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单词 Complex
释义

complex


There are some polynomial equations with real coefficientsMathworldPlanetmath that don’t have real solutions.  Examples of these are  x2+5=0,  x2+x+1=0.  Mathematically we express this by saying that is not an algebraically closed field.

In to solve that kind of equation, we have to “extend” our number system by adjoining a number i that has the property that  i2=-1.  In this way we extend the field of real numbers to a field whose elements are called .  A formal construction can be seen at [http://planetmath.org/node/471complex numbersMathworldPlanetmathPlanetmath] (cf. the field adjunction).  The field is algebraically closedMathworldPlanetmath: every polynomialMathworldPlanetmathPlanetmath with complex coefficients, and especially every polynomial with real coefficients, (and with positive degree) has at least one complex zero (which might be real as well).

Any complex number can be written as  z=x+iy (with x,y). Here we call x the real part of z and y the imaginary part of z. We write this as

x=Re(z),y=Im(z).

Real numbers are a subset of complex numbers, and a real number r can be written also as r+i0.  Thus, a complex number is real if and only if its imaginary part is equal to zero.

By writing x+iy as  (x,y) we can also look at complex numbers as ordered pairs.  With this notation, real numbers are the pairs of the form  (r, 0).

The rules of addition and multiplication for complex numbers are:

(a+ib)+(x+iy)=(a+x)+i(b+y)(a,b)+(x,y)=(a+x,b+y)
(a+ib)(x+iy)=(ax-by)+i(ay+bx)(a,b)(x,y)=(ax-by,ay+bx)

(to see why the last identityPlanetmathPlanetmathPlanetmathPlanetmath holds, expand the first productMathworldPlanetmath and then simplifyby using  i2=-1).

We have also the negatives (http://planetmath.org/OppositeNumber):  -(a,b)=(-a,-b)  and the multiplicative inversesMathworldPlanetmath:

(a,b)-1=(aa2+b2,-ba2+b2).

Seeing complex numbers as ordered pairs also let us give the structure of vector space (over ).  The norm of  z=x+iy  is defined as

|z|=x2+y2.

Then we have  |z|2=zz¯  where z¯ is the conjugate of z=x+iy and it’s defined as  z¯=x-iy.  Thus we can also characterize real numbers as those complex numbers z such that z=z¯.

ConjugationMathworldPlanetmath obeys the following rules:

z1+z2¯=z1¯+z2¯
z1z2¯=z1¯z2¯
z¯¯=z

The real and imaginary parts of a complex number may be expressed with the conjugate as

Re(z)=z+z¯2,Im(z)=z-z¯2i.

The ordered-pair notation lets us visualize complex numbers as points in the plane; this is called the complex plane, often also the z-plane.  As well, we can also describe complex numbers with polar coordinates.

Using this representation, we see that the real numbers are located at the abscissa (horizontal) axis, which is then known as the real axis.  The ordinate (vertical) axis is known as the imaginary axis, since it consists of all complex numbers with real part equal to zero.

If  z=a+ib  is represented in polar coordinates as  (r,t)  we call r the of z and t its argument.

If  r=a+ib=(r,t),  then  a=rsint  and  b=rcost.  So we have the following expression, called the polar form of complex number z:

z=a+ib=r(cost+isint)

Multiplication of complex numbers can be done in a very neat way using polar coordinates:

(r1,t1)(r2,t2)=(r1r2,t1+t2).

Remark.  The adjective  complex  qualifying such nouns as “number”, “root” and “solution” is in the English ambiguous; it may mean that it is a question of a element belonging to either or to , i.e. the complex  may either have its basic sense or mean ‘non-real’.

Titlecomplex
Canonical nameComplex
Date of creation2013-03-22 11:57:12
Last modified on2013-03-22 11:57:12
Ownerdrini (3)
Last modified bydrini (3)
Numerical id43
Authordrini (3)
Entry typeDefinition
Classificationmsc 12D99
Classificationmsc 30-00
Synonymcomplex number
Related topicPolynomial
Related topicArgandDiagram
Related topicRealNumber
Related topicComplexNumber
Related topicComplexConjugate
Related topicNthRoot
Related topicRiemannZetaFunction
Related topicImaginary
Related topicImaginaryUnit
Related topicRegion
Related topicUnitDisk
Related topicUpperHalfPlane
Related topicZeroesOfAnalyticFunctionsAreIsolated
Related topicRiemannSphere
Related topicSquareRoot
Related topicCardanosFormulae
Related topicFundamenta
Definescomplex plane
Definesz-plane
Definesreal axis
Definesimaginary axis
Definesreal part
Definesimaginary part
Definesconjugate
Definesargument
Definespolar form
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更新时间:2025/5/4 19:53:37