Stereometrica apparently mistakenly considers the volume of a pyramid trunk with an impossible solution, reaching the end ${displaystyle {sqrt {81-144}}=3i{sqrt {7}}}$ in its calculations, although no negative amounts were granted in hellenic mathematics and Heron simply replaced it with the same positive value ( ${displaystyle {sqrt {144-81}}=3{sqrt {7}}}$ ).

The interest in studying complex numbers as a subject itself arose for the first time in the centuryXVIwhen Italian mathematicians discovered algebraic solutions for the roots of cubic and quartum polynomials (see Niccolò Fontana Tartaglia and Gerolamo Cardano). They soon realized that these formulas, even if they were only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. As for example, in the Tartaglia formula for a cubic equation of form ${displaystyle x^{3}=px+q}$ the solution to the equation $x 3 = x$ in the form

{displaystyle {tfrac {1}{sqrt {3}}}left(left({sqrt {-1}}right)^{1/3}+left({sqrt {-1}}right)^{-1/3}right).}

At first glance, this looks like a nonsense. However, formal calculations with complex numbers show that the equation $z 3 = i$ solutions $- i$ , ${displaystyle {tfrac {sqrt {3}}{2}}+{tfrac {1}{2}}i}$ and ${displaystyle {tfrac {-{sqrt {3}}}{2}}+{tfrac {1}{2}}i}$ . Replace these in turn for ${displaystyle {{sqrt {-1}}^{1/3}}}$ in the cubic formula of Tartaglia and simplifying, 0, 1 and -1 are obtained as the solutions of $x 3 - x = 0$ . Of course, this particular equation can be solved in plain view, but it illustrates that when general formulas are used to resolve cubic equations with real roots, then, as the subsequent mathematicians demonstrated rigorously, the use of complex numbers is inevitable. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of the cubic equations, and developed the rules for the complex arithmetic that tries to solve these problems.

The term "imaginary" for these amounts was coined by René Descartes in 1637, striving precisely to emphasize its imaginary nature.

[...] sometimes only imaginary, that is, one can imagine as many as already said in each equation, but sometimes there is no amount that matches what we imagine. ([...] quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginr autant que j'ai dit en chaque équation, mais qu'il n'y a quelquefois aucune quantité qui corresponden à celle qu'on image.)

Another source of confusion was that the equation ${displaystyle {sqrt {-1}}^{2}={sqrt {-1}}{sqrt {-1}}=-1}$ seemed to be capriciously inconsistent with algebraic identity ${displaystyle {sqrt {a}}{sqrt {b}}={sqrt {ab}}}$ which is valid for non-negative real numbers $a$ and $b$ , and that was also used in calculations of complex numbers with one of $a$ or $b$ positive and the other negative. Incorrect use of this identity (and related identity ${displaystyle {tfrac {1}{sqrt {a}}}={sqrt {tfrac {1}{a}}}}$ ) in the case that $a$ and $b$ be negative, he even worried Euler. This difficulty eventually led to the convention to use the special symbol $i$ instead of √−1 to protect against this mistake. Even so, Euler considered it natural to present the students complex numbers long before what is done today. In his textbook of elemental algebra, Elements of Algebra, he introduced these numbers almost immediately and then used them naturally.

In the 18th century, complex numbers gained wider use, as it was noted that the formal manipulation of Complex expressions could be used to simplify calculations involving trigonometric functions. For example, in 1730 Abraham de Moivre observed that the complicated identities relating the trigonometric functions of an integer multiple of an angle to the powers of the trigonometric functions of that angle could simply be restated by the following well-known formula that bears his name, the formula from De Moivre:

{displaystyle (cos theta +ioperatorname {sen} theta)^{n}=cos ntheta +ioperatorname {sen} ntheta.}

In 1748 Leonhard Euler went further and obtained Euler's Formula for complex analysis:

{displaystyle cos theta +ioperatorname {sen} theta =e^{itheta }}

formally manipulating complex power series, and observed that this formula could be used to reduce any trigonometric identities to much simpler exponential identities.

The idea of a complex number as a point in the complex plane was first described by Caspar Wessel in 1799, although it had been anticipated as early as 1685 in De Algebra tractatus by John Wallis.

The Wessel Memoirs appeared in the Proceedings of the Royal Danish Academy of Fine Arts, but went unnoticed. In 1806, Jean-Robert Argand independently issued a booklet on complex numbers and provided a rigorous proof of the Fundamental Theorem of Algebra. Carl Friedrich Gauss had previously published an essentially topological proof of the theorem in 1797, but expressed doubts to him at the time about & # 34; the true metaphysics of the square root of −1 & # 34; . It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. In the early 19th century, other mathematicians independently discovered the geometric representation of complex numbers: Buée, Mourey, Warren, Français, and his brother, Bellavitis.

The English mathematician Godfrey Harold Hardy commented that Gauss was the first mathematician to use complex numbers "in a really safe and scientific way", although mathematicians like Niels Henrik Abel and Carl Gustav Jakob Jacobi necessarily used them routinely before Gauss published his 1831 treatise.

“If this subject has been considered so far from the wrong point of view and, therefore, wrapped in mystery and surrounded by darkness, it is largely due to inadequate terminology that must be blamed. Yes to +1, -1 and √−1, instead of being called positive, negative and imaginary unity (or worse yet impossible), they would have been given the names of direct, reverse and lateral unity, would hardly have spread such darkness.” - Gauss

Augustin Louis Cauchy and Bernhard Riemann provided fundamental insights into complex analysis, raising it to a high state of completion, beginning around 1825 in the case of Cauchy.

The common terms used in theory are mainly due to its founders. Argand called "direction factor" to ${displaystyle cos phi +isin phi }$ and "module" to ${displaystyle r={sqrt {a^{2}+b^{2}}}}$ ; Cauchy (1828) called ${displaystyle cos phi +isin phi }$ the "reduced form" (l'expression réduite) and apparently introduced the term "argument"; Gauss used $i$ for $sqrt{-1}$ , introduced the term "complex number" for $a + bi$ and called $a 2 + b 2$ the "norma." The expression Direction coefficientoften used for ${displaystyle cos theta +ioperatorname {sen} theta }$ is due to Hankel (1867), and absolute value for module It's because of Weierstrass.

Subsequent classic writers on general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Amandus Schwarz, Karl Weierstrass, and many others.

Complex numbers linked to analytical functions or complex variables have allowed the concept of calculus to be extended to the complex plane. The complex variable calculus has several notable properties that entail properties that can be used to obtain various useful results in applied mathematics.

Definition

Each complex number is defined z like an orderly pair of real numbers: z =a, b). In turn the first element a defined as real part of z, it denotes ${displaystyle a={text{Re}}(z)}$ the second element b defined as imaginary part of z, it denotes ${displaystyle b={text{Im}}(z)}$ . Then in group C of complex numbers, three operations are defined and the relation of equality:

Equality

(a,b)=(c,d)iff a=cland b=d

Number $(a,0)$ is called actual complex number and as between the set of these and the R set of the actual numbers is established an isomorphism is assumed that all real number is a complex number. To the complex number $(0,b)$ is called pure imaginary number. Post $(a,0) + (0,b) = (a, b)$ it is said that a complex number is the sum of a real number with a pure imaginary number.

Rational Operations

Addendum

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

Output by climbing

r(a,b)=(ra,,rb)

Multiplication

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

From these operations we can deduce others such as the following:

Resta

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

Division

{displaystyle {frac {(a,b)}{(c,d)}}={(ac+bd,,bc-ad) over c^{2}+d^{2}}=left({ac+bd over c^{2}+d^{2}},{bc-ad over c^{2}+d^{2}}right)}

Imaginary unit

A special complex number is defined, especially in algebra, of great relevance, the number i (j in physics), called imaginary unit, defined as

{mathrm {i}}=(0,1),!

Which satisfies the following equality:

{mathrm {i}}^{2}={mathrm {i}}cdot {mathrm {i}}=(0,1)cdot (0,1)=(-1,0)=-1

Taking into account that $(a,0)cdot (0,1)=(0,a)$ I.D.

(a, 0) cdot (0, 1) = amathrm{i} = (0, a)

In elementary texts it is defined that i² equals -1. It is also one of the roots of the equation x² + 1 = 0.

Absolute value or module, argument and conjugate

Absolute value or module of a complex number

The formula of Euler illustrated in the complex plane.

The absolute value, module or magnitude of a complex number z is given by the following expression:

$|z|={sqrt {zz^{*}}}={sqrt {{hbox{Re}}^{2}(z)+{hbox{Im}}^{2}(z)}}$

If we think of the Cartesian coordinates of the complex number z as some point in the plane; We can see, by the Pythagorean theorem, that the absolute value of a complex number coincides with the Euclidean distance from the origin of the plane to that point.

If the complex is written in exponential form z = r e^iφ, then |z| = r. It can be expressed in trigonometric form as z = r (cosφ + isenφ), where cosφ + isenφ = e^iφ is the well-known Euler formula.

We can easily check these four important properties of absolute value

left|zright|=0Longleftrightarrow z=0

left|z+wright|leq |z|+|w|

left|zwright|=|z||w|

{displaystyle left|z-wright|geq ||z|-|w||}

for any complex z and w.

By definition, the distance function reads as follows d(z, w) = |z - w| and it provides us with a metric space with the complexes thanks to which we can speak of limits and continuity. Complex addition, subtraction, multiplication, and division are continuous operations. Unless otherwise stated, it is assumed that this is the metric used on complex numbers.

Plot or phase

Main article: Argument (complex analysis)

The main argument or phase of a generic complex number $z=x+yi,$ Where ${displaystyle x=Re(z)}$ e ${displaystyle y=Im(z)}$ , it's the angle $phi$ forming the axis of abscises OX and the vector OMwith M(x,and). It is given by the following expression:

$phi =operatorname {Arg}(z)=operatorname {atan2}(y,x)$

where atan2(y,x) is the arctangent function defined for the four quadrants:

0\arctan left({frac {y}{x}}right)+pi &qquad ygeq 0,x<0\arctan left({frac {y}{x}}right)-pi &qquad y<0,x0,x=0\-{frac {pi }{2}}&qquad yatan2 (and,x)={arctan (andx)x▪0arctan (andx)+π π and≥ ≥ 0,x.0arctan (andx)− − π π and.0,x.0+π π 2and▪0,x=0− − π π 2and.0,x=0indefinitelyand=0,x=0{cHFFFFFF}{cH00FF00}{cH00FF00}{cH00FF00}{cH00FF00}{cH00FF00}{cH00FFFF00}{cH00FFFF00}{cH00FF00}{cH00FF00}{cH00FFFFFF00}{cH00}{cH00FFFFFFFFFF00}{cH00}{cH00}{cH00}{cH00FFFFFFFFFFFFFFFF00}{cH00}{cH00}{cH00}{cH00FFFFFFFFFFFF00FFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cH00}{cH00}{cH00}{cH00}{cH00}{cH00FFFF00}{cH00}{cH00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0\arctan left({frac yx}right)+pi &qquad ygeq 0,x<0\arctan left({frac yx}right)-pi &qquad y<0,x0,x=0\-{frac {pi }{2}}&qquad y

Or also: $operatorname {atan2}(y,x)={frac pi 2}operatorname{sgn}(y)-arctan left({frac xy}right)quad forall x,yin {mathbb R}$ Being:

<math alttext="{displaystyle operatorname {sgn}(y)={begin{cases}1qquad ygeq 0\-1qquad ysgn (and)={1and≥ ≥ 0− − 1and.0{displaystyle operatorname {sgn}(y)={begin{cases}1qquad ygeq 0qquad yqquad y \end{cases}}}}}}<img alt="operatorname{sgn}(y)={begin{cases}1qquad ygeq 0\-1qquad y

the sign function.

The argument has periodicity 2π, with what ${displaystyle arg z=operatorname {arg} z+2kpi }$ being $k$ any whole number. The angle Arg z is the main value of arg z which verifies the conditions -π Δ Arg z ≤ π described above.

Conjugate of a complex number

Two binomials are called conjugates if they only differ in their central sign. In this way, the conjugate of a complex z (denoted as ${bar {z}}$ or $z^{*},!$ ) is a new complex number, defined as:

{displaystyle {bar {z}}=a-bmathrm {i} Longleftrightarrow z=a+bmathrm {i} }

It is observed that both differ in the sign of the imaginary part. With this number the properties are fulfilled:

overline {z+w}={bar {z}}+{bar {w}}

z+overline {z}=2cdot {hbox{Re}}(z)

z-overline {z}=2icdot {hbox{Im}}(z)

overline {zw}={bar {z}}{bar {w}}

zin {mathbb {R}}Longleftrightarrow {bar {z}}=z

{displaystyle |z|^{2}=z{bar {z}}geq 0}

zneq 0Rightarrow {frac {1}{z}}={frac {{bar {z}}}{|z|^{2}}}

This last formula is the method chosen to calculate the inverse of a complex number if it is given in rectangular coordinates.

Representations

Binomial representation

A complex number represented as a point (red) and a position vector (blue) in an Argand diagram;

a+bi

is the expression binomial from the point.

A complex number is represented in binomial form as:

z=a+bi,

The real part of the complex number and the imaginary part can be expressed in several ways, as shown below:

a={hbox{Re}}(z)=Re (z)

b={hbox{Im}}(z)=Im (z)

Polar Representation

The argument φ and module r locate a point in an Argand diagram;

r(cos phi +isin phi)

re^{{iphi }}

is the expression Polar from the point.

In this representation, $textstyle {r}$ It's him. module complex number and angle $textstyle {phi }$ It's him. argument of the complex number.

textstyle {phi }=arctan left({frac {b}{a}}right)=arctan left({frac {{hbox{Im}}(z)}{{hbox{Re}}(z)}}right)=-arctan left(-{frac {{hbox{Im}}(z)}{{hbox{Re}}(z)}}right)

cos phi ={frac {a}{r}}\ sin phi ={frac {b}{r}}

Collecting a and b in the previous expressions and, using the binomial representation, it results:

{displaystyle z=a+mathrm {i} b;;z=rcos {phi }+mathrm {i} roperatorname {sen} {phi }}

Taking out common factor r:

z=rleft(cos {phi }+{mathrm {i}}sin {phi }right)

Often, this expression is conveniently abbreviated as follows:

z=r;operatorname {cis};{phi }

which only contains the abbreviations of the trigonometric cosine ratios, the imaginary unit and the sine ratio of the argument respectively.

According to this expression, it can be seen that to define a complex number both in this way and with the binomial representation, two parameters are required, which can be the real and imaginary part or module and argument, respectively.

According to Euler's Formula:

{displaystyle cos {phi }+ioperatorname {sen} {phi }=e^{mathrm {i} phi };;z=re^{iphi }}

In angular notation, often used in Electrotecnia is represented to the module fasor $r$ and argument $phi$ like:

{displaystyle z=rangle phi.}

However, the angle $phi$ is not uniquely determined by z, because there can be infinite complex numbers that have the same value represented on the plane, which are differentiated by the number of revolutions, whether they are of anti-hour (positive) or time (negative) meaning which are represented by integers $kin {mathbb {Z}}$ as the Euler formula implies:

{displaystyle forall {k}{in }mathbb {Z} quad z=re^{mathrm {i} (phi +2pi {}k)}}

For this, usually $phi$ is restricted to the [-π, π) and to this $phi$ restricted is called main argument of z and denotes φ=Arg(z). With this agreement, the coordinates are uniquely determined by z.

Operations in polar form

Multiplying complex numbers is especially easy with polar notation:

z_{1}z_{2}=rse^{{{mathrm {i}}(phi +psi)}}Leftrightarrow z_{1}z_{2}=re^{{{mathrm {i}}phi }}se^{{{mathrm {i}}psi }}

Division:

{frac {z_{1}}{z_{2}}}={frac {r}{s}}e^{{{mathrm {i}}(phi -psi)}}

Empowerment:

{displaystyle z^{n}=r^{n}e^{mathrm {i} phi n}Leftrightarrow z^{n}=left(re^{iphi }right)^{n}z^{n}=(a+bmathrm {i})^{n}=}

z^{n}=(a+b{mathrm {i}})^{n}={n choose 0}a^{n}+{n choose 1}a^{{n-1}}b{mathrm {i}}+{n choose 2}a^{{n-2}}left(b{mathrm {i}}right)^{2}+ldots +{n choose {n-1}}aleft(b{mathrm {i}}right)^{{n-1}}+{n choose n}left(b{mathrm {i}}right)^{n}

Nth root of a complex number

Square root

Given the complex number z we will say ${displaystyle {sqrt {z}}={w_{1},w_{2}}}$ and is fulfilled ${displaystyle z=w_{i}^{2},,i=1,2}$

Example

{displaystyle {sqrt {i}}=left{cos {frac {pi }{4}}+operatorname {sen} {frac {pi }{4}},,cos {frac {3pi }{4}}+operatorname {sen} {frac {3pi }{4}}right}}

Representation in the form of matrices of order 2

In the ring of second-order matrices over the field of real numbers, a subset can be found that is isomorphic to the field of complex numbers. Well, a correspondence is established between each complex number a+bi with the matrix

{displaystyle {begin{pmatrix}a&-b\b&a\end{pmatrix}}.}

In such a way, a bi-curring correspondence is obtained. The sum and product of two of this matrices has again this form, and the sum and product of complex numbers corresponds to the sum and product of such matrices. In particular the matrix ${displaystyle {bigl (}{begin{smallmatrix}0&-1\1&0end{smallmatrix}}{bigr)}}$ fulfills the role of imaginary unity.

Plane of complex numbers or Argand Diagram

Main article: Complex plan

The concept of complex plane allows to interpret complex numbers geometrically. Addition of complex numbers can be related to addition with vectors, and multiplication of complex numbers can be expressed simply using polar coordinates, where the magnitude of the product is the product of the magnitudes of the terms, and the angle counted from the real axis. of the product is the sum of the angles of the terms and can be seen as the transformation of the vector that rotates and changes its size simultaneously.

Multiplying any complex by i corresponds to a 90º counterclockwise rotation. Likewise, the fact that (-1)·(-1)=+1 can be understood geometrically as the combination of two 90° rotations, obtaining a 180º rotation (i squared = -1), resulting in a sign change upon completing one turn.

Argand diagrams are often used to show the positions of the poles and zeros of a function in the complex plane.

Complex analysis, the theory of complex functions, is one of the richest areas of mathematics, which finds application in many other areas of mathematics as well as in physics, electronics, and many other fields.

Properties

Field of complex numbers

The set ℂ of complex numbers satisfies the laws of axiomatics that define a field:

Commutative property: z+w = w+z; zw= wz.
Associative property: v+(w+z♪v+w)+ z; v(wz♪vw)z
Distributive property: v(w+z) = vw+vz(w+z)v = wv+zv
Existence of identities:

Additive identity, zero: z+ 0 = 0+z = z; multipliative identity, 1:

{displaystyle zcdot 1=1cdot z=z}

Inverses: Each complex number has its reverse additive -z such as z +(-z) = 0 and each complex number, different from zero, has its reverse multiplier z^-1, such that z·z^-1 = 1.

If we identify the real number a with the complex (a, 0), the field of real numbers R appears as a subfield of C. Furthermore, C forms a 2-dimensional vector space over the reals. Complexes cannot be ordered like, for example, real numbers, so C cannot be converted to an ordered field in any way.

Vector space

The set ℂ with the addition of complex numbers and considering the real numbers as scalars, can be defined ℂ as a vector space. That is:

Yeah. z,w are complex numbers, then z+w It's a complex number. This internal operation defines an additive group structure.
Yeah. r It's real number and z It's a complex number, then rz, called multiple climbing zIt's also a complex number. The two operations satisfy the axiomatic of a vector or linear space.

Applications

In mathematics

Solutions of Polynomial Equations

A root or zero of the polynomial p is a complex z such that p (z)=0. An important result of this definition is that all polynomial (algebraic) equations of degree n have exactly n solutions in the field of complex numbers, that is, it has exactly n complexes z satisfying the equality p(z)=0, counted with their respective multiplicities. This is known as the Fundamental Theorem of Algebra, and it shows that the complexes are an algebraically closed field; For this reason, mathematicians consider complex numbers to be more natural numbers^{[citation needed]} than the real numbers a when solving equations.

It also holds that if z is a root of a polynomial p with real coefficients, then the complex conjugate of z is also a root of p.

Complex variable or complex analysis

Main article: Complex analysis

The study of functions of complex variables is known as Complex Analysis. It has a myriad of uses as a tool in applied mathematics as well as in other branches of mathematics. Complex analysis provides some important tools for proving theorems even in number theory; while the real functions of a real variable need a Cartesian plane to be represented; functions of complex variables require a four-dimensional space, which makes them especially difficult to represent. Colored illustrations in three-dimensional space are often used to suggest the fourth coordinate or 3D animations to represent all four.

Differential Equations

In differential equations, when the solutions of the linear differential equations with constant coefficients are studied, it is customary to first find the roots (generally complex) $lambda ,$ of the characteristic polynomial, which allows to express the general solution of the system in terms of functions of base of the form: $f(x)=e^{{lambda x}},$ .

Fractals

Main article: Fractal

Many fractal objects, such as the Mandelbrot set, can be obtained from convergence properties of a sequence of complex numbers. Analysis of the domain of convergence reveals that such sets can have enormous self-similar complexity.

In Physics

Complex numbers are used in electronic engineering and other fields for an appropriate description of variable periodical signals (see Fourier Analysis). In an expression of the kind $z=re^{{iphi }},$ We can think of $r,$ like the breadth and the $phi ,$ like the phase of a sinusoidal wave of a given frequency. When we represent an alternating current or voltage (and therefore with sinusoidal behavior) as the actual part of a complex variable function of the form $f(t)=ze^{{iomega t}},$ where ω represents the angular frequency and the complex number z gives us the phase and the amplitude, the treatment of all the formulas that govern the resistances, abilities and inducers can be unified by introducing imaginary resistances for the last two (see electric networks). Electrical and physical engineers use the letter j for imaginary unity instead of i which is typically intended for current intensity.

The complex field is equally important in quantum mechanics whose underlying mathematics uses infinite-dimensional Hilbert spaces over C (ℂ).^{[citation needed]}

In special relativity and general relativity, some formulas for the metric of spacetime are much simpler if we take time as an imaginary variable.^{[citation needed]}

Generalizations

Complex numbers can be generalized giving rise to hypercomplete numbers. The body of complex numbers is a commutative subbody of the quaternionic algebra $scriptstyle {mathbb {H}}$ , which in turn is a sub-algebra of other more extensive algebras (octoniones, sedeniones):

${mathbb {C}}subset {mathbb {H}}subset {mathbb {O}}subset {mathbb {S}}$

Another possible generalization is to consider the completeness of hyperreal numbers:

${mathbb {C}}subset {}^{*}mathbb{R} (i)$

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