Complex Division

Made simple

Complex division can be done in two ways. The easy way and the other way. So let's start with the easy way, prove our answers with the other way and then run it through the online calculator.

Polar Conversion

The simplest way to divide complex numbers is convert any Cartesian (rectangular) numbers into polar form.

Just in case you forgot how to do this, we need to find the moduli by using a little Pythagoras theorem:

On a Cartesian number "A+jB" we find the square root of (A² + B²). This will give us the modulus (r).
Now all we need is the argument and we are ready. using the SOHCAHTOA memory aid, the Cosine of the angle is Adjacent over Hypotoneuse, or A/r.

Let us throw in some values so we are all clear so far.

We will complete the sum (3+j4)/(4+j3)=?

Using our conversion above, r₁ = sqr(3²+4²) = 5.
The argument then, Cos=(3/5) >> Cos=(0.6) >> Arg=(arccosine(0.6)) = 53.13 degrees

So (3+j4) = 5L53.13degrees

The second part (4+j3) = 5L36.87degrees

Remember this:

To Divide two Polar Numbers, we Divide their Moduli and subtract their arguments.

Using our figures from above we divide 5 by 5 (=1), and 53.13 - 36.87 = 16.26degrees.
Therefore our answer in polar form is 1L16.26degrees.
Convert back to Cartesian using SOHCAHTOA.

Adjacent=Cosine(16.26) X Hypotoneuse = 0.96 X 1 = 0.96

Opposite=Sine(16.26) X Hypotoneuse = 0.28 X 1 = 0.28

Solution in Cartesian = 0.96+j0.28

The Other Way

This way is a little more involved but has some advantages

It looks more impressive
Math lecturers like it
It's really hard to explain it in HTML without using MathML and images (But let's have a go).

With our above sum of (3+j4)/(4+j3)=?, we need to rationalise the denominator (that simply means, lose the j bits from the bottom part [that is 4+j3]).

To rationalise, we use the Conjugate of the denominator. Conjugate of 4+j3 is 4-j3 (kind of opposite). The result will be a whole number.

Think about a real world scenario, we have a resistance with inductive reactance as one impedance and now we are going to apply another resistance with a capacitive reactance to cancel out the reactances, leaving just a a REAL value of resistance.

Before we go any further note that we must apply this rule to both the numerator and the denominator. First we will apply it to the denominator

Denominator
4+j3 X 4-j3 =
16+j12-j12-j²9 =
16-j²9 =
16-(-1)9 = (remember j²=-1)
16+9 =
25 (our real denominator)

Numerator
3+j4 X 4-j3 =
12+j16-j9-j²12 =
12+j7-(-1)12 =
12+j7+12 =
24+j7 Put these together and we have 24+j7/25 which is the same as
(24/25) + j(7/25) =
0.96+j0.28
Not so hard really, just a little more to remember.

Now check your answers on the Cartesian calculator.

The kusashi calculators are free online tools for general use.
All calculators are scripted on the server and require no special browser settings or plug-ins. Some pages contain additional information in MathML. Browser support for this varies.