Complex Numbers

Complex number is a couple of two real numbers (x, y).
We can think about complex numbers as points in the coordinate system.
Let z is a complex number.
z = (x,y)
x is the real part of z, and y is the imaginary part of z.

Complex numbers form C - the field of complex numbers.
The field of real numbers is its subset.

If we have two complex numbers z1 = (x1, y1) and z2 = (x2, y2) then:

z1 = z2 <=> x1 = x2
z1 ± z2 = (x1, y1) ± (x2, y2) = (x1 ± x2, y1 ± y2)
z1z2 = (x1, y1)(x2, y2) = (x1x2 - y1y2, x1y2 + y1x2)
z1
z2
=
(x1, y1)
(x2, y2)
=
x1x2+y1y2
x22+y22
,
x2y1-x1y2
x22+y22

Another way to write z is: z = x + iy,
x is the real part of z,
y is the imaginary part and
i is the imaginary unit. i2 = -1, i = √-1.

Each complex number z = x + iy has its complex conjugate z = x - iy.

  • z + z = 2x - real number;
  • z - z = i2y - imaginary number;
  • z.z = x2 + y2 = |z|2 - real number

Each complex number (x, y) have a relevant point in the coordinate system. We can not say point A > B, because of that we can not say for two complex numbers (x1, y1) > (x2, y2) It means that complex number have no ordering.

Addition, multiplication and division of complex numbers

Complex numbers addition:

(a + bi) + (c + di) = (a + c) + i(b + d)

Complex numbers subtraction:

(a + bi) - (c + di) = (a - c) + i(b - d)

Complex numbers multiplication:

(a + bi)(c + di) = (ac - bd) + i(ad + bc)

Complex numbers division:

(a + bi)/(c + di) = ((ac + bd) + i(bc - ad))/(c2 + d2)

Polar form

complex number

The polar form of a complex number is:

z = |z|(cos(θ) + i⋅sin(θ)) = |z|e or
z = r(cos(θ) + isin(θ)) = re

Here, |z|(or r) is known as the complex modulus(it is equal ot the measure of OM) θ is known as complex argument or phase. Тhe dashed circle above represents the complex modulus |z| of z and the angle θ represents its complex argument.

Let we have two complex numbers z1 and z2 in polar form:
z1 = r1(cos(θ1) + i.sin(θ1))
z2 = r2(cos(θ2) + i.sin(θ2))

then

z1⋅z2 = r1⋅r2[cos(θ1 + θ2) + i⋅sin(θ1 + θ2)]

$\frac{z_1}{z_2}=\frac{r_1}{r_2}[cos(\theta_1-\theta_2)+i\cdot sin(\theta_1-\theta_2)]$

Moivre's formulaes

The powers of a complex number:
zn = rn(cos(nθ) + i⋅sin(nθ))

Finding the root of a complex number:
$\sqrt[n]{z}=\sqrt[n]{r}(cos(\frac{\theta+2k\pi}{n})+i\cdot sin(\frac{\theta+2k\pi}{n}))$
k = 0, 1, 2,..., n-1

Complex numbers forum


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