complex

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Mathematical concept

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A number in the form of a+bi (both a and b are real numbers) is a complex number, where a is called real part , b is called imaginary part, and i is imaginary number Company. Complex numbers are usually represented by z, that is, z=a+bi. When the imaginary part of z b=0, z is a real number; When z imaginary part When b ≠ 0 and the real part a=0, z is often called Pure imaginary number 。

The complex number field is the algebraic closure of the real number field, that is, any complex coefficient polynomial There is always a root in the complex field.

The plural is defined by Italy Milan scholar Cartan It was first introduced in the 16th century after Darumbel 、 Desmoff 、 Euler 、 Gaussian This concept is gradually accepted by mathematicians.

Chinese name: complex
Foreign name: complex number
Presenter: Heron of Alexandria
Proposed time: 1st century AD
Applicable fields: mathematics , Programming

Applied discipline: mathematics 、 physics 、 computer science
Correlation theorem: Euler's formula, Timofer's theorem
Collection: Unordered set
Named by: René Descartes
General form: Z=a+bi (both a and b are real numbers)

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history

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The earliest literature on complex square roots dates from the first century AD Greece mathematician Helen He considered the impossibility of the flat topped pyramid.

Complex plane

Italian Milan scholar in the 16th century Caldano (Jerome Cardan, 1501-1576) published the general solution of the unary cubic equation in the book Important Art published in 1545, which was called“ Cardan formula ”。 He was the first mathematician to write the square root of negative numbers into the formula, and when discussing whether it is possible to divide 10 into two parts so that their product is equal to 40, he wrote the answer as

Although he thought that and these two expressions were meaningless, imaginary and ethereal, he still divided 10 into two parts and made their product equal to 40. French mathematician gave the name of "imaginary number" Descartes (1596~1650), he《 geometry 》(published in 1637). Since then, imaginary numbers have spread.

The discovery of a new star in the number system, the imaginary number, has caused a lot of confusion in the mathematical world, and many mathematicians do not recognize the imaginary number. German mathematician Leibniz (1646-1716) said in 1702: "The imaginary number is the subtle and strange refuge of the gods, and it is probably an amphibian between existence and illusion.". However, what is really rational can stand the test of time and space, and eventually occupy its own place. French mathematician Darumbel (1717-1783) pointed out in 1747 that if the imaginary number is operated according to the four rules of polynomial operation, its result is always in the form of a+bi (both a and b are real numbers). In 1722, French mathematician Timothy (1667-1754) discovered the famous Timofer's theorem 。 Euler discovered the famous relationship in 1748, and it was the first time he used i to represent the square root of - 1 in his article Differential Formula (1777), and he pioneered the use of the symbol i as the unit of imaginary numbers. The "imaginary number" is actually not imagined, but it does exist. Norwegian surveyor Wessel (1745-1818) tried to give an intuitive geometric interpretation of this imaginary number in 1797, and first published his practice, but he did not get the attention of the academic community.

Representation of complex numbers^[3]

At the end of the 18th century, the plural was gradually accepted by most people Kaspar Wessel It is proposed that complex number can be regarded as a point on the plane. A few years later, Gauss put forward this idea again and vigorously promoted it, and the study of complex numbers began to develop at a high speed. Surprisingly, as early as 1685 John Wallis Already in De Algebra tractatus Put forward this view.

Kaspar Wessel Published in 1799《 Proceedings of the Copenhagen Academy 》In terms of contemporary standards, it is also quite clear and complete. He also considered the sphere and concluded that Quaternion And put forward a complete Spherical trigonometry Theory. In 1804, Abb é Bu é e also independently put forward a view similar to Wallis, that is, to represent the unit line segment perpendicular to the real axis in the plane. In 1806, Bu é e's article was officially published. In the same year, Jean Robel Algan also published similar articles, and Algan's complex plane became the standard. In 1831, Gauss thought that the complex number was not popular enough. The next year, he published a memorandum, which established the position of the complex number in the field of mathematics. The efforts of Cauchy and Abel swept away the last scruple about the use of plural numbers, and the latter was the first to be famous for the study of plural numbers.

Complex numbers have attracted the attention of many famous mathematicians, including Kummer (1844) Kronecker (1845), Scheffler (1845, 1851, 1880), Bellavitis (1835, 1852), George Picuk (1845) and De Morgan (1849). Mobius He published a lot of short articles on complex geometry, and John Peter Dirichlet extended many real number concepts, such as prime numbers, to complex numbers.

Agande (1777-1855), a German mathematician, published the graphical representation of complex numbers in 1806, that is, all real numbers can be represented by a number axis, and complex numbers can also be represented by a point on a plane. stay Rectangular coordinate system In, take point A corresponding to real number a on the horizontal axis, and point B corresponding to real number b on the vertical axis, and lead straight lines parallel to the coordinate axis through these two points, their intersection points C Means plural. Like this, the plane whose points correspond to complex numbers is called“ Complex plane ”Later, it was also called "Forrest Gump's Plane". In 1831, Gauss used real number groups to represent complex numbers, and established some operations of complex numbers, making some operations of complex numbers as "algebraic" as real numbers. In 1832, he put forward the term "plural" for the first time, which will also refer to two different methods for the same point on the plane—— Cartesian coordinate method And polar coordinate method. It is unified in the algebraic expression and trigonometric expression that represent the same complex number, and the point on the number axis corresponds to the real number one by one, expanding to the point on the plane corresponds to the complex number one by one. Gauss regards complex numbers not only as points on the plane, but also as a vector And using the one-to-one correspondence between complex numbers and vectors, this paper expounds the geometric addition and multiplication of complex numbers. So far, the complex number theory has been established completely and systematically.

Through the long-term unremitting efforts of many mathematicians and the profound exploration and development of the complex number theory, the ghost who has been wandering in the field of mathematics for 200 years - imaginary number has lifted the veil of mystery and revealed its true nature. The imaginary number has become a member of the number family, so the set of real numbers has expanded to the set of complex numbers.

With the progress of science and technology, the complex number theory has become more and more important. It is not only of great significance to the development of mathematics itself, but also plays an important role in proving the basic theorem of lift on wings, demonstrating its power in solving the problem of dam seepage, and providing an important theoretical basis for the establishment of huge hydropower stations.

definition

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Number set expanded to real number In the range, there are still some operations that cannot be carried out (such as opening the negative number to the even power). In order to make the equation have a solution, we will expand the number set again.

Define the binary ordered pair z=(a, b) on the real number field, and specify that there are operations "+", "×" (denote z one =(a,b)，z two =(c,d)）：

z one +z two =(a+c，b+d)

z one ×z two =(ac-bd，bc+ad)

It is easy to verify that all ordered pairs defined in this way form a field under the addition and multiplication of ordered pairs, and for any complex number z, we have:

z=(a,b)=(a,0)+(0,1)×(b,0)

Let f be the mapping from the real number field to the complex number field, f (a)=(a, 0), then this mapping preserves addition and multiplication on the real number field, so the real number field can be embedded in the complex number field, and can be regarded as a subfield of the complex number field.

If i=(0,1), then according to the operation we define, (a, b)=(a, 0)+(0,1) × (b, 0)=a+bi, i × i=(0,1) × (0,1)=(- 1,0)=- 1, which solves the problem of the existence of the imaginary number unit i only through real numbers.

In the form of

The number of is called complex number, where i is specified as imaginary unit, and

(a and b are arbitrary real number ）

We will plural

The real number a in is called the real part of the complex number z and recorded as Re z=a; The real number b is called the complex number z imaginary part (imaginary part), recorded as Im z=b.

When a=0 and b ≠ 0, z=bi, we call it Pure imaginary number 。

Plural aggregate use C The set of real numbers is represented by R It is obvious that, R yes C Of Proper subset 。

The complex number set is an unordered set, and the size order cannot be established.

concept

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Module of complex number

Combine the real part of complex number with the imaginary part square Positive of sum square root The value of is called the module of the complex number and recorded as.

That is, for complex numbers

, its module

。

Conjugate complex

For complex numbers

, called complex number

= a - b I is z Conjugate complex 。 That is, two real parts are equal, imaginary part Mutual Inverse number The complex numbers of are conjugate complex numbers. complex z The conjugate complex number of

。

On the complex plane, two conjugate The plural points are symmetric about the x axis, which is the source of the word "conjugation" - two oxen pull a plow in parallel, and their shoulders should be equipped with a beam, which is called "yoke".

If z is used to represent x+yi, a horizontal line above the letter z will represent its conjugate complex x-yi^[1] 。

Conjugate complex numbers have some interesting properties:^{[5 ]}

Complex argument

stay Complex function In, the argument z can be written as

, r is the module of z, that is, r=| z |; θ is the argument of z, which is recorded as Arg (z). The argument angle in the interval [- π, π] is called Principal value of spoke angle , recorded as arg (z) (lower case A).

Any complex number that is not zero

The argument angle of has an infinite number of values, and these values differ by an integer multiple of 2 π. The value of argument angle θ suitable for - π≤ θ≤ π is called the principal value of argument angle and recorded as arg (z). The principal value of the spoke angle is unique.

Index form:

。

Arithmetical property

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Commutativity For all α , β ∈ C All have α ＋ β ＝ β ＋ α ， αβ ＝ βα 。

Associativity For all α , β , λ ∈ C All have（ α ＋ β )＋ λ ＝ α ＋( β ＋ λ )，( αβ ) λ ＝ α ( βλ )。

Units For all λ ∈ C All have λ ＋0＝ λ ， λ 1＝ λ 。

Additive inverse For each α ∈ C There are unique β ∈ C bring α ＋ β ＝0。

Multiplicative inverse For each α ∈ C ， α ≠ 0 all have unique β ∈ C bring αβ ＝1。

Distributive property For all λ , α , β ∈ C All have λ ( α ＋ β )＝ λα ＋ λβ 。^[4]

Algorithm

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addition rule

The addition rule of complex numbers: let z one =a+bi，z two =C+di is any two complex numbers. The real part of the sum is the sum of the original two complex real parts, and its imaginary part is the sum of the original two imaginary parts. The sum of two plural numbers is still plural.

I.e

。

product rule

Multiplication rule of complex number: multiply two complex numbers, similar to two polynomial Multiply, i in the result two =-1. Combine the real part and the imaginary part respectively. The product of two complex numbers is still a complex number.

I.e

。

Division rule

Definition of complex division: meet

Plural of

It is called the quotient of complex number a+bi divided by complex number c+di.

Operation method: multiply the numerator and denominator by the denominator Conjugate complex , and then use the multiplication rule,

I.e

(Denominator real number).

Square root rule

, then

（k＝0，1，2，3，…，n-1）。

Operational law

Commutative law of addition: z one +z two =z two +z one

Commutative law of multiplication: z one ×z two =z two ×z one

Associative law of addition: (z one +z two )+z three =z1+(z two +z three )

Associative law of multiplication ：(z one ×z two )×z three =z one ×(z two ×z three )

Distribution law: z one ×(z two +z three )=z one ×z two +z one ×z three

Power rule of i

i 4n+1 =i,i 4n+2 =-1,i 4n+3 =-i,i 4n =1（n∈ Z ）

Timofer's theorem

For the complex number z=r (cos θ+isin θ), there is z to the nth power z n =r n [cos (n θ)+isin (n θ)] (where n is positive integer ）

Number series expansion

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In mathematics, the study of "quantity" starts with numbers, starting with familiar natural numbers and integers and rational and irrational numbers described in arithmetic.

Specifically:

Due to the need of counting, human beings have abstracted natural numbers 0, 1, 2, 3,... from real things, which is the starting point of all "numbers" in mathematics.

Since the natural number is not closed to the subtraction operation (that is, the smaller natural number minus the larger natural number, the result is not a natural number), in order to close the subtraction operation, we expand the natural number to an integer;

Since the division operation of an integer is not closed (that is, an integer cannot be divided by another integer, and the result is not an integer), in order to close the division operation, we expand the integer to a rational number;

Since rational numbers are not closed to square root operations (that is, rational numbers are open to a positive integer power, and the results can not be rational numbers), we expand rational numbers to some algebraic numbers in order to close square root operations. "Algebraic number" is defined as the root of univariate polynomial equation with integral coefficient (or rational coefficient), which includes part of real number and part of imaginary number. Complex numbers that are not algebraic numbers are called "transcendental numbers", such as π and e. In addition, there are some algebraic numbers that cannot be represented by the four arithmetic operations and square root operations of rational numbers in a finite number of steps. They cannot be represented as algebraic forms of rational numbers.

On the other hand, rational numbers are not closed to limit operations. In order to close limit operations, we extend rational numbers to real numbers. Thus, limit, calculus and infinite series operations can be operated well. In other words, limit, definite integral, multiple integral, infinite series, infinite product and other operations are performed on functions defined in the real number field. As long as they are not divergent, their simplification results are within the real number range.

Finally, in order to avoid negative numbers being unable to open even power operations within the range of real numbers, we extend real numbers to complex numbers. Complex number is the smallest algebraic closed field containing real number. We conduct four operations and square root for any complex number, and the reduction results are complex numbers.

From the above discussion, there are two ways to divide rational numbers and complex numbers:

{Rational}={Integer} →{Fraction}={Positive Rational} →{Zero} →{Negative Rational},
{complex number}={real number} →{imaginary number}={algebraic number} →{transcendental number}.

classification

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After the number classification is expanded to the complex number range, we classify the number set of the complex number range as follows:

Plural（ a + b i) -- Set symbol C
—	Real number (complex number b =0) -- collective symbol R
—	—	Rational Number - Set Symbol Q （ p / q ）
—	—	—	① Positive Rational Number - Set Symbol Q +
—	—	—	—	Positive Integer - Set Symbol N + or N *
—	—	—	—	—	one
—	—	—	—	—	Prime number
—	—	—	—	—	Composite number
—	—	—	—	Positive fraction
—	—	—	①0
—	—	—	① Negative Rational Number - Set Sign Q -
—	—	—	—	Negative Integer - Set Symbol Z -
—	—	—	—	Negative fraction
—	—	—	② Integer - set symbol Z
—	—	—	—	Odd number
—	—	—	—	even numbers
—	—	—	② Score
—	—	Irrational number
—	—	—	Positive irrational number
—	—	—	Negative irrational number
—	Imaginary number (b ≠ 0)
—	—	Pure imaginary number （a=0）
—	—	Mixed imaginary number (a ≠ 0)

Note: ① ② represents two different classifications of "rational number".

application

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systems analysis

stay systems analysis In, the system often passes Laplace transform from time domain Transform to frequency domain 。 Therefore, the system's pole And zero. Analyzing system stability Root locus method Nyquist plot, Nyquist plot and Nichols plot are carried out on the complex plane.

Whether the poles and zeros of the system are in the left half plane or the right half plane, the root locus method is very important. If the system poles are located in the right half plane, the causal system is unstable; If both lie in the left half plane, the causal system is stable; On the imaginary axis, the system is critical stable. If all the zeros and poles of the system are in the left half plane, then this is a minimum phase system. If the poles and zeros of the system are symmetric about the imaginary axis, then this is All-pass system 。

signal analysis

The use of complex numbers in signal analysis and other fields can easily represent periodic signals. Modulus| z |Indicates the amplitude of the signal, Radial angle arg( z) For a given frequency sine wave Of phase 。

utilize Fourier transformation The real signal can be expressed as the sum of a series of periodic functions. these ones here Periodic function It is usually expressed by the real part of the complex function in the following form:

Where ω corresponds to Angular frequency , plural z It contains amplitude and phase information.

In the circuit analysis, the introduction of the imaginary part of capacitance and inductance related to frequency can facilitate the use of simple linear The equation is expressed and solved. (Sometimes letters are used j As an imaginary number unit, so as not to be associated with the current symbol i Confusion.)

Anomalous integral

At the application level, Complex analysis Abnormal functions commonly used to calculate some real values are obtained by complex valued functions. There are many methods, see Contour integral method 。

quantum mechanics

quantum mechanics The complex number is very important because its theory is based on the infinite dimension of the complex number field Hilbert space 。

relativity

It can be simplified if the time variable is regarded as an imaginary number narrow sense and General relativity Space time metric equation in.

applied mathematics

In practical application, solve the given Difference equation The system of the model is usually found first Linear difference equation Corresponding characteristic equation All complex characteristic roots of r , and then take the system as f ( t )= e Linear combination representation of the basis function of.

fluid mechanics

Complex functions can describe two-dimensional potential flow in fluid mechanics.

Fractals

some Fractals as Mandelbrot set and Julia Collection (Julia set) is based on a point on the complex plane.

Real variable elementary function

We put mathematical analysis The basic real variable elementary functions in are extended to the complex variable elementary functions, so that the various complex variable elementary functions defined are the same as the corresponding real variable elementary functions when z becomes the real variable x (y=0).

Note that according to these definitions, when z is any complex variable,

① Which corresponding properties of real variable elementary functions are preserved,

② Which corresponding properties of real variable elementary functions are no longer valid,

③ What new properties do not exist in the corresponding real variable elementary functions.