Mathematics for Engineering

# Complex numbers

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:

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:

${z}_{1}–{z}_{2}=\left({x}_{1}–{x}_{2}\right)+i*\left({y}_{1}–{y}_{2}\right)$

Complex numbers z and $\overline{z}$ are complex conjugated if . Module or absolute value of the complex number z=x+i*y is a real number .

Quotient of two complex numbers z1 and z2, (z2≠0), z, where z*z2=z1. If

The absolute value of the complex number states that: $|z*w{|}^{2}=\left(z*w\right)*\left(\overline{z*w}\right)=\left(z*w\right)*\left(z̅*w̅\right)=\left(z*z̅\right)*\left(w*w̅\right)= |z{|}^{2}*|w{|}^{2}$, then .

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 .

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 $|z|=\sqrt{{x}^{2}+{y}^{2}}$.

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 $|{z}_{1|}+|{z}_{2}|\le |{z}_{1}+{z}_{2}|$.

Let’s look at the following sum:

Important corollary

Trigonometric form of a complex number $z\ne 0$, is the following:

where φ is an argument of the z number, and is described by the statements . Angle φ always exists, because $\left(\frac{x}{|z|}{\right)}^{2}+\left(\frac{y}{|z|}{\right)}^{2}=\frac{{x}^{2}+{y}^{2}}{|z{|}^{2}}=1$. The argument of a complex number 0 does not exist.

All possible arguments are ${\phi }_{1}=\phi +2\mathrm{\pi k}$, 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:

Complex numbers z1 = z2 are equal, when .

The formula of multiplication and division of complex numbers is the following:

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 .

When n belongs to the range of natural numbers,

Then the Moivre’s formula states:

, 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 $\sqrt[n]{w}$.

Theorem. Equation zn = w, has n different complex roots $\mathrm{w\ne 0}$, 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:

The $|z{|}^{n}=|w|,\mathrm{\zeta =}\frac{\theta +2\mathrm{\pi k}}{n}$ 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

, where k=0,1,2,…n-1.

All n of zk roots belong to the circle with the radius $\sqrt[n]{w}$, with the centre 0. They divide the circle by n parts with the angle $\frac{2\pi }{n}$.