Argand Plane

Argand plane is used to represent a complex number in a two-dimensional plane. The argand plane is similar to a coordinate plane, and a complex number z = x + iy is plotted as a point (x, y). The point in an argand plane can also be represented in polar form (r, θ), where r is the modulus of the complex number, and θ is the angle made the line, connecting the point (x, y) and the origin, with respect to the positive x-axis.

Let us learn more about the argand plane, polar representation in an argand plane, properties of the argand plane, with examples, FAQs.

Argand plane is used to represent a complex number. A complex number of the form z = x + iy is represented as a point (x, y) in the argand plane. The modulus of a complex number z = x + iy is |z| = \(\sqrt{x^2 + y^2}\), and it represents the distance of the point (x, y) from the origin O, of the argand plane. The complex numbers z₁= -x + iy, z₂ = -x - iy, z₃ = x - iy, corresponds to the points (-x, y), (-x, -y), (x, -y) in the argand plane.

The point on the x-axis of the argand plane corresponds to the complex number a + i0, and it represents the real part of the complex number. And the point on the y-axis of the argand plane corresponds to the complex number 0 + ib and it represents the imaginary part of the complex number.

The x-axis and the y-axis of the coordinate axis are identified as the real axis and the imaginary axis of the argand plane. The complex number z = x + iy, and it's conjugate complex number z = x - iy is represented as the points (x, y) and (x, -y) in the argand plane. Further, these points (x, -y) is the mirror image of the point (x, y), with respect to the real axis of the argand plane.

Polar coordinates are the coordinates of a point represented in the argand plane. The complex number Z = a+ ib represents a point P(a, b) in the argand plane, and the distance of this point P from the origin is OP, and OP = r = |z| = \(\sqrt {a^2 +b^2}\). Here r = \(\sqrt{a^2 + b^2}\) is called the modulus of the complex number. The line OP makes an angle θ with the positive direction of the x-axis (real axis) of the argand plane. Here θ = \(Tan^{-1}\frac {b}{a}\), and θ is called the argument of the complex number.

The complex number having a modulus of 'r', and an argument θ, is represented as (r, θ) and this representation is called the polar coordinates of the point. Further the complex number z = a + ib is also represented as z = rCosθ + irSinθ = r(Cosθ + iSunθ), which is the polar representation of the complex number in the argand plane.

The following properties of the argand plane help in a better understanding of the argand plane.

• The argrand plane has axes similar to the regular coordinate axes.
• The point of intersection of the real and imaginary axis of the argand plane is the origin.
• The real and the imaginary axis of the argand plane are perpendicular to each other.
• Similar to the coordinate axis, the real and the imaginary axis of the argand plane divides it into four quadrants.
• The formulas of distance and midpoint are the same in the argand plane, as in the coordinate axes.
• The points in the argand plane are either represented as cartesian coordinates or polar coordinates.

Related Topics on Argand Plane:

Please check the following links to help us easily learn the argand plane.

• Cartesian Coordinate System
• Coordinate Plane
• Polar Coordinates
• X and Y Coordinates

What Is Argand Plane?

Argand plane is used to represent a complex number. The argand plane is similar to the coordinate plane, and the x-axis is the real part of the complex number, and the y-axis represents the imaginary part of the complex number. The complex number z = x + iy is represented as the point (x, y) and it can also be represented in the polar form with its polar coordinates.

How Do You Represent A Point In A Argand Plane?

The complex number z = a + ib is represented as a point P(a, b) in the argand plane. The points are plotted in the argand plane, similar to the points in the coordinate plane. The x-axis represents abscissa or the real part of the complex number and the y-axis represents the ordinate or the imaginary part of the complex number.

What Are The Polar Coordinates In An Argand Plane?

The complex number z = x + iy can also be written in the argand plane using polar coordinates as (r, θ). Here r is called the module of the complex number, which is equal to r = \(\sqrt {x^2 + y^2}\). And θ is called the argument of the complex number, which is equal to θ = \(Tan^{-1}\frac {b}{a}\).

What Is The Use Of Argand Plane?

The argand plane is primarily used to represent complex numbers in a geometric format. The complex number z = x + iy can be represented as a point (x, y), with the x coordinate representing the real part of the complex number and the y coordinate representing the imaginary part of the complex number.

How To Find the Argument In An Argand Plane?

The argument of the complex number in an argrand plane is θ = \(Tan^{-1}\frac {b}{a}\). It is an angle value, and is equal to the inverse of the tangent of the imaginary part, divided by the real part of the complex number.