Tuesday, 12 August 2008

Quick Mental Conversion Of Complex Number Forms

This post covers simple terms in complex numbers that can be mentally converted from its Rectangular form to Polar form, and vice versa.

There are times where complex number terms warrant the use of detailed computational steps to perform cross-conversion, or the need of calculator to do the task. But not all need this detail steps.

If you understand the principles of angle and magnitude, you may find some short-cuts.

Below are some tips to bypass the tedious conversion.

  • 1 <==> 1 Ð00

  • -1 <==> 1 Ð1800

  • i <==> 1 Ð900

  • -i <==> 1 Ð-900

The trick is the understanding that the "i" is plotted in the vertical axis while the Real number is at the horizontal axis. This makes the "i" 900 away from the Real number.

(The diagram plotted, by the way, is the Argand Diagram.)

Let's do some examples to see their usefulness.

Example A: 8 + 1 Ð1800

To solve the math question, we need to convert the polar term to rectangular since the 2 terms cannot be added directly as they are of a different form.

8 + 1 Ð1800 ==> 8 + (-1) = 7 (Answer).

It can be done without the aid of calculator or detailed conversion steps.

Example B: 3 + i3 in polar form

Mentally, we know that i3 has amplitude "3" and "argument" = +900 or 3 Ð900 .

When the horizon length is 3 units and the vertical length is also the same, we know that the angle formed will be 450. This will be the resultant argument for the 3 + i3.

The amplitude for the 3 + i3 can be gotten from the Pythagorean Theorem,
==> sqrt(32 + 32) = sqrt(18).

Therefore, 3 + i3 = sqrt(18) Ð450 .

This is again done without detailed conversion but with the understanding that the imaginary "i3" is vertically "up".

Example C: 2 / (1 - i) in polar form

Instead of complex conjugate method, we can use the above mental conversion to simplify.

2 => 2 Ð00

1 - i ==> sqrt(2)Ð-450. 1 unit on the horizontal and 1 unit pointing down vertically.

Note: "-i" means vertically down, therefore angle is -900.

Moving forward, 2 / (1 - i) = [2 /sqrt(2)] Ð0-(-45)0 = [2 /sqrt(2)] Ð450. Visit this link to review Polar Number Division.

From the above 3 examples, you can see that knowing the simple conversion simplifies the mathematical computation of "complex number" questions.


No comments: