# Complex Numbers 5: Multiplying Complex Numbers

#### Nirvana

Joined Jan 18, 2005
58
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.

Nirvana.