Complex multiplication is **not** a difficult operation to understand from either an algebraic or a geometric point of view. We can do it algebraically first, and let’s take specific complex numbers to multiply, say 3 + 2i and 1 + 4i. Each has two terms, so when we multiply them, we’ll get four terms:

(3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i2.

Now the 12i + 2i simplifies to 14i, of course. What about the 8i2? Remember we introduced i as an abbreviation for √–1, the square root of –1. In other words, i is something whose square is –1. Thus, 8i2 equals –8. Therefore, the product (3 + 2i)(1 + 4i) equals –5 + 14i.

In general, if you multiply complex numbers of the form **(x+iy)(a+ib)** you will get **(xa – yb) (xb+ya)i**

**Note: The multiplication of a complex number with its conjugate will always give ONLY real part.**

eg Given **2+3i **and its conjugate is **2-3i, **The multiplication will give **2^2 +3^2 ***which is a Real part*

The following image explains more.