Tuesday, 12 August 2008

Division Of Complex Number

Division of Complex Number can be done the same way as normal algebra. However, depending on the divisor, if it is a Complex Number, then the use of Conjugate is appropriate.

Let's look at an example of division of complex number by a real integer.

Example 1: (a + ib) / 2

This is a simple math operation similar to basic algebra.

(a + ib) / 2 = (a/2) + i(b/2).

We obtained the solution by separately dividing the numerator by the integer 2.

Example 2: (a + ib) / ( c + id)

Note here that the denominator (divisor) is another complex number!

In this example, we need to apply the concept of Conjugate Multiplication.

For a review of Conjugate, please click here.

The denominator is a complex number that makes division difficult to perform.

To solve this obstacle (complex number denominator), we can change it to a real number by the use of Conjugate multiplication.

(c + id) (c - id) = c2 + d2 ==> a real numer

** The above is a useful step in Complex Number Division.

Moving forward, by multiplying the denominator by its conjugate, we need to also multiply the numerator by this conjugate of the denominator without changing the meaning of the question of Example 2.

(a + ib) / (c + id) = (a + ib) (c - id) / (c + id) (c - id)

The product of the numerator can be obtained through normal multiplication.
(a + ib) (c - id) = (ac + bd) + i(bc - ad).

Click this link to review complex number multiplication.

(a + ib)/(c + id) = [(ac + bd) + i(bc - ad)]/ k
where k = (c + id) (c - id) = c2 + d2.

Finally the answer is:
(a + ib)/(c + id) = [(ac + bd) + i(bc - ad)]/ k ==> (ac + bd)/k + i(bc - ad)/k

The above answer is of the form x + iy, the simple complex number format.

So through the use of conjugate multiplication, we can easily perform division of complex number by another complex number.

Hurray! :)

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