Definition of complex number

A Complex number is a pair of real numbers (x;y). Its algebraic form is z=x+i*y, where i is an imaginary number. Its algebraic form is , where  is an imaginary number. I – is a formal symbol, corresponding to the following equability i2 = -1.

Two complex numbers (x1;y1) and (x2;y2) are equal, when x1 = x2,y1 = y2 .

The sum of the complex numbers (x1;y1) and (x2;y2) is a complex number (x1 + x2; y1 + y2).

The product of complex numbers (x1;y1) and (x2;y2) is a complex number (x1x2 – y1y2; x1y2 + x2y1).

Complex numbers have the following features:

1. z1+z2=z2+z12. (z1+z2)+z3=z1+(z2+z3)3. z1*z2=z2*z14. (z1*z2)*z3=z1*(z2*z3)5. (z1+z2)*z3=z1*z3+z2*z3

The Residual of complex numbers  and  is a complex number z + z2 = z1. The residual of complex numbers is z1 = x1 + i * y1 and z2 = x2 + i * y2 always exist and is defined by the formula:


Complex numbers z and z¯ are complex conjugated if z=x+i*y and z̅=xi*y. Module or absolute value of the complex number z=x+i*y is a real number x2+y2=|z|. And z*z̅=x2+y2.

Quotient of two complex numbers z1 and z2, (z2≠0), z, where z*z2=z1. If z1=x1+i*y1,z2=x2+i*y2, then z1z2=x1*x2+y1*y2x22+y22+i* x2*y1x1*y2x22+y22=z1*z2¯z2*z2¯=z1*z2¯|z2|2

The absolute value of the complex number states that: |z*w|2=(z*w)*(z*w¯)=(z*w)*(z̅*w̅)=(z*z̅)*(w*w̅)= |z|2*|w|2, then then |z*w|=|z|*|w|.Ifz2≠0, then |z1|=z1z2*z2=z1z2*|z2|,|z1||z2| =z1z2.

The real part of a complex number is: z=x+i*y, is x=Re(z).
The imaginary part of a complex number is: z=x+i*y, is y=Im(z).
Where Re(z)=z+z¯2, Im(z)=zz¯2i.

Complex numbers on the coordinate plane
Let’s consider the complex number z=x+i*y (Picture 1). Put the point on the coordinate plane with coordinates (x;y), it’s radius-vector z, and it’s value |z|=x2+y2.

The Complex plane is a plane for representing complex numbers. X axis is a real axis, Y axis is an imaginary axis. As far as complex numbers are concerned z1,z2 and z3 correspond to the points on the complex plane so we can assume they are the same.

Definitions of sum and residual complex numbers mean that complex numbers sum up and subtract as vectors. Vector interpretation of sum and residual complex numbers are represented in Picture 2.

Let’s look at the triangle with the peaks 0, z1 and z1 + z2. The length of the triangle sides are |z1|+|z2|≤|z1+z2|.

AvermentIf z2= 0,then the inequality is valid.If z2≠0, then |z1+1|2=(z1+1) (z1¯)+1)=|z12 |+z1+(z1¯)+1.If z1=x1+iy, then z1+(z1¯)=2x1x12+y12 =2|z1|.|z1+1|2≤|z1|2+2|z1|+1=(|z1|+1)2. Then |z1+1|≤ |z1|+1.

Let’s look at the following sum:

|z1+z2|=|z2|*z1z2+1≤|z2|*z1z2+1), then |z1+z2|≤|z1|+|z2|.

Important corollary

As z1=-z2+z1+z2, and |z2|=|-z2|, then according to the triangle inequality |z1|≤|-z2|+|z1+z2| ,then |z1+z2|≥|z1|-|z2|, and ||z1+z2||≥||z1|-|z2||

Trigonometric form of a complex number z0, is the following:

z=|z|*(cosφ+i sinφ)

where φ is an argument of the z number, and is described by the statements cosφ=x|z|, sinφ=y|z|. Angle φ always exists, because (x|z|)2+(y|z|)2=x2+y2|z|2=1. The argument of a complex number 0 does not exist.

All possible arguments are φ1=φ+2πk, where k is an integer. Usually φ=Arg z belongs to the angle range (-π;π).

When z=x+iy, the arg z can be found from the following equalities:

Arg z=tan1 yx, x>0 Arg z=tan1 yx+π, x<0 ,y0Arg z=tan1 yxπ,x<0, y<0Arg z=π2,x=0,y>0 Arg z=π2,x=0,y<0

Complex numbers z1 = z2 are equal, when |z1|=|z2|,arg z1=arg z2.

The formula of multiplication and division of complex numbers is the following: z1*z2=|z1|(cos φ1+i sinφ1)*|z2|(cos φ2+isin φ2)=|z1|*|z2|*(cos φ1+φ2) +i sin (φ1+φ2) z1z2=|z1|(cos φ1+i sin φ1)|z2|(cos φ2+i sin φ2)=|z1||z2|*cos φ1+i sin φ1cos φ2+i sin φ2=|z1||z2| *cos (φ1φ2)+i sin (φ1φ2)

It means that when we multiply complex numbers their modules multiply and arguments sum up; when divided, the modules divide, and arguments subtract.



Moivre’s formula
Multiplication and division of complex numbers can lead us to the rule of complex numbers construction to an integer power (the rule is called the Moivre’s formula):

When z≠0, z0=1.

When n belongs to the range of natural numbers,  zn=|z|n(cos φ+i sin φ)n=|z|n(cos nφ+i sin nφ), z0If zn=1zn, then for m=n<0, z0, the following statement is true:zm=1zn=1|z|n(cos nφ+i sin nφ)=1|z|n*cos nφi sin nφcos nφ)2+sin nφ)2=zn*cos(nφ)+i sin(nφ)

Then the Moivre’s formula states:

zm=|z|m*cos +i sin , where m is an element of M unity.


Roots extraction from complex numbers
N-power root of a complex number  is a complex number , where zn = w. N-th root unity of is marked as wn.

Theorem. Equation zn = w, has n different complex roots w≠0, n belongs to N range.
Averment. Let’s suggest w=|w|*(cos⁡θ+i sin⁡θ). Then the complex number z should be:

z=|z|*(cos⁡ζ+i sin⁡ζ)

Let’s use an equation zn = w and Moivre’s formula:

|z|n*(cos +i sin )=|w|*(cos θ+i sinθ)

The |z|n=|w|,ζ=θ+2πkn where k belongs to unity Z. With k=0,1,2,…,n-1 there are different root values. When k=n, root value is equal to the one with k=0. When k=n+1, the root value is equal to one with k=1 etc.

So the number of different root values is n, and

zk=nw*(cos θ+2πkn+i sin θ+2πkn), where k=0,1,2,…n-1.

All n of zk roots belong to the circle with the radius wn, with the centre 0. They divide the circle by n parts with the angle 2πn.

#2 Complex numbers

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