Coldwelded Math

Basics of Complex Numbers and Functions

Complex Numbers has a very important role in many mathematical calculations and it is very important for graphing Two Dimensional and Three Dimensional in the “Complex Plane”. Complex Numbers consists of both Real and Imaginary Numbers. Real Numbers are those which can be represented in a number line, whatever axis it may be, you can graph a Real number along with the axis you’ve chosen. Now things get little trickier about Imaginary Numbers. Just to recall, an Imaginary number are those numbers which do not have a proper or REAL square root.

Let’s Consider an Example:-

For Example, consider √-49 ( square root of -49). It would be easy for us to say the square root of 49 or simply √49 which is 7 but how to find the square root of √-49. Pretty hard to think, right? Well, the answer involves a special concept called “The Imaginary Unit”. Here’s where a Complex number starts to trace its existence because a Complex number is a combination of both Real and Imaginary Numbers.

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Imaginary Unit:-

An Imaginary Unit is nothing but a constant but not just any other constant but a special one. It is referred as “i” and it has a value of √-1.

Or simply,

Complex Numbers
Value of “i”

Yes, if you have a careful look at the imaginary number we are talking about – √-49 it is nothing but

√49 x √-1 or simply √49which can be further simplified as 7i. Now, coming to the concept numbers, we generally say “A Complex Number has two parts – a real one and an imaginary one, Both of the parts are added to give a complex number” This is further explained below,

Composition of Complex Numbers:-

Complex Numbers
The composition of a Complex Number with its parts mentioned

As mentioned in the image above, “Z” is our required Complex Number, “X” is a Real and number so as “Y” while “i” is the imaginary unit which turns “Y” an Imaginary Number.

Argand Diagram:-

Now, moving on with its graphing, A Complex number cannot be graphed in a normal x-y plane. It has a different system of graphing. We call the graphs of Complex Numbers as Argand Diagrams, named after the scientist “Jean Robert Argand”. In this method of graphing, we have two axes namely, the real axis – one which is horizontal and the imaginary axis – one which is vertical. It is obvious that the Real axis is for marking real part and the Imaginary axis is for marking the imaginary part. Here, the imaginary part includes both “i” and “y”. Let us discuss this more with an example,

Complex Numbers
An Example Graph (Argand Diagram) of a Complex Number

 

In the above graph, we have a complex number whose value is 5+9i. To plot this number in the imaginary plane so as to obtain its argan diagram, we have to follow these steps:-

Step 1:-

Draw a graph with two axes, where the horizontal axis represents the real part and the vertical axis is the imaginary part.

Step 2:-

Then mark a point 5 units (since the value of the real part is 5) away from the origin of the positive side ( First Quadrant ) and draw a perpendicular at this point.

Step 3:-

Now Let’s mark 9 units (since the value of the imaginary part is 9 but here i is not considered) above the origin along the imaginary axis and Let’s draw a perpendicular at this point.

Step 4:-

The point where both the perpendiculars meet is the required point of the complex number.

Some Important Formulae which we need to consider is:-

  • The magnitude of the Complex Number that is |Z|
  • Angle of inclination

Magnitude Formula (|Z|):-

As we have shown in the image above, the Magnitude of a complex number is represented as |Z| and it is equal to the square root of the sum of the squares of x and y. Here, Our  x is the real part and y is the real number from the imaginary part and here we exclude I because it cannot be considered for magnitude calculation as it only defines the character and has nothing to do with magnitude…

Angle of Inclination (θ) and Slope (m):-

Complex Numbers
Slope and Angle of Inclination Formula

The angle of Inclination and Slope is also an important concept because unlike real numbers, complex numbers require both the axes for marking a complex number in the complex plane. We generally derive the angle of Inclination from the slope of the line formula. In other words, Slope is nothing but the tan of the angle of inclination which is equal to the quotient of y divided by x and the angle of inclination is the tan inverse of the slope.

Other Related Articles:-

  • Polar form and Basic Operations (Under Moderation)
  • Analytic Functions and Complex Differentiation – Topic (Under Moderation)
  • Complex Integration – Topic (Under Moderation)

 

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