Complex Numbers

Imaginary Numbers
z = a + ib is a complex number. Real part of z is denoted as Re(z) = a & Imaginary part is denoted by Im(z) = b

i = √(-1)

z1± z2= a1 ± a2 ± ib1 ± ib2

z1z2 = (a1a2 - b1b2)+ i(a1b2+ a2b1)

If z1 = z2

a1 + ib1= a2+ ib2

a1=a2 & b1=b2 ;thus one complex equation is equivalent to two real equations.

Complex Numbers
A Complex Number is a number consisting of a real and an imaginary part.

e.g. a + ib is a Complex number.

Equality of Complex Numbers
Complex Numbers can be equated but operations such as greater than, lesser than cannot be performed with imaginary parts.

Complex Conjugate
If a + bi is a complex number, then it's complex conjugate is a - bi.

z = a + bi     z' = a - bi

Polar Representation
z = |z| {cos(amp z) + i sin(amp z)}

i.e. z = r[cosθ + i sinθ]  where r = |z|  ; θ = amp z

Modulus of a Complex Numbers
Modulus of a complex number is the same as the modulus of a vector. The modulus of a complex number is its magnitude of it sqrt (a2 + b2)


 * z| = √(a2+b2)

amp(z) = arg(z) = tan-1(y/x)

amplitude is the angle subtended by line joining the point (a,b) and the positive X-axis.

Properties

(z1 ± z2)' = z1' ± z2'

(z1z2)' = z1'z2'

(z1/ z2)' = z1' / z2'

amp(z1z2) = amp z1 + amp z2

amp(z1/ z2) = amp z1 - amp z2

amp z2 = 2 amp z

amp [z/z'] = 2 amp z

Argand's Diagram
Argand's Diagram is the graphical representation of complex numbers. a & b represent the axes of the graph.

Complex Cube Root of Unity
Cuberoot of -1

Tips & Tricks
1) In order to get rid of i term, you can rationalize the question to convert i into i2 . Prefer to reach i2.