* * * York complex number, where two real parts are equal and imaginary parts are opposite, is * * * York complex number. When the imaginary part is not zero, the complex number of yoke means that the real part is equal and the imaginary part is opposite. If the Ruo Xu part is zero, its * * * yoke complex number is itself (also called * * * yoke imaginary number when the imaginary part is not equal to 0). The * * * yoke of the complex number z is marked as z (with a horizontal line on it), and sometimes it can be expressed as Z*. At the same time, the complex number Z (plus a horizontal line) is called the complex yoke of the complex number Z.

Chinese name

Complex number of * * * yoke

Foreign name

conjugate complex number

kind

law

type

concept

subject

mathematics

quick

navigate by water/air

Algebraic characteristics

Operating characteristics

Operational attributes of the module

formula

According to the definition, if z=a+bi(a, b∈R), then = a-bi (a, b ∈ r). * * * The points corresponding to the complex number of the yoke are symmetrical about the real axis (see the attached figure for details). 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 cows pull a plow in parallel with a beam on their shoulders. This beam is called a "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:

There are also about four operational attributes.

Algebraic characteristics

( 1)| z | = | |；

(2)z+=2a (real number), z-= 2bi;

(3) z = | z | 2 = A2+B2 (real number).

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, (a+bi) (c+di) = (a c)+(b d) i. [1]

Subtraction rule

The difference between two complex numbers is the difference between real number and imaginary number (multiplied by I)

Namely: z1-z2 = (a+ib)-(c+id) = (a-c)+(b-d) i.

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.

Namely: z1z 2 = (a+bi) (c+di) = AC+ADI+BCI+BDI 2 = (AC-BD)+(BC+AD) I.

Division rule

Definition of complex number division: the complex number x+yi(x, y∈R) satisfying (c+di)(x+yi)=(a+bi) is called the quotient operation method of dividing the complex number a+bi by the complex number c+di: multiply the numerator and denominator by the * * * yoke complex number of the denominator at the same time, and then operate according to the multiplication rule.

Namely:

Prescription rule

If Zn = r(cosθ+isθ), then (k = 0, 1, 2,3 ... n-1)

* * * Yoke rules

* * * Yokes with z=x+iy, marked with z* indicates the number of * * * yokes, z*=x-iy.

Namely: ZZ * = (x+iy) (x-iy) = x2-xyi+xyi-y2i2 = x2+y2.

That is, when a complex number is multiplied by its yoke number, the result is a real number.

Z=x+iy and z*=x-iy are called * * * yoke pairs.