Graphical Representation of Complex Numbers

numbers is called complex numbers. It is the extension of real numbers that contains all the polynomial roots that have a degree. Suppose you have to define the solution of x² = -1, then complex numbers are composed in a set in the form of a+ib. The set can be written as : {a + ib la, b ∈ R}

 Important Definitions in the concept of Complex Numbers

 ●  Equality of Complex Numbers

 Two complex numbers are equal if the real part of that complex number and the imaginary part is equal. For example y₁ = c₁ + id₁ and y₂ = c₂ + id₂. We say y₁ and y₂ are equal if c₁ = c₂ and d₁ = d₂.

●  Purely Real Complex Numbers

 A complex number is said to be real purely when it does not have any imaginary part. When we take y = c₁ + id₁ as a complex number. Y will be considered a purely real complex number if it’s imaginary part d₁=0.

 ●  Purely Imaginary Complex Number 

A complex number is considered purely imaginary when it does not have any real part. We take the example of y = c₁ + id₁ as a complex number. Y will be considered a purely imaginary complex number if it’s c₁=0. 

●  Zero Complex Numbers

 A complex number is said to be a zero complex number if both the real part and the imaginary part are zero. For example, y = c₁ + id₁ is a zero complex number when c₁=0 and d₁=0.

  How to represent complex numbers graphically?

 We can easily represent any complex number on a complex plane by following some basic steps. While plotting complex numbers graphically, you use the horizontal part of the coordinate axis to plot the real part and the vertical part of the coordinate axis for the imaginary part. The geometrical extension is the concept of one dimension line extending to the two-dimension line. Complex using the plane for the horizontal line for the real part and vertical line.

 Plotting a Complex Number

 To plot complex numbers a+ib thought of as point(a,b), we take the S plane, which is the same as the x-y axis relating to a normal cartesian system. S plane is also known as Argand plane or Complex-plane or Argand diagram, which was named after Jean-Robert Argand. He plots the real and imaginary values as a and b at a point P. The horizontal axis is called the real axis and the vertical axis is known as the Imaginary axis. Consider (1,4) in a complex plane. The term can be defined as 1+4i in complex numbers plotted in the first quadrant as both real and imaginary numbers are positive. Moreover, we can have -1+3 as (-1,3) in the second quadrant. It is written as -1-4i and 1-4 in the fourth quadrant. You can similarly understand the graphical representation of motion or vectors with the help of this concept.

 How to calculate the Powers of Iota?

Iota is defined as the negative root of unity.

That is i = √−1 

When we are properties as

 i² = i x i = (√−1)x(√−1)

 = -1   

 i³ = i x i x i = √−1 x√ −1 x √−1

 = -1 x i = -i 

i⁴ = i x i x i x i = √−1x√ −1x √−1x√ −1

 = (-1) x (-1) = 1  

This also means, 

in

 = i4k

  for some value of n as well as k,

 = 1

 in

 = i4k+1

 for some value of n as well as k,

 = i

in

 = i4k+2

 for some value of n as well as k,

 = -1

in

 = i4k+3

 for some value of n as well as k,

 = -i

Important Complex Numbers Properties

Conjugate of a complex number

 Take a complex number z = a + ib. Then its conjugate is expressed as z* = a – ib. It is known as the reflection of z about the real axis.

 Conjugating a complex number that has twice resulted in the complex number itself, i.e.,

 z** = z.

Consider z = a+ib. then, z* = a – ib and z** = a + ib; thus z** = z is justified.

Moreover, Re(z*) = Re(z) and |z*| = |z|.

Similarly, Im(z*) = -Im(z) and arg(z*) = -arg(z).

zz* = |z|2 = |z*|2

Re(z) = (z+z*)/2

Im(z) = (z-z*)/2i

Arithmetic Operation on Complex Numbers

Considerz1= a + ib and z2= c + id;

Addition: z1+ z2= (a + c) + i(b + d)

Subtraction: z1– z2= (a – c) + i(b – d)

Multiplication: z1* z2= ac – bd + i(ad + bc).

Complex Number Commutative Property

z1+ z2= z2+ z1

z1* z2= z2* z1

|z1 + z2| ≤ |z1| + |z2 |

 Examples

 Problem 1: Your task is to represent the given complex number, z = 1+ i√3 in the polar form.

Answer:

Let us take r cos θ = 1,

r sin θ = √3.

Then we follow the process of squaring and then adding both the sides, the output is

r2 (cos2  θ +sin2 θ ) = 3+1

r2 = 4

Hence, r = 2

By Substituting the value of  r in r cos θ = 1

Now, we know that cos θ = 1/2

Also we know, sin θ = √3/2

⇒ θ = π/3

Hence the answer of the polar form of the complex number z is = 2 (cos  π/3 + i sin π/3).

Problem 2: Let a complex number be defined by z = 3 + 4i and another complex number a = 3 + xi and b = y + 2i. Given that z = a + b. Then, find the values of x and y?

Solution: z = a + b => (3 + 4i) = (3 + xi) + (y + 2i)

=> 3 = 3 + y and 4 = x + 2

=> Solving, we have x = 2 and y = 0.

Problem 3: Find the conjugate of z1 if z2 + z3 = 0 and z1 = 3 * z3; given z2 = 5 + 5i and z3 = x + yi? Also, find the magnitude and argument of the conjugate of z2?

Solution: z2 + z3 = 0 implies 5 + x = 0 and 5 + y = 0.

Solving we have x = – 5 and y = -5.

Thus, z3 = -5 – 5i.

Given z1 = 3 * z3

=> 3(-5 – 5i) = -15 -15i.

We need to find the complex conjugate of z1.

Hence, z1* = -15 + 15i.

Now, z2 * = 5 – 5i.

|z2 |= √(25 + 25) = 7.07

And Ɵ = tan-1 (y/x) = tan-1 (-5/5) = tan-1 (-1) = –π/4.

Conclusion

 Complex numbers are those numbers whose roots are needed to extend real numbers. That extension is known as complex numbers. Thus a complex number can be defined as a+bi, which can be identified as a point i.e., is P(c,d) on a complex plane. The complex plane can be represented in a polar form associated with each complex number with a distance from the origin as its magnitude with a particular angle is called as argument of the complex number. Hope you have got all the important information about Graphical Representation of Complex Number.