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).
= 3(2+i) + 2i(2+i)
= 6+3i + 4i+2i2
= 6+3i +4i-2 as i=√-1
Example 2: Find the product (1+i)2.
= (1+i) (1+i)
= 1(1+i) + i(1+i)
= 1+i +i+i2
= 1+2i -1
So 1+i whole square is equal to 2i.
Q1: How to multiply two complex numbers?
Answer: Two complex numbers a+ib and c+id are multiplied by the rule (a+ib)(c+id) = (ac-bd) +i (ad+bc).