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 +i^{2} bd = ibc – bd [as we know that i^{2}=-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+2i^{2}

= 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+i^{2}

= 1+2i -1

= 2i

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

**Also Read:**

**FAQs**

### 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).