Written By Gnanambigai G S
Last Modified 27-01-2023

Geometrical Representation of Complex Numbers: Gaussian Plane, Conjugate, Negation

Geometrical Representation of Complex Numbers: In combination with real numbers, imaginary numbers are also called complex numbers that have several applications in real life. Beyond mathematics, they are commonly used in physics, engineering, astrophysics, and many other fields too. Complex numbers are widely used in quantum physics to study periodic motions such as light waves and alternating currents. Hence, it is vital to have a great grip on the basics of complex numbers for future applications.

In this article, let us deal with complex numbers from a geometric perspective and learn how to represent complex numbers geometrically.

What are Complex Numbers?

Complex numbers are of the form \(a + ib,\) where \(a\) and \(b\) are real numbers.

There are two parts in a complex number.

\(a \to \)real part

\(b \to \)imaginary part

In a complex number, \(i\) is called the imaginary unit and has a value of\(\sqrt { – 1} .\) Hence, we can say that \({i^2} = \, – 1.\)

Geometrical Representation of Complex Numbers

Like any pair of numbers, a complex number, \(a + ib,\) can also be represented as a point on a coordinate plane by plotting \(a\) on the \(x\) -axis and \(b\) on the \(y\) -axis. The plane that is used to interpret complex numbers is called a complex plane. It is denoted by the letter \(C.\)

Complex Numbers on Gaussian Plane (Argand Plane)

A complex plane is also called the Gaussian plane or the Argand plane. It is similar to a cartesian coordinate system, where:

1. The \(x\)-axis is called the real axis – formed by real numbers
2. The \(y\)-axis is called the imaginary axis – formed by imaginary numbers

When plotting the complex number \(a + ib,\) it corresponds to the ordered pair \(\left( {a,\,b} \right)\) as shown below.

How to plot complex numbers on a complex plane?

Step 1: Identify the real part and imaginary part of the complex number.
Step 2: Move along the real axis as much as the real part.
Step 3: Move parallel to the imaginary axis as much as the imaginary part.
Step 4: The point you arrive at is the required complex number representation on the Gaussian plane.

Observe how the complex number \( – 2 + 3i\) is plotted on the complex plane.

How to plot complex numbers on a complex plane?

Modulus of Complex Numbers

When a complex number \(a + ib\) is plotted on an Argand plane, the distance of the point from the origin \(\left( {0,\,0} \right)\) is called the modulus of that complex number. It is also called the magnitude or absolute value of a complex number.

Consider the complex number \(z = x + iy\) plotted on a complex plane. It forms a right triangle \(OXY\) with the \(x\)-axis with the base as \(x,\) and the perpendicular as \(y.\) The distance of the point from the origin is the length of the hypotenuse, \(r.\)

Using the Pythagorean theorem, we get, \(r = \sqrt {{x^2} + {y^2}} .\)

The modulus of a complex number is represented as \(\left| z \right| = r = \sqrt {{x^2} + {y^2}} .\)

Observe that for a complex number, \(\left| z \right| = \sqrt {{\rm{Real}}\,{\rm{par}}{{\rm{t}}^2} + {\rm{Imaginary}}\,{\rm{par}}{{\rm{t}}^2}} \)

Argument of Complex Numbers

The argument of a complex number is the angle formed by the line that connects the point to the origin with the positive \(x\)-axis or the real axis. It is denoted by \(\arg \) or \(\theta .\)

For a complex number, \(z = x + iy,\)

\(\theta = {\tan ^{ – 1}}\left( {\frac{y}{x}} \right)\)

The polar form of a complex number is written using modulus and argument of that complex number. First, plot the complex number on the Argand plane as shown.

How are argument and modulus related to complex numbers?

From the figure, we can write,
\(\sin \,\theta = \frac{x}{r} \Rightarrow x = r\,\sin \,\theta \)
\(\cos \,\theta = \frac{y}{r} \Rightarrow y = r\,\cos \,\theta \)
\(\therefore x + iy = r\left( {\cos \,\theta + i\,\sin \,\theta } \right)\)
Hence, the polar form of a complex number is written as \(r\left( {\cos \,\theta + i\,\sin \,\theta } \right).\)

Conjugate of Complex Numbers

Every complex number has a complex conjugate. Conjugate of a complex number is its mirror image across the \(x\)-axis or the real axis. The conjugate of a complex number, \(z,\) is represented by \(\overline z .\)

From the figure, observe that the complex conjugate of \(x + iy\) is \(x – iy.\) The complex conjugate is identified by the coordinate \(\left( {x,\, – y} \right).\) Also, the argument of the complex number and the conjugate is the same \(\theta ,\) but in opposite directions.

Algebraically, the complex conjugate of a complex number is obtained by changing the signs of the imaginary part of the complex number.

Complex Number	Complex Conjugate
\(a + ib\)	\(a – ib\)
\( – p – iq\)	\( – p + iq\)
\(x – iy\)	\(x + iy\)

Argument and Modulus of a Complex Conjugate

If the argument of a complex number is \(\theta ,\) then the argument of its complex conjugate is \( – \theta .\)

The magnitudes of the complex number and its complex conjugate will be equal, as magnitude is independent of direction.

Let us now see how the operations on complex numbers are represented geometrically.

Graphical Addition of Complex Numbers

Graphical Subtraction of Complex Numbers

Negation of a complex number will be \({180^{\rm{o}}}\) rotation of the complex number on the complex plane. For example, the negation of \(a + bi\) is \( – a – bi.\) The subtraction of complex numbers also uses the parallelogram rule. The subtrahend is negated first and then added to the minuend to get the difference between the two complex numbers.

Graphical Multiplication of Complex Numbers

For two complex numbers, say \({z_1}\) and \({z_2},\) the magnitude and argument of their product are defined as:

\(\left| {{z_1}{z_2}} \right| = \left| {{z_1}} \right|\left| {{z_2}} \right|\)

\(\arg \left( {{z_1}{z_2}} \right) = \arg \left( {{z_1}} \right) + \arg \left( {{z_2}} \right)\)

The product \(wz\) is graphically represented as shown here.

Graphical Division of Complex Numbers

For two complex numbers, say \({z_1}\) and \({z_1},\) the magnitude and argument of their ratio are defined as:

\(\left| {\frac{{{z_1}}}{{{z_2}}}} \right| = \left| {\frac{{{z_1}}}{{{z_2}}}} \right|\)

\(\arg \left( {\frac{{{z_1}}}{{{z_2}}}} \right) = \arg \left( {{z_1}} \right) – \arg \left( {{z_2}} \right)\)

The ratio \({\frac{{{z_1}}}{{{z_2}}}}\) graphically looks like:

Solved Examples on Geometrical Representation of Complex Numbers

Q.1. What is \(\arg \,z + \arg \,{\overline z } \)?
Ans:
Let’s first plot \(z\) and \( {\overline z } \) on a complex plane. As these are complex conjugate numbers, their arguments are equal at \( {\overline z } ,\) in opposite directions.

\(\arg \,z + \arg \, {\overline z } = \theta + \theta = 2\theta \)

Q.2. In which quadrant of the complex plane does \(1 – 4i\) lie?
Ans:
The complex number \(1 – 4i\) can be represented by the coordinate \(\left( {1,\, – 4} \right)\) on the complex plane.

The complex number \(1 – 4i\) lies in the fourth quadrant.

Q.3. Plot the complex number, its negation, and its conjugate on the graph.
Ans:

Type	Rectangular Form	Coordinates
Complex Number \(\left( z \right)\)	\(x + iy\)	\(\left( {x,\,y} \right)\)
Negation \(\left( { – z} \right)\)	\( – x – iy\)	\(\left( { – x,\, – y} \right)\)
Complex Conjugate \(\left( {\overline z } \right)\)	\(x – iy\)	\(\left( {x,\, – y} \right)\)
Complex Conjugate (Negation) \(\left( {\overline { – z} } \right)\)	\( – x + iy\)	\(\left( { – x,\,y} \right)\)

Q.4. Find the argument and modulus of \(1 + i\sqrt 3 .\)
Ans:
Let \(z = 1 + i\sqrt 2 \)

Modulus of a Complex Number	Argument of a Complex Number
\(\left\| z \right\| = \sqrt {{\rm{Real}}\,{\rm{par}}{{\rm{t}}^2} + {\rm{Imaginary}}\,{\rm{par}}{{\rm{t}}^2}} \) \(\left\| z \right\| = \sqrt {{1^2} + {{\left( {\sqrt 3 } \right)}^2}} \) \(\left\| z \right\| = \sqrt {1 + 3} \) \(\therefore \left\| z \right\| = 2\)	\(\theta = {\tan ^{ – 1}}\left( {\frac{y}{x}} \right)\) \(\theta = {\tan ^{ – 1}}\left( {\frac{{\sqrt 3 }}{1}} \right)\) \(\theta = {60^{\rm{o}}}\)

Argument of \(\left( { – 1 + i} \right)\)	Argument of \(\left( {4 – 6i} \right)\)
\(\theta = {\tan ^{ – 1}}\left( {\frac{y}{x}} \right)\) \(\theta = {\tan ^{ – 1}}\left( {\frac{1}{{ – 1}}} \right)\) \(\theta = \, – {45^{\rm{o}}}\)	\(\theta = {\tan ^{ – 1}}\left( {\frac{y}{x}} \right)\) \(\theta = {\tan ^{ – 1}}\left( {\frac{{ – 6}}{4}} \right) = {\tan ^{ – 1}}\left( { – \frac{3}{2}} \right)\) \(\theta = \, – {56.3^{\rm{o}}}\)

In the multiplication of complex numbers, we know that,
\(\arg \left( {wz} \right) = \arg \left( w \right) + \arg \left( z \right)\)
\(\arg \left( { – 1 + i} \right)\left( {4 – 6i} \right) = \arg \left( { – 1 + i} \right) + \arg \left( {4 – 6i} \right)\)
\(\arg \left( { – 1 + i} \right)\left( {4 – 6i} \right) = \, – {45^{\rm{o}}} – {56.3^{\rm{o}}}\)
\(\arg \left( { – 1 + i} \right)\left( {4 – 6i} \right) = \, – {101.3^{\rm{o}}}\)

Summary

The article starts with defining complex numbers as having two parts – real and imaginary. Then, it explains the geometrical representation of a complex number. It also goes on to elaborate on the geometrical representations of various operations such as addition, subtraction, multiplication, and division of two complex numbers. It also explains how the modulus and argument are related to the complex number. The article also explains the modulus and argument of complex numbers, their products, and ratios. The solved examples help us understand the concepts and the calculations involved in the operations of complex numbers.

FAQs on Geometrical Representation of Complex Numbers

Q.1. What is the geometrical representation of a complex number?
Ans: Geometrical representation of a complex number is marked on a complex plane. The complex plane is similar to a coordinate plane except that the horizontal axis has the real numbers and the vertical axis has the imaginary numbers. The complex number \( – 2 – 3i\) is plotted on the complex plane by the coordinate \(\left( { – 2,\, – 3} \right).\)

Q.2. How do you geometrically divide a complex number?
Ans: For two complex numbers, say \(w\) and \(z,\) the magnitude and argument of their ratio are defined as:
\(\left| {\frac{w}{z}} \right| = \frac{{\left| w \right|}}{{\left| z \right|}}\)
\(\arg \left( {\frac{w}{z}} \right) = \arg \left( w \right) – \arg \left( z \right)\)
The ratio \(\frac{w}{z}\) is graphically represented as shown here.

Q.3. What is the square of a complex number?
Ans: A complex number has two parts – real and imaginary. Apply the FOIL approach to multiply the two:
1. First terms
2. Outer terms
3. Inner terms
4. Last terms
Example: Let \(z = a + ib\)
\({z^2} = \left( {a + ib} \right)\left( {a + ib} \right)\)
\({z^2} = {a^2} + iab + iab + {i^2}{b^2}\)
\({z^2} = \left( {{a^2} – {b^2}} \right) + i\left( {2b} \right)\)

Q.4. How do you add complex numbers?
Ans: The geometrical addition of two complex numbers follows the parallelogram rule. Steps to add two complex numbers \({z_1},\) and \({z_2}\) geometrically:
Step 1: Plot \({z_1}\) and \({z_2}\) on the complex plane.
Step 2: Construct a parallelogram with coordinates of \({z_1}\) and \({z_2}\) as opposites vertices.
Step 3: Draw the diagonal from the origin as the resultant vector.
Step 4: The sum of the two complex numbers is the fourth vertex of the parallelogram.

Q.5. What is the product of two complex numbers?
Ans: For two complex numbers, say \(w\) and \(z,\) the magnitude and argument of their product are defined as:
\(\left| {wz} \right| = \left| w \right|\left| z \right|\)
\(\arg \left( {wz} \right) = \arg \left( w \right) + \arg \left( z \right)\)
The product \(wz\) is graphically represented as shown here.

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