How Are Complex Numbers Used ?

In math, there are many "hidden" ideas and concepts.
One of them is the application of complex number. Read on to discover more .....

Complex Number can be expressed in 3 forms.

One of them, the Polar form, can clearly demonstrate that the information provided is "amplitude" and "argument".

Polar form: A ÐK⁰

where

"A" represents the length and normally termed "Amplitude" or "Modulus", and

"K" representing the angle it makes with a horizontal axis and termed "Argument".

With this understanding, let me proceed to quote an example of its application.

The example I have in mind is related to the electronic engineering field, and will therefore be a bit technical. However, do not despair, as I will focus on the mathematical application.

In electronics, we basically deal with 2 types of electronic component.

One type has voltage (source of energy) and current (flow of electrons) within it that mingle in harmony, that is, they are in synchronisation. They are, therefore, said to be "in-phase" or going maximum /minimum in value at the same time.

In complex number notation, we classify it as "Real" since the "argument" of their characteristics is ZERO.

The second type is not "in-phase" and has a time difference between the voltage and current within them, and normally at 90⁰ angle delay.

This type, in complex number expression, is attached with the "i" notation since there is an angle difference of 90⁰.
(NOTE: It can also means "-i").
They are the "Imaginary" part.

When the 2 types of components are placed together, they are either added up (connected in series) or multiplied (connected in parallel).

Example: Z1 = R, Z2 = iX

When in series, Z1 + Z2 = R + iX

When in parallel, (1/Z1) + (1/Z2) = (Z1 x Z2) / (Z1 + Z2) <== Just understand they are more complicated when connected in parallel.

From the above expressions, we can see that we have to use all the 4 basic mathematical operations in complex number (+, -, /, x).

In conclusion, complex number comes into play when the target we need to solve has angle (argument) information within it. Complex number contains both amplitude and argument information, and thus, is a suitable mathematics tools that we can apply.

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