Conjugate complex number

- Mar 31, 2018 -

Conjugate complex numbers, two real parts are equal, complex numbers whose imaginary parts are inverse numbers are conjugate complex numbers. When the imaginary part is not zero, the conjugate complex number is the real part and the imaginary part is the opposite. If the imaginary part is zero, the conjugate complex number is itself. (When the imaginary part is not equal to 0, it is also called a conjugate imaginary number.) The conjugated complex number z is denoted by zˊ. At the same time, complex z ˊis called a complex conjugate of complex z.[1]

By definition, if z=a+bi(a,b∈R) then zˊ=a-bi(a,b∈R). The point corresponding to the conjugate complex number is symmetric about the real axis (see the drawing for details). Two complex numbers: x+yi and x-yi are called conjugate complex numbers, their real parts are equal, and the imaginary parts are opposite numbers. On the complex plane, the points representing the two conjugate complex numbers are symmetrical about the X axis. One point is the source of the word "conjugation." Two cows pull a plow in parallel. They have a beam on their shoulders. This beam is called a "yoke". If you use X to represent X+Yi, then in Z A "one" above the word indicates X-Yi, or vice versa.

Some interesting properties of conjugate complex numbers:

There are also some four arithmetic properties.


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

(3) z·z′=|z|^2=a^2+b^2 (real number);

Addition rule

Complex addition rule: Let z1=a+bi and z2=c+di be any two complex numbers. The real part of the two sums 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 is still plural. That is, (a+bi)±(c+di)=(a±c)+(b±d)i.[2]

Subtraction rule

The difference between two complex numbers is the difference between real numbers plus the difference between imaginary numbers (multiplied by i)

That is: z1-z2=(a+ib)-(c+id)=(a-c)+(b-d)i

Multiplication rule

Multiplicative multiplication rule: Multiply two complex numbers, similar to two polynomial multiplications. In the result, i^2 = -1, combine the real part and the imaginary part. The product of two complex numbers is still a complex number.

That is: z1z2=(a+bi)(c+di)=ac+adi+bci+bdi2=(ac-bd)+(bc+ad)i.

Division rule

Complex division definition: The complex number x+yi(x,y∈R) that satisfies (c+di)(x+yi)=(a+bi) is called quotient of the complex number a+bi divided by the complex number c+di: The numerator and denominator are simultaneously multiplied by the conjugate complex number of the denominator and then multiplied by the multiplication rule.

which is: 

Open method

If z^n=r(cosθ+isinθ), then z=n√r[cos(2kπ+θ)/n+isin(2kπ+θ)/n] (k=0, 1, 2, 3...n -1)

Conjugation rule

The conjugate of z=x+iy, labeled z* is the conjugate number z*=x-iy

That is: zz*=(x+iy)(x-iy)=x2-xyi+xyi-y2i2=x2+y2

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

z=x+iy and z*=x-iy are called conjugate pairs

Operation feature editing


(2) (z1-z2)'=z1'-z2'

(3) (z1·z2)′=z1′·z2′

(4) (z1/z2)'=z1'/z2' (z2≠0)

Summary: The conjugate of sum (difference, product, quotient) is equal to the sum of conjugates (difference, product, quotient).

Modulo operation property editing

1 | z1·z2| = |z1|·|z2|

2|z1|-| z2|┃≤| z1+z2|≤| z1|+| z2|

3|z1-z2| = | z1z2|, is a distance formula between two points in a complex plane. From this geometrical meaning, the equations of a straight line, a circle, a hyperbola, an ellipse in a complex plane, and a parabola can be derived.

PS: z' represents a conjugate complex number of complex z (the actual form is a horizontal plane on z), and z'' represents a conjugate complex number of a complex conjugate of complex z (two transverse planes on z), ie z〃=z.[1 ]