Summation of Polar Complex Numbers

Complex numbers can be added (or subtract) as normal algebra math operation. However, in complex number, there are not as straight forward depending on which form is chosen for the addition. To view the types of form used to represent complex numbers, click this link.

When the complex numbers are added using the Rectangular form, normal algebra method can be applied. (But do note that Real terms add to Real term, and Imaginary term to Imaginary term; do not intermix Real and Imaginary terms for addition or subtraction.)

When trigonometric form is used, it has to be converted to Rectangular form before normal algebra method can be used.

When the terms are in Polar form, careful consideration has to be taken.

They cannot be added directly in their Polar form!

Common mistake

Example: 5 Ð 0⁰ + 4 Ð180⁰

The answer given is (5 + 4 ) Ð(0⁰ + 180⁰). This is wrong!

Let me explain.

Let's say, we take 5 steps in the direction of right, and continue with another 4 steps but in the direction of left, we end up with 1 step in the direction of right from the initial starting point. (This is because we came back 4 steps after moving 5 steps.)

If we express the above in math,

5 steps to the right ==> 5 Ð0⁰, and
4 steps to the left ===> 4 Ð180⁰,

we are actually doing the 5 Ð0⁰ + 4 Ð180⁰ math computation.

From the logic deduction above, the answer should be 1 Ð0⁰ or 1 step to the right.

How then is 5 Ð0⁰ + 4 Ð180⁰ = 1 Ð0⁰ ?

The catch in adding Polar complex number is to convert this Polar form to Rectangular form! This is because length "travelled" in different directions cannot be added up directly. Angle consideration has to be taken.

Thus,

5 Ð0⁰ ==> 5 (cos 0⁰ + i sin 0⁰) = 5, and
4 Ð180⁰ ==> 4 (cos 180⁰ + i sin 180⁰) = -4.

5 Ð0⁰ + 4 Ð180⁰ = 5 - 4 = 1 ==> 1 Ð0⁰ , that is, 1 step to the right, which tallies with the logical deduction done in the first place.

In conclusion, to add Polar complex numbers, we must convert them to Rectangular form before performing the normal algebraic addition of its real and imaginary terms.

Note:
1) Here summation also implies to subtraction of polar complex numbers.
2) Add or subtract are operators of rectangular format, be it algebraic or complex numbers.

These are principles of mathematics that any maths learners have to know. It will be too shameful if we make this type of mistake. Maths will turn sour after that!

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