The Info List - Complex Numbers

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A COMPLEX NUMBER is a number that can be expressed in the form a + bi, where a and b are real numbers , and i is the imaginary unit (which satisfies the equation i2 = −1). In this expression, a is called the real part of the complex number, and b is called the imaginary part. If z = a + b i {displaystyle z=a+bi} , then we write Re ( z ) = a , {displaystyle operatorname {Re} (z)=a,} and Im ( z ) = b . {displaystyle operatorname {Im} (z)=b.}

Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex 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 whose real part is zero is said to be purely imaginary , whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers are a field extension of the ordinary real numbers, in order to solve problems that cannot be solved with real numbers alone.

As well as their use within mathematics, complex numbers have practical applications in many fields, including physics , chemistry , biology , economics , electrical engineering , and statistics . The Italian mathematician Gerolamo Cardano is the first person known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century.


* 1 Overview

* 1.1 Definition * 1.2 Notation * 1.3 Complex plane * 1.4 History in brief

* 2 Relations

* 2.1 Equality * 2.2 Ordering

* 3 Elementary operations

* 3.1 Conjugate * 3.2 Addition and subtraction * 3.3 Multiplication and division * 3.4 Reciprocal * 3.5 Square root

* 4 Polar form

* 4.1 Absolute value
Absolute value
and argument * 4.2 Multiplication and division in polar form

* 5 Exponentiation

* 5.1 Euler\'s formula * 5.2 Natural logarithm * 5.3 Integer
and fractional exponents

* 6 Properties

* 6.1 Field structure * 6.2 Solutions of polynomial equations * 6.3 Algebraic characterization * 6.4 Characterization as a topological field

* 7 Formal construction

* 7.1 Formal development * 7.2 Matrix representation of complex numbers

* 8 Complex analysis
Complex analysis

* 8.1 Complex exponential and related functions * 8.2 Holomorphic functions

* 9 Applications

* 9.1 Control theory
Control theory
* 9.2 Improper integrals * 9.3 Fluid dynamics
Fluid dynamics
* 9.4 Dynamic equations * 9.5 Electromagnetism
and electrical engineering * 9.6 Signal analysis * 9.7 Quantum mechanics
Quantum mechanics
* 9.8 Relativity

* 9.9 Geometry

* 9.9.1 Fractals * 9.9.2 Triangles

* 9.10 Algebraic number theory * 9.11 Analytic number theory
Analytic number theory

* 10 History * 11 Generalizations and related notions * 12 See also * 13 Notes

* 14 References

* 14.1 Mathematical references * 14.2 Historical references

* 15 Further reading * 16 External links


Complex numbers allow solutions to certain equations that have no solutions in real numbers . For example, the equation ( x + 1 ) 2 = 9 {displaystyle (x+1)^{2}=-9,}

has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the imaginary unit i where i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: ( ( 1 + 3 i ) + 1 ) 2 = ( 3 i ) 2 = ( 3 2 ) ( i 2 ) = 9 ( 1 ) = 9 , {displaystyle ((-1+3i)+1)^{2}=(3i)^{2}=(3^{2})(i^{2})=9(-1)=-9,} ( ( 1 3 i ) + 1 ) 2 = ( 3 i ) 2 = ( 3 ) 2 ( i 2 ) = 9 ( 1 ) = 9. {displaystyle ((-1-3i)+1)^{2}=(-3i)^{2}=(-3)^{2}(i^{2})=9(-1)=-9.}

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


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

A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i2 = −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 part of a + bi. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginary part. The real part of a complex number z is denoted by Re(z) or ℜ(z); the imaginary part of a complex number z is denoted by Im(z) or ℑ(z). For example, Re ( 3.5 + 2 i ) = 3.5 Im ( 3.5 + 2 i ) = 2. {displaystyle {begin{aligned}operatorname {Re} (-3.5+2i)&=-3.5\operatorname {Im} (-3.5+2i) width:22.967ex; height:6.176ex;" alt="{begin{aligned}operatorname {Re} (-3.5+2i)&=-3.5\operatorname {Im} (-3.5+2i)">z is equal to Re ( z ) + Im ( z ) i {displaystyle operatorname {Re} (z)+operatorname {Im} (z)cdot i} . This expression is 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 number bi 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 − 4i instead of 3 + (−4)i.

The set of all complex numbers is denoted by ℂ, C {displaystyle mathbf {C} } or C {displaystyle mathbb {C} } .


Some authors write a + ib instead of a + bi, particularly when b is a radical. In some disciplines, in particular electromagnetism and electrical engineering , j is used instead of i, since i is frequently used for electric current . In these cases complex numbers are written as a + bj or a + jb.


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

A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system
Cartesian coordinate system
called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001 ), named after Jean-Robert Argand . The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as 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 are emphasized in a complex number's polar form . Using the polar form of the complex number in calculations may lead to a more intuitive interpretation of mathematical results. Notably, the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to 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 to rotating the position vector counterclockwise by a quarter turn (90° ) about the origin: (a+bi)i = ai+bi2 = -b+ai.


Main section: History

The solution in radicals (without trigonometric functions ) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called casus irreducibilis ). This conundrum led Italian mathematician 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 that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus 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 . A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton
William Rowan Hamilton
, who extended this abstraction to the theory of quaternions .



Two complex numbers are equal if and only if both their real and imaginary parts are equal. In symbols: z 1 = z 2 ( Re ( z 1 ) = Re ( z 2 ) Im ( z 1 ) = Im ( z 2 ) ) . {displaystyle z_{1}=z_{2},,leftrightarrow ,,(operatorname {Re} (z_{1})=operatorname {Re} (z_{2}),land ,operatorname {Im} (z_{1})=operatorname {Im} (z_{2})).}


Because complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linear ordering on the set of complex numbers.

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



See also: Complex conjugate Geometric representation of z and its conjugate z {displaystyle {bar {z}}} in the complex plane

The complex conjugate of the complex number z = x + yi is defined to be x − yi. It is denoted by either z {displaystyle {bar {z}}} or z*.

Formally, for any complex number z: z = Re ( z ) Im ( z ) i . {displaystyle {bar {z}}=operatorname {Re} (z)-operatorname {Im} (z)cdot i.}

Geometrically, z {displaystyle {bar {z}}} is the "reflection" of z about the real axis. Conjugating twice gives the original complex number: z = z {displaystyle {bar {bar {z}}}=z} .

The real and imaginary parts of a complex number z can be extracted using the conjugate: Re ( z ) = 1 2 ( z + z ) , {displaystyle operatorname {Re} ,(z)={tfrac {1}{2}}(z+{bar {z}}),,} Im ( z ) = 1 2 i ( z z ) . {displaystyle operatorname {Im} ,(z)={tfrac {1}{2i}}(z-{bar {z}}).,}

Moreover, a complex number is real if and only if it equals its own conjugate.

Conjugation distributes over the standard arithmetic operations: z + w = z + w , {displaystyle {overline {z+w}}={bar {z}}+{bar {w}},,} z w = z w , {displaystyle {overline {z-w}}={bar {z}}-{bar {w}},,} z w = z w , {displaystyle {overline {zw}}={bar {z}}{bar {w}},,} ( z / w ) = z / w . {displaystyle {overline {(z/w)}}={bar {z}}/{bar {w}}.,}


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

Complex numbers are added by separately adding the real and imaginary parts of the summands. That is to say: ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i . {displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i. }

Similarly, subtraction is defined by ( a + b i ) ( c + d i ) = ( a c ) + ( b d ) i . {displaystyle (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 interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram , three of whose vertices are O, A and B. Equivalently, X is the point such that the triangles with vertices O, A, B, and X, B, A, are congruent .


The multiplication of two complex numbers is defined by the following formula: ( a + b i ) ( c + d i ) = ( a c b d ) + ( b c + a d ) i . {displaystyle (a+bi)(c+di)=(ac-bd)+(bc+ad)i. }

In particular, the square of the imaginary unit is −1: i 2 = i i = 1. {displaystyle i^{2}=itimes i=-1. }

The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit. Indeed, if i is treated as a number so that di means d times i, the above multiplication rule is identical to the usual rule for multiplying two sums of two terms. ( a + b i ) ( c + d i ) = a c + b c i + a d i + b i d i {displaystyle (a+bi)(c+di)=ac+bci+adi+bidi}  (distributive property ) = a c + b i d i + b c i + a d i {displaystyle =ac+bidi+bci+adi}  (commutative property of addition—the order of the summands can be changed) = a c + b d i 2 + ( b c + a d ) i {displaystyle =ac+bdi^{2}+(bc+ad)i}  (commutative and distributive properties) = ( a c b d ) + ( b c + a d ) i {displaystyle =(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, and real division. When at least one of c and d is non-zero, we have a + b i c + d i = ( a c + b d c 2 + d 2 ) + ( b c a d c 2 + d 2 ) i . {displaystyle ,{frac {a+bi}{c+di}}=left({ac+bd over c^{2}+d^{2}}right)+left({bc-ad over c^{2}+d^{2}}right)i.}

Division can be defined in this way because of the following observation: a + b i c + d i = ( a + b i ) ( c d i ) ( c + d i ) ( c d i ) = ( a c + b d c 2 + d 2 ) + ( b c a d c 2 + d 2 ) i . {displaystyle ,{frac {a+bi}{c+di}}={frac {left(a+biright)cdot left(c-diright)}{left(c+diright)cdot left(c-diright)}}=left({ac+bd over c^{2}+d^{2}}right)+left({bc-ad over c^{2}+d^{2}}right)i.}

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


The reciprocal of a nonzero complex number z = x + yi is given by 1 z = z z z = z x 2 + y 2 = x x 2 + y 2 y x 2 + y 2 i . {displaystyle {frac {1}{z}}={frac {bar {z}}{z{bar {z}}}}={frac {bar {z}}{x^{2}+y^{2}}}={frac {x}{x^{2}+y^{2}}}-{frac {y}{x^{2}+y^{2}}}i.}

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


See also: Square roots of negative and complex numbers

The square roots of a + bi (with b ≠ 0) are ( + i ) {displaystyle pm (gamma +delta i)} , where = a + a 2 + b 2 2 {displaystyle gamma ={sqrt {frac {a+{sqrt {a^{2}+b^{2}}}}{2}}}}

and = sgn ( b ) a + a 2 + b 2 2 , {displaystyle delta =operatorname {sgn}(b){sqrt {frac {-a+{sqrt {a^{2}+b^{2}}}}{2}}},}

where sgn is the signum function. This can be seen by squaring ( + i ) {displaystyle pm (gamma +delta i)} to obtain a + bi. Here a 2 + b 2 {displaystyle {sqrt {a^{2}+b^{2}}}} is called the modulus of a + bi, and the square root sign indicates the square root with non-negative real part, called the PRINCIPAL SQUARE ROOT; also a 2 + b 2 = z z {displaystyle {sqrt {a^{2}+b^{2}}}={sqrt {z{bar {z}}}}} , where z = a + b i {displaystyle z=a+bi} .


Main article: Polar coordinate system Figure 2: The argument φ and modulus r locate a point on an Argand diagram; r ( cos + i sin ) {displaystyle r(cos varphi +isin varphi )} or r e i {displaystyle re^{ivarphi }} are polar expressions of the point.


An alternative way of defining a point P in the complex plane, other than using the x- and y-coordinates, is to use the distance of the point from O, the point whose coordinates are (0, 0) (the origin ), together with the angle subtended between the positive real axis and the line segment OP in a counterclockwise direction. This idea leads to the polar form of complex numbers.

The absolute value (or modulus or magnitude) of a complex number z = x + yi is r = z = x 2 + y 2 . {displaystyle textstyle r=z={sqrt {x^{2}+y^{2}}}.,}

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 P representing the complex number z to the origin. The square of the absolute value is z 2 = z z = x 2 + y 2 . {displaystyle textstyle z^{2}=z{bar {z}}=x^{2}+y^{2}.,}

where z {displaystyle {bar {z}}} is the complex conjugate of z {displaystyle z} .

The argument of z (in many applications referred to as the "phase") is the angle of the radius OP with the positive real axis, and is written as arg ( z ) {displaystyle arg(z)} . As with the modulus, the argument can be found from the rectangular form x + y i {displaystyle x+yi} : = arg ( z ) = { arctan ( y x ) if x > 0 arctan ( y x ) + if x ( y x ) if x 2 if x = 0 and y 0\arctan({frac {y}{x}})+pi &{mbox{if }}x

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