How to Multiply Complex Numbers with Examples

Complex numbers can be multiplied like real numbers, but the rule is a bit different. In this post, we will learn how to multiply complex numbers with an example.

Note that a complex number can be represented by a+ib where a and b are real numbers and i=√-1 is an imaginary complex number.

Method of Multiplying Complex Numbers

Let z=a+ib and w=c+id be two complex numbers. To find the multiplication of z and w, that is, to get the value of zw, we need to follow the below steps:

Step 1: Write the two complex numbers side by side as follows (a+ib)(c+id).

Step 2: Multiply a with c+id. Also, multiply ib with c+id. Thus we get the numbers a(c+id) and ib(c+id).

Step 3: Calculate the two numbers as follows: a(c+id) = ac + i ad. Also, ib(c+id) = ibc +i² bd = ibc – bd [as we know that i²=-1].

Step 4: Add the two numbers obtained in the previous step, that is, in Step 3. This gives us a(c+id)+ ib(c+id) = (ac + i ad) + (ibc – bd) = (ac-bd) +i (ad+bc).

Step 5: The quantity obtained in Step 4 is the product zw.

So the multiplication of two complex numbers a+ib and c+id is given by the rule:

(a+ib)(c+id) = (ac-bd) +i (ad+bc)

Lets us now understand the above multiplication rule of complex numbers with an example.

Example 1: Find the product (3+2i)(2+i).

Solution:

(3+2i)(2+i)

= 3(2+i) + 2i(2+i)

= 6+3i + 4i+2i²

= 6+3i +4i-2 as i=√-1

= 4+7i.

Example 2: Find the product (1+i)².

Solution:

(1+i)²

= (1+i) (1+i)

= 1(1+i) + i(1+i)

= 1+i +i+i²

= 1+2i -1

= 2i

So 1+i whole square is equal to 2i.

Also Read:

Solve x⁶-1=0

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