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.
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