The topic of mathematical complex numbers

We call a number in the form of z = a+bi (both a and b are real numbers) a complex number, where a is called the real part, b is called the imaginary part, and I is called the imaginary part. When the imaginary part of z is equal to zero, z is often called a real number; When the imaginary part of z is not equal to zero and the real part is equal to zero, z is often called pure imaginary number. Complex number field is an algebraic closure of real number field, that is, any polynomial with complex coefficients always has roots in complex number field. Complex number was first put forward by Cardan, a scholar in Milan, Italy, in the16th century. Through the work of D'Alembert, De Moivre, Euler and Gauss, this concept was gradually accepted by mathematicians.

When the number set is extended to the real number range, there are still some operations that cannot be performed (for example, driving a negative number to an even power). In order to make the equation have a solution, we extend the number set again.

Define binary ordered pair z=(a, b) in real number field, and specify that there are operations "+"and "x" between ordered pairs (note z 1=(a, b), z2=(c, d))；);

z 1 + z2=(a+c，b+d)

z 1 × z2=(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. For any complex number Z, we have

z=(a，b)=(a，0)+(0， 1) × (b，0)

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

Remember that (0, 1)=i, then according to our defined operation, (a, b) = (a, 0)+(0, 1) × (b, 0) = a+bi, i× i = (0,1.

shape

The number of is called a complex number, where I is defined as an imaginary unit and

(A and B are arbitrary real numbers)

We will put the plural number

The real number A in is called the real part of the complex number Z, and is recorded as REZ = A.

The real number b is called the imaginary part of the complex number z, and is recorded as imz = B.

When a=0 and b≠0 and z=bi, we call it pure imaginary number.

The set of complex numbers is represented by C, and the set of real numbers is represented by R. Obviously, R is the proper subset of C.

The complex set is out of order, and the size order cannot be established.

Modulus of complex number

The value of the positive square root of the sum of squares of the real part and imaginary part of a complex number is called the module of a complex number, and is expressed as ∣z∣.

That is, for complex numbers

, its module

For complex numbers

, called the plural.

=a-bi is the * * * yoke complex number of z, that is, two complex numbers with equal real parts and opposite imaginary parts are conjugate complex numbers. The complex number of the yoke of the complex number z is written as

. By definition, if

(a, b∈R), then

=a-bi(a, b∈R).* * * The point corresponding to the complex number of the yoke is symmetrical about the real axis. Two complex numbers: x+yi and x-yi are called * * * yoke complex numbers. Their real parts are equal, but their imaginary parts are opposite. On the complex plane, the points representing the complex numbers of two yokes are symmetrical about X, which is the origin of the word "yoke"-two trays on the tray balance need to be fitted with a horizontal beam, which is called "yoke". If x+yi is represented by z, then adding a "one" above the word z is x-yi, and vice versa.

* * * yoke complex numbers have some interesting properties:

In a complex function, the independent variable z can be written as

, r is the modulus of z, that is, r = | z |θ is the radial angle of z, which is denoted as Arg(z). The divergence angle between-π and π is called the principal value of divergence angle, which is recorded as arg(z) (lowercase a).

Any complex number that is not zero

The radial angle of has an infinite number of values, which are different by integer multiples of 2π. Applicable to-π≤θ

Exponential form:

Addition rule

The addition rule of complex numbers: let z 1=a+bi and z2=c+di be any two complex numbers. The real part of 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 complex numbers or a complex number.

that is

Multiplication rule

Complex multiplication rule: two complex numbers are multiplied, similar to two polynomials. In the result, i2=-1, and the real part and imaginary part are merged respectively. The product of two complex numbers is still a complex number.

that is

Division rule

Definition of complex division: satisfying

plural

Divide the complex number a+bi by the complex number c+di.

Operation method: the numerator and denominator are multiplied by the * * * yoke complex number of the denominator at the same time, and then operated according to the multiplication rule.

that is

Prescription rule

If Zn = r(cosθ+isθ), then

(k=0， 1，2，3…n- 1)

Arithmetic law

Additive commutative law: z 1+z2=z2+z 1.

Multiplicative commutative law: z 1×z2=z2×z 1.

Additive associative law: (z1+z2)+z3 = z1+(z2+z3)

The law of multiplicative association: (z1× z2 )× z3 = z1× (z2× z3)

Distribution law: z/kloc-0 /× (z2+z3) = z/kloc-0 /× z2+z/kloc-0 /× z3.

My power rules

I4n+ 1 = i, i4n+2 =- 1, i4n+3 =-i, i4n = 1 (where n∈Z).