Argand Plane
Argand plane is used to represent a complex number in a twodimensional 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 xaxis.
Let us learn more about the argand plane, polar representation in an argand plane, properties of the argand plane, with examples, FAQs.
1.  What Is An Argand Plane? 
2.  Polar Represent In Argand Plan 
3.  Properties Of Argand Plane 
4.  Examples On Argand Plane 
5.  Practice Questions 
6.  FAQs On Argand Plane 
What Is An Argand Plane?
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 z_{1}= x + iy, z_{2} = x  iy, z_{3} = x  iy, corresponds to the points (x, y), (x, y), (x, y) in the argand plane.
The point on the xaxis 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 yaxis of the argand plane corresponds to the complex number 0 + ib and it represents the imaginary part of the complex number.
The xaxis and the yaxis 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 Representation In 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 xaxis (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.
Properties Of 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.
Examples on Argand Plane

Example 1: Find the modulus and argument of the complex number z = \(\sqrt 3  i\) in the argand plane.
Solution:
The given complex number is z = \(\sqrt 3  i\). Comparing this with z = a + ib, we have a = \(\sqrt 3\), and b = 1.
Modulus of the complex number is z = \(\sqrt {a^2 + b^2}\) = \(\sqrt{(\sqrt 3)^2 + (1)^2}\) = \(\sqrt{3 + 1^2}\) = \(\sqrt{3 + 1}\) =\(\sqrt 4\) = 2
The argument of the complex number = θ = \(Tan^{1}\frac {b}{a}\) = \(Tan^{1}\frac {\sqrt 3}{1}\) = \(Tan^{1}\sqrt 3\) = \(Tan^{1}\sqrt 3\) = 60º
Therefore, the modulus of the complex number is 2, and the argument of the complex number is 60º.

Example 2: Represent the complex number z = \(1 + i\sqrt 3\) in polar form in the argand plane.
Solution:
The given complex number is z = \(1 + i\sqrt 3\)
The modulus of the complex number = r = \(\sqrt{1^2 + (\sqrt 3)^2}\) = \(\sqrt {1 + 3}\) = \(\sqrt 4 \) = 2
Cosθ = 1/2 = Cos60º, Sinθ = \(\frac{\sqrt 3}{2}\)=Sin60º.
The polar form is Z = r(Cosθ + iSinθ)
Z = 2(Cos60° + iSin60º).
Therefore, the polar form of the complex number is Z = 2(Cos60° + iSin60º).
FAQs on Argand Plane
What Is Argand Plane?
Argand plane is used to represent a complex number. The argand plane is similar to the coordinate plane, and the xaxis is the real part of the complex number, and the yaxis 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 xaxis represents abscissa or the real part of the complex number and the yaxis 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.
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