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, y1) ± (x2, y2) = (x1 ± x2, y1 ± y2)
z1.z2 = (x1, y1).(x2, y2) = (x1.x2 - y1.y2, x1.y2 + y1.x2)
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:
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.
More about complex numbers in the maths forum