Complex Numbers 5: Multiplying Complex Numbers

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Joined Jan 18, 2005
What about multiplying two complex numbers together - well lets have a look. Let the first complex number be
r1(Cosθ1 + jSinθ1) and the second complex number be
r2(Cosθ2 + jSinθ2), then to multiply these together we get;
r1(Cosθ1 + jSinθ1) x r2(Cosθ2 + jSinθ2) =
(r1 x r2) [(Cosθ1 + jSinθ1)(Cosθ2 + jSinθ2)] =
Cosθ1Cos θ2 + jSinθ2Cosθ1 + jSinθ1Cosθ2 +
j(squared)Sinθ1 Sin θ2.
Now remembering that j squared (j x j) is equal to -1 we get;
Cosθ1 Cosθ2 + jSinθ2Cosθ1 + jSinθ1Cosθ2 -
Sinθ1 Sinθ2.
Seperating this out into real and imaginary components we get;
[Cosθ1 Cos θ2 - Sinθ1 Sin θ2] + jSinθ2Cosθ1 + jSinθ1Cosθ2.
Now taking j out as a common factor gives;
[Cosθ1 Cos θ2 - Sinθ1Sin θ2] + j(Sinθ2Cosθ1+Sinθ1Cosθ2).
Remembering our trigonometric identities we can re-write the above as;
Cos (θ1 + θ2) + jSin(θ1 +θ2) and bringing back our real numbers r1 x r2 .

So basically , to multiply two complex numbers together convert each complex number in to the form ; r (Cos θ + jSin θ) (known as Polar Form), then multiply the r’s together (sometimes known as the modulus of the complex number) and add together the angles θ, then put it back into Polar Form.
So to multiply together; 5(Cos20 + jsin40) and 6(Cos40 + jSin50) we get
5 x 6 (Cos (20 + 40) + jSin(40 + 50)) = 30(Cos60 + jSin90).

This works all the time you can have as many Polar forms as you wish being multiplied together and the result is just the same – multiply the r’s and add the angles – that’s it!.

It is also very easy to multiply together complex numbers in the form of (a + jb), you simply multiply out the brackets, for example lets multiply together (3 + j6) and (2 + j4)
So we have: (3 + j6) (2 + j4) = 6 + j24 + j12 + j(squared) 24 = 6 + j36 – 24 (as j squared is -1 remember). Notice when multiplying we don’t keep the real and imaginary numbers letters like we do in addition or subtraction, we can multiply a real number by a complex number – the same holds true when dividing which we will get onto shortly.

If there are any problems don’t hesitate to ask.