The Conjugate of Complex Numbers on Argand plane

It’s easy to plot a real number on a number line, for instance, see below where we have plotted 3 and -4, .

Now let us see how we represent an ordered set of real numbers Z (2,3). This will be presented as 2 units on the X-axis and 3 units on the Y-axis. This gives us the P(2,3). To plot the ordered pair of real numbers, we use the X, Y plane, which has real numbers on both the axes. This is called the cartesian plane.

Now, look at the complex number 3+ 4i. This has 3 as a real number and 4i as the imaginary part. Let’s check how this can be represented. For this, we will assume the X-axis as the real number part and the Y-axis as the imaginary number part. Based on this, we can plot the P (3,4)

To plot the set of complex numbers consisting of a real and imaginary part, where X is the real axis and Y is the imaginary axis. This is called the argand plane.

You will find brief information on the concept of the argand plane, a thorough explanation of properties of the conjugate of complex numbers on the argand plane, its functions and so on. So, let us start with the definition of the conjugate of complex numbers on the argand plane.

Definition of the Conjugate of Complex Numbers on the Argand plane

One of the fundamental laws of mathematical algebra is complex numbers’ geometrical presentation. A complex number Z = x+ yi, where x is a real number and y is the imaginary part. This can be presented as p (x, y) in a plotting plane called the Argand plane. Where the value of i= √-1.

The Conjugate of Complex Numbers on the Argand plane

The conjugate of a complex number is keeping the real number of the equation as it is and changing the imaginary unit with its inverse. The conjugate is represented by Z* or ˉz sign mathematically.

For example, if Z = x + iy, then its conjugate would be

ˉz = x – iy

Reverse image of Z = x + iy along with the real axis shows the conjugate of the pair of the complex numbers.

Let’s understand the concept of representation of complex numbers on an argand plane diagram.

Example 1

Plot Z = 5+3i on an Argand Plane.

Solution: An Argand plane is where a vertical line or Y axis represents the imaginary part and horizontal line or X axis represents the real part of complex numbers.

Now, to find the conjugate of the above complex number, we need to follow the below procedure.

To represent the conjugate of a complex number on an argand diagram, we will be reflecting it on the real -axis. In the above diagram, we can represent the conjugate of the complex number by reflection on the horizontal real axis.

Let’s understand another example.

Example 2

Represent Z = -3 + 2i on an argand plane

Solution: In the above complex number Z = -3 + 2i, where -3 is the real number and 2i is the imaginary number, they will be plotted on the horizontal X axis and vertical Y axis respectively. As we have studied, the X axis denotes the real axis, where real number sets are plotted and the Y axis shows the imaginary part where imaginary units are plotted. This explanation gives us the below argand diagram.

To find out the conjugate of the above complex number, we have to reverse the imaginary number which will be denoted as ˉz = -3-2i.

The conjugate of a complex number on an argand diagram will be a reflection on the real-axis. In the above diagram, we can represent the conjugate of the complex number by reflecting the coordinates on the horizontal real axis.

Properties of the Argand Plane

For better understanding of the argand plane, let us walk through the below properties of the argand plane.

• The argand plane has axes similar to the regular coordinate/cartesian axes.

• The intersection point of the real and imaginary axis on the argand plane is the point of origin.

• The real and the imaginary axis are perpendicular to each other on an argand plane.

• The real and the imaginary axis of the argand plane divides it into four quadrants, similar to regular cartesian axes.

• The formulas of distance and midpoint are the same in the argand plane, as in the coordinate axes.

• The argand plane point is represented as cartesian coordinates or polar coordinates.

Conclusion

In this article, describing the complex numbers, we studied the concept of the conjugate of complex numbers on the argand plane and its basic properties in length. We covered several other topics such as, the functions that can be performed with the conjugate of complex numbers on the argand plane, and other related topics along with solved examples.