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Graphing Complex Numbers

In this page we are going to discuss about graphical representation of complex numbers concept. Before that learn what is complex number.

Complex number:

Complex number is defined as a number which including two parts such as real part and imaginary part. These numbers are consisted the real numbers as ordinary. It can be extended by adding the extra numbers.

The general form of complex number is (a + bi). Here a represents real part of complex number, b represents imaginary part of complex number with the standard imaginary unit of i.

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Complex Number Graph

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The complex numbers are graphing with complex plane. On that plane, the horizontal axis denotes the real part of complex number and vertical axis denotes the imaginary part of the complex number.

Graphing Complex Numbers

The complex number Z = x + iy may be represented graphically by the point P whose rectangular co-ordinates are (x, y). Thus each point in the plane is associated with a complex number. In the figure, P defines Z = x + iy. It is customary to choose x-axis as real axis and y-axis as imaginary axis.

Draw PM perpendicular to x-axis.


P`hatQ` M = `theta` , OM = X, PM = y, OP = r

r = $\sqrt{x^2 + y^2}$, x = r cos $\theta$

Here the positive value of r is called the modulus of Z or absolute value of Z. $\theta$ is called the amplitude or argument of Z.

$\therefore$ r = | Z | = | x + iy | = $\sqrt{x^2 + y^2}$

$\theta$ = amp Z = arg Z = tan-1 $\frac{Y}{X}$

Usually complex numbers have some basic properties as follows,

  • The group of every complex numbers is generally indicated by C.
  • If z = x + yi, the real part x is indicated Re (z), and the imaginary part y is indicated Im (z). Here the imaginary unit i gives the property as i^2 = -1.
  • The real number x is combined with the complex number as x + 0i.
  • The imaginary number y is combined with the complex number as 0 + yi.
  • The equality property of complex numbers states that, let two complex numbers are a + bi and c + di. From the equality property it can be written as a = c and b = d.


Generally, the complex numbers are used in following fields,

  • Engineering
  • Electromagnetism
  • Quantum physics
  • Applied mathematics
  • Chaos theory
→ Read More

Graphing Complex Numbers Examples

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Few of examples based on graphing complex numbers are as follow:

Solved Examples

Question 1: Find the modulus and argument of the complex number: 1 + i
Let Z = 1 + i
Compare the given complex number with its general form i.e. Z = x + iy
=> x = 1 and y = 1
Modulus = r = $\sqrt{x^2+y^2}$ = $\sqrt{1^2+1^2}$ = $\sqrt{2}$
Argument = $\theta$ = Tan$^{-1}$($\frac{y}{x}$) = Tan$^{-1}$($\frac{1}{1}$)
= Tan$^{-1}$(1)
= $\frac{\pi}{4}$


Question 2: Graph the following complex numbers on the Complex coordinate system:
5 + 6i, -3 - 2i and 4 - 4i.

To graph the given complex numbers on the complex system, represent real parts along horizontal axis (real axis) and imaginary part along vertical axis (imaginary axis).
5 + 6i (Re = 5 and Im = 6)
-3 - 2i (Re = -3 and Im = -2) and
4 - 4i (Re = 4 and Im = -4)


Question 3: Add 3 + 2i and 5 - i. Plot the resultant on the graph.

Given complex numbers are: 3 + 2i and 5 - i
3 + 2i + 5 - i = (3 + 5) + i(2 - 1) = 8 + i


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