Pacific University Cheat Sheet - Complex Numbers

Rectangular form:

Definitions: z = a+ i bz∗ = a− i b

Real part: Re(z) = aImaginary part:a Im(z) = b

aNote, there is no i in the imaginary part.

Polar form:

Definition: z = Aeiφ

Magnitude/Modulus: |z|= AArgument/Phase: arg(z) = φ

Conversions:

tanφ = ba

|Z |=p

a2+ b2

a = Acosφ b = Asinφ

Rationalization:

Multiply the fraction (numerator and de-nominator) by the complex conjugate of thedenominator. For example:

Z =iA

1+ iB(1)

=iA

1+ iB

�

1− iB

1− iB

�

(2)

=iA− (iB)(iA)

1− iB+ iB− (iB)(iB)(3)

=iA+ BA

1+ B2 (4)

which means

z =BA

1+ B2 +A

1+ B2 i (5)

or

Re(z) =BA

1+ B2 (6)

and

Im(z) =A

1+ B2 (7)

The complex plane:

Plot Re(z) on the horizontal axis, and Im(z)on the vertical axis. Using the conversionfrom rectangular to polar forms, we see thatthe angle φ corresponds to the angle be-tween the real axis (horizontal) and the linefrom the origin to the point z.

φ

z = a + ib

Im(z)

Re(z)a

b|z|

Figure 1: The complex plane

Operations:

Addition, multiplication, subtraction, anddivision are done just as they are with realnumbers. The only difference is that youwill have i’s running around in your alge-bra. Use the fact that i2 = −1 to simplifyyour result. Hint: some operations are sim-pler in one form or another (polar vs. rect-angular).

1