Argand Diagram

Introduction 

The Argand diagram is a graphic representation of complex numbers of the form ‘x + iy’, where ‘x’ and ‘y’ are real integers, and ‘i’ is the square root of ‘-1’. It was invented about 1806 by Swiss mathematician Jean Robert Argand. Caspar Wessel, a Danish surveyor, offered an equal representation in 1797, although it was not recognised until later. The pure imaginary numbers (those consisting solely of the iy component) are represented on one axis, while the real numbers are represented on the other, only x-values. This enables the plotting of complex numbers as points in the plane defined by the two axes. 

Let    Z=x+iy, where Re(z)=x and Im(z)=y.

A point will be represented by the ordered pair (x,y) on the Argand plane. This point represents our complex number z. We draw a directed line from the origin to the point P(x,y). Let θ be the angle formed by this line with the “Real Axis’ positive direction. As a result, the “Imaginary Axis” forms an angle of (90-θ).

The idea for the argand diagram comes from the knowledge that a complex number is nothing more than an ordered pair of real numbers. As a result, any complex number can be expressed geometrically as points on a plane.  

Argument of z

On the Argand Plane, every Complex number has a representation. Complex numbers are closed under our Algebra’s operation; hence this is true. Take two points z1=5+6i and z2=5-6i. If you map the two points (5, 6) and (5, -6) are symmetrically above and below the real axes. We refer to them as mirror copies of one another. 

The Argument of  z1 and z2 is a new quantity that we introduced to differentiate these two points. It’s the angle “θ” formed by the line connecting point P and the origin with the positive direction of the Real Axis. Each complex number has its sense of direction or orientation on the Argand Plane as a result of this. As a result, we can represent every point on the Argand Plane in a unique way.

Complex numbers on a number line

In the same manner that real numbers may be plotted on a number line, complex numbers cannot. We can, however, depict them graphically. To represent a complex number, we must handle both of the number’s components. The argand plane graph is a coordinate system in which the x-axis represents the real component, and the vertical axis represents the imaginary component. Complex numbers are ordered pairs (a, b) of points on the plane, where ‘a’ represents the horizontal axis coordinate and ‘b’ represents the vertical axis coordinate.

Complex numbers exist, and they play an essential role in mathematics. For example, the real number line is just the real axis on the argand plane or argand diagram, yet it encompasses more. To plot numbers on the argand diagram, you can use a complex plane graph.

Stereographic projections

It’s helpful to conceive the argand diagram as if it were a sphere’s surface. Place the centre of a unit-radius sphere at the origin of the argand plane, positioned so that the sphere’s equator corresponds with the plane’s unit circle and the north pole is “above” the plane.

The points on the sphere’s surface minus the north pole and the points in the argand plane have a one-to-one connection. Draw a straight line linking a point in the plane to the sphere’s north pole. That line will meet the sphere’s surface at precisely one other point.

The sphere’s south pole will be projected using Re(z) = 0. Because the unit circle’s interior is contained within the sphere, the entire region will be mapped to the southern hemisphere. The area of the unit circle will be mapped into the northern hemisphere, minus the north pole, and the exterior of the unit circle (|z| = 1) will be mapped onto the equator. This technique is reversible: we can draw a straight line from any place on the sphere’s surface that is not the north pole to the north pole, intersecting the flat plane in precisely one spot. 

The north pole is not connected to any point in the argand plane in this stereographic projection. We complete the one-to-one correspondence by extending the argand plane by one additional point, the point at infinity, and associating it with the sphere’s north pole. The extended argand plane is a topological space that includes the argand plane and the point at infinity. When talking about sophisticated analysis, we use the term “point at infinity.” There are two infinity points (positive and negative) on the real number line, while in the extended argand plane, there is only one infinity point (the north pole).

Argand diagram polar form

Different types of coordinate systems exist. The Polar Coordinate System is one of them. It’s a collection of lines that are perpendicular to each other. The Pole is the name given to the point of origin. Any point’s position is determined by measuring the length of the line that connects it to the origin and the angle between the line and a given axis. For example, we can locate ‘P’ if we know the values of ‘θ’ and ‘r’. This is because the polar coordinates are ‘r’ and ‘θ’.

Use of the argand plane in control theory

The ‘s-plane’ is a control theory application of the argand plane. It’s used to graphically represent the equation’s roots that describe a system’s behaviour, and it’s the characteristic equation. The equation is usually represented as a polynomial in the Laplace transform parameter ‘s’ therefore the name ‘s’ plane. 

The Nyquist stability criterion is another application of the complex plane. A Nyquist plot of the open-loop magnitude and phase response as a function of frequency in the complex plane can be used to determine the stability of a closed-loop feedback system.

Conclusion

The x-axis and the y-axis are referred to as the “real axis” and the “imaginary axis,” respectively, in an argand plane, a modified version of the Cartesian plane. Because it is used in Argand diagrams, a complex plane is also referred to as the Argand plane. The argand plane or argand diagram is named after Jean-Robert Argand (1768–1822), a Paris-based amateur mathematician.