The 3 forms are namely:
- Rectangular form
- Trigonometric form
- Polar form
These 3 forms serve different purpose in math computation.
The first form (rectangular) is the most common presentation.
Example of complex number rectangular form: Z = a + ib
The second form is the Trigonometric form.
By its name, you can suspect that it is related to angle and its trigonometric ratio.
You are not wrong!
If you can recall what trigonometric ratio is about, you will know that any number can be represented by a trigonometric function. Example: cos A = 0.2 or 0.2 = cos A.
This second form can be linked to the rectangular form through the above understanding.
Therefore, a + i b ==> Cos A + i Sin A.
A numerical example: 0.5 + i 0.866 = cos 600 + i sin 600
How did we get the value of the angle (in the above case, 600)?
Use the real and imaginary number.
From the value of 0.5 and 0.866, and applying K = tan-10.866 / 0.5, we get the answer for the angle as K = 600.
For more detail on obtaining the angle from the real and imaginary term, see this link.
This leads us to the third presentation, Polar form.
Example of Polar form: 5 Ð 600
In this polar form, the interpretation for "5" is the length of the vector line and 600 is the tilt angle of this vector line with the horizontal reference.
(NOTE: a + ib forms a vector line with a certain direction in Argand diagram)
Polar form can also be converted to Trigonometric form easily through the below relation:
5 Ð 600 = 5 (cos 600 + i sin600)
Note: A Ð -600 = A (cos 600 - i sin600), look at the "minus" sign!
And from the Trigonometric form, you can also easily convert it to Rectangular form.
When to use which complex number form depends on the math question and the available information.
If the question given has only pure numbers for its real and imaginary term, then it would be appropriate to use the Rectangular form to proceed.
If angle information is within the question, then you can either select the Trigonometric or Polar form.
However, if division of complex number is involved, we can use either the Conjugate method or the Polar form.
Trigonometric form will be hard to handle in complex number division as we cannot manipulate the trigonometric terms directly without conversion to pure numbers.
By knowing the various merits and demerits of the three formats of complex numbers, we are able to decide what is better for us. This is nothing difficult and applies to any things we encounter in our daily life (issues other than maths).