Plotting the Complex Plane – Graphing Complex Numbers – Argand Diagrams

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The Argand diagram is a special representation of the complex number in a plane. As already stated, a complex number is the addition of real numbers and imaginary numbers. This means if a complex number is to be plotted, it will be in the form of the real number line and the imaginary number line.

Now, if you remember correctly, the real number line is a horizontal line starting comprising negative and positive integers. However, when the imaginary number gets involved in the equation, their plotting is represented in a vertical line and comprises negative infinite imaginary numbers (like – 55i, -231i), positive infinite imaginary numbers (like 23i, 565i) and every other value in between.

Therefore, with the above, a complex number, when represented in its number line, will be a mix of real numbers and imaginary numbers. The graphical representation of these lines is what is known as an argand diagram

The Argand Diagram: How to Graph a Complex Number

The Argand Diagram is the representation of complex numbers in their plane. This is known as the complex plane. Since a complex number has a real number and an imaginary number, graphing an Argand Diagram requires you to create the imaginary number line and the real number line.

Now since the real number line is always horizontal, it will remain so in the argand diagram. That leaves the imaginary number line as vertical. Now, this is how the real number and positive lines will look.

Now the Argand diagram is a mix of these two lines. This means we will have to join them together with the two lines interesting at zero to form a cross. As such, the Argand diagram will be

The above is about 60% of an Argand Diagram. Now since the above diagram has been plotted. We will have to label them. The positive side of the number line after the zero is called the positive X-axis. The negative side of the same real number line is called the negative x-axis.

The Positive Side of the imaginary vertical line before the zero is known as the positive y-axis, while the negative side of the same imaginary number line is known as the negative y-axis.

This will make our Developing Argand Diagram become

Now the argand diagram is almost complete and what remains is how to plot a Complex number on it. Once a complex number has been plotted on the diagram above, then we have a complete Argand Diagram.

How to Plot an Argand Diagram

Step 1: write out the basic form of the initial Argand Diagram

Step 2: identify the a and b of the complex numbers. Recall that z = a + bi. a is for the X-axis, and b is for the y axis. 

Step 3: If a is positive, it will be plotted in the positive X axis. It will be plotted on the negative x axis if it is negative. In the same vein, if b is positive, it will be plotted in the y axis of the positive, and if it is negative, it will be plotted in the negative y axis

Example 1

Plot an Argand diagram of z = 4 + 2i

Solution

Step 1: write out the basic form of the initial Argand Diagram

The basic form of the initial argand diagram is given as

Step 2: identify the a and b of the complex numbers. Recall that z = a + bi. a is for the X axis, and b is for the y axis. 

z = 4 + 2i

as such a = 4 and b = 2

Step 3: If a is positive, it will be plotted in the positive X axis, if it is negative, it will be plotted in the negative x axis. In the Same vein, if b is positive, it will be plotted in the y axis of the positive and if it is negative, it will be plotted in the negative y axis

since a = 4 is positive, it will be plotted on the x axis

and,

b = 2 is positive it will be plotted in the y axis

AS such, our argand diagram will be in this form

Clearly, since a = 4, it will be on the number of the 4 of the positive axis while b = 2 will be on the 2 of the positive y axis

The intersecting line where they meet is the red spot in the diagram below

Now that the intersection of the real and imaginary numbers is drawn, we have successfully determined the argand diagram completely.

Now, what if we drew a line from the x axis so that it touched the intersection point. We will have something of the form

Now, if we cross that intersection with another dotted line, we will have a right angle whose base or adjacent is 4, and its opposite is 2i. This is what we are saying

Now r is the modulus of the complex number, which is given as z = 4 + 2i

  • The modulus is very important for determining the polar form of a standard complex number which is an entirely different topic of its own.
  • The Argand diagram can also be used to find the argument of the complex number, which is again also important for finding the Polar form of a standard complex number

As Such, Finding the Polar form of the complex number is heavily dependent on the Argand diagram. It also means the Argand Diagram is simply a means to an end, not an end.

Example 2

Now, we can see that from this example, the plotted Argand diagram is in an entirely different part quarter compared to the first example, which was in the quarter above this one. These quarters are known as quadrants and are usually termed 1st quadrant, second quadrant, third quadrant and fourth quadrant.

This particular complex number is in the fourth quadrant. That is diagrammatical;

Example 3

Plot the Argand diagram of the complex number z = -3i

Solution

Based on the standard form of the complex number,

a = 0, and b = -3

this means that the Argand diagram will not have an x mark or an intersection. It will just be like this below

Scaling in The Argand Diagram

Generally, plotting x,y axes requires the need to use a common scale to create an equal distance among every point or number. However, for the argand diagram, that is not the case. The x, Y axes are simply number lines and are only relevant for finding the quadrant that a complex number fall in with the ultimate aim of finding the polar form of the expression.

Therefore, students can simply mark the numbers, plot their complex number in the argand diagram, and get the quadrant, modulus and argument. If a student decides to use a measured distance to separate the points, it will make the work neater, but that is all there is to it.

Conclusion

As stated in this article, the aim of plotting the Argand Diagram is to find the polar expression of a standard complex number. This piece explicitly outlined the importance of the argand diagram in complex number theory. Every step and process of plotting the diagram was explained throughout the way, so students could know how to plot the angles themselves without any confusion.

It is recommended that students continually plot the Argand Diagram of different complex numbers until they master it. If they run into any difficulty, they can always come back to check the steps outlined in this article. Once the Argand diagram has been fully mastered, students can now advance to learn the modulus and arguments of a complex number. Once the two have been successfully mastered, they will also finally advance to learning how to write the polar form of the complex number.

Plotting the Complex Plane – Graphing Complex Numbers – Argand Diagrams

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