Argand Plane of Complex Numbers

A complex number z = x + iy is represented as a point on the argand plane, which is analogous to a coordinate plane (x, y). The point in an argand plane can alternatively be written in polar form (r,𝚹 ), where r is the complex number’s modulus and 𝚹  is the angle formed by the line connecting the point (x, y) as well as the origin with regard to the positive x-axis.

What Does an Argand Plane Look Like?

A complex number is represented using the Argand plane. A point (x, y) in the argand plane represents a complex number of the type z = x + iy. The distance of the point (x, y) from the argand plane’s origin O is represented by the modulus of a complex number z = x + iy, which is |z| = x2+y2. The complex numbers z1=-x + iy, z2 = -x – iy, and z3 = x – iy correspond to the argand plane points (-x, y), (-x, -y), and (x, -y).

The complex number a + i0 corresponds to the point on the argand plane’s x-axis, and so it represents part of the complex number. And the imaginary part of the complex number is represented by the point on the y-axis of the argand plane that corresponds to the complex number 0 + ib.

The real axis and imaginary axis of the argand plane are recognised as the x-axis and y-axis of the coordinate axis, respectively. The points (x, y) & (x, -y) in the argand plane represent the complex number z = x + iy and its conjugate complex number z = x – iy. Furthermore, with regard to the real axis of the argand plane, these points (x, -y) are the mirror image of the point (x, y).

In the Argand Plane, Polar Representation

The coordinates of a point in the argand plane are called polar coordinates. A point P (a, b) in the argand plane is represented by the complex number Z = a+ ib, and its distance from the origin is OP, with OP = r = |z| =√( a2+b2).The modulus of complex numbers is defined as r =√( a2+b2). The line OP forms an angle with the positive direction of the argand plane’s x-axis (real axis). Here, = Tan1ba, and is referred to as the complex number’s argument.

The polar coordinates of the point are expressed as (r,) for a complex number with a modulus of ‘r’ and an argument. Furthermore, z = a + ib is a complex number.

The polar coordinates of the point are expressed as (r,) for a complex number with a modulus of ‘r’ and an argument. Furthermore, the complex number z = a + ib can be written as z = rCosϴ + irSinϴ = r (Cosϴ + iSinϴ), which is the complex number’s polar representation in the argand plane.

Argand Plane Properties

• The features of the argand plane shown below can help you comprehend it better

• The axes of the argand plane are comparable to the axes of ordinary coordinates

• The origin is the place where the real and imaginary axes of the argand plane intersect

• The argand plane’s real and imaginary axes are perpendicular to one other

• The real and imaginary axes of the argand plane divide it into four quadrants, similarly to the coordinate axis

• The distance and midpoint formulas are the same in the argand plane as they are in the coordinate axes

• Cartesian coordinates or polar coordinates are used to represent points in the argand plane

CONCLUSION

The Argand plane is similar to the XY plane or the Cartesian plane, except the x-axis is considered the real axis and the y-axis is considered the imaginary axis. As a result, the argand plane is used to graphically locate complex numbers. The argand plane as well as the polar notation of complex numbers will be discussed in depth in this article.

The distance between any two points and the origin is called the Modulus, and it is symbolised by |z| for any complex number z.

If the complex number is simply a real number, that is, its imaginary part is zero or (b = 0), it is purely positioned on the real axis on the argand plane, either towards the right or left.

This means that any point on the real axis will have the formula z = a + i0. While the real component of a wholly imaginary number is zero, or (a = 0), it is placed on the imaginary axis either upwards or downwards of the origin on the argand plane, depending on the sign of the number’s imaginary part. This implies that any point on the imaginary axis has the form z = 0 + ib.