# Complex Number

Complex number (x, y) is two ordered real numbers where x and y are real numbers. If z = (x,y) - z is a complex number, x is the real part of z, and y is imaginary part of z, because of thet complex numbers does not have ordering.

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)
z1.z2 = (x1, y1).(x2, y2) = (x1.x2 - y1.y2, x1.y2 + y1.x2)
 z1 z2
=
 (x1, y1) (x2, y2)
=
 x1x2 + y1y2 x22 + y22
,
 x2y1 - x1y2 x22 + y22

The complex numbers are the field of numbers. The field of complex numbers includes the field of real numbers as a subfield. Other 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 have its complex conjugate = x - iy.

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

The phasor form of the complex number is:

z = |z|(cosθ + isinθ) = |z|e

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

#### Moivre's formulaes

zn = rn(cosnθ + sinnθ)

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

complex subtraction:

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

complex multiplication:

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

complex division:

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

#### More about complex numbers in the maths forum

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