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Argand Plane
In the earlier classes, you read about the number line. It is a convenient way to represent real numbers as points on a line. Similarly, you read about the Cartesian Coordinate System. It is a set of three mutually perpendicular axes and a convenient way to represent a set of numbers (two or three) or a point in space.
Let us begin with the number line. Imagine that you are some kind of a mathematics god and you just created the real numbers. It so happened that you drew another line perpendicular to the real axis. What will this line be? It is definitely not real. Hence, it must be imaginary or the complex line.
Argand Plane and Polar Representation | Algebra - Mathematics
Thus we have a way to represent any imaginary number graphically. All we need to do is find its real part and an imaginary part. Secondly, we represent them on the two mutually perpendicular number lines. The point of intersection, as shown above, is the origin of our Plane.
The Plane so formed is known as the Argand Plane and is a convenient way to represent any imaginary number graphically. Let z = x + iy. Then Re(z) = x and Im(z) = y.
The ordered pair (x,y) represented on the Argand plane will represent a point. This point corresponds to our complex number z. We draw a directed line from O to the point P(x,y) which represents z. Let θ be the angle that this line makes with the positive direction of the “Real Axis”. Therefore, (90 – θ) is the angle which it makes with the “Imaginary Axis”. This is somewhat important, so keep it handy!
Argand Plane and Polar Representation | Algebra - Mathematics

Argument of z
As already established, every Complex number can be represented somewhere on the Argand Plane. This follows from the fact that under the operation of our Algebra, Complex numbers are closed. Imagine you represent two numbers, z1 = 2 +3i and z2 = 2 – 3i. We can see that |z1| = |z2|. Oops! What have we done? If you plot the two points (2, 3) and (2, -3), you will find they are symmetrical above and below the real axes. We call them the mirror images of each other.
How do we tell the difference between them? We introduce another quantity called the Argument of z1 and z2. It is defined as the angle ‘θ’ that the line joining the point P (representing our complex number) and the origin O, makes with the positive direction of the “Real Axes”. This gives each complex number a unique sense of a direction or orientation on the Argand Plane. Hence we can uniquely represent every point on the Argand Plane.

Modulus of A Complex Number
In an earlier section we defined the modulus of an imaginary number z = a+ib as|z|=Argand Plane and Polar Representation | Algebra - Mathematics . Here we shall see that this definition fits perfectly with the geometrical representation of the complex numbers.
Argand Plane and Polar Representation | Algebra - Mathematics
In the above figure, suppose the arrowhead is P (a, b), where P represents the number z = a + ib. Then the length of OP can be found out by using the distance formula as = (a–0)2+
Argand Plane and Polar Representation | Algebra - Mathematics
Hence we can say that OP =Argand Plane and Polar Representation | Algebra - Mathematics . So the modulus is the length of the line segment joining the point, corresponding to our complex number, with the origin of the Argand Plane. As you can see it is always positive, hence we call it the modulus. It all falls into place now, doesn’t it?

Polar Representation
We have different types of Coordinate systems. One of them is the Polar Coordinate system. It is just a set of mutually perpendicular lines. The origin is called the Pole. We measure the position of any point by measuring the length of the line that connects it to the origin and the angle the line makes with a specified axis. For example, if we know the value of φ and r we can locate P. These are the polar coordinates, r and φ.Argument
Argand Plane and Polar Representation | Algebra - Mathematics
Similarly, if we know the Argument of a complex number in the Argand Plane and the length OP, we can locate the said number. Let r = OP. We also know that OP = |z| = r ; where z = x + iy
Argand Plane and Polar Representation | Algebra - Mathematics
The coordinates of P are (x, y). In the right angled triangle we see that x = r cos(θ) and y = r sin(θ). So we can write, z = r cos(θ) + r sin(θ)  = r [cos(θ) + sin(θ)]. This, my dear friends is the Polar representation of our complex number z = x + iy with:
Arg(z) = θ and |z| = r
Now y/x =  r sin(θ)/r cos(θ) = tan θ
Therefore, θ = tan-1(y/x)
Using this relation, we can find the argument of a complex number.

Solved Examples For You
Question: If z = −2(1+2i)/(3 + i) where i= \( Argand Plane and Polar Representation | Algebra - Mathematics), then the argument θ(−π < θ ≤ π) of z is:(A)3 π/4
(B) π/4
(C)  5 π/6
(D) -3 π/4
Solution: (D) As z = −2(1+2i)/(3 + i)
Multiplying and dividing by (3 – i), we get
z = -2(1+2i)×(3 – i)/(3 + i)×(3 – i)  = -1 – i
Comparing this to z = x + iy, we have x = -1 and y = -1
Therefore, θ = tan-1(y/x) = tan-1(1) = -3 π/4
Why not π/4 ? Well because, both x and y are negative. This means that the point P is in the third quadrant now. Therefore, θ = -3 π/4.

The document Argand Plane and Polar Representation | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Argand Plane and Polar Representation - Algebra - Mathematics

1. What is the Argand Plane in mathematics?
Ans. The Argand Plane is a complex plane where complex numbers are represented. It consists of a horizontal x-axis (real axis) and a vertical y-axis (imaginary axis). The complex numbers are represented as points on this plane, with the real part as the x-coordinate and the imaginary part as the y-coordinate.
2. What is the polar representation of a complex number?
Ans. The polar representation of a complex number represents the number in terms of its magnitude (distance from the origin) and argument (angle with the positive real axis). It is usually written in the form r(cosθ + isinθ), where r is the magnitude and θ is the argument.
3. How can we convert a complex number from the Argand Plane to polar representation?
Ans. To convert a complex number from the Argand Plane to polar representation, we can use the formula: r = √(x^2 + y^2) θ = arctan(y/x) where x is the real part and y is the imaginary part of the complex number. The magnitude r is calculated using the Pythagorean theorem, and the argument θ is calculated using the arctan function.
4. What are the advantages of using polar representation of complex numbers?
Ans. The polar representation of complex numbers has several advantages: 1. It provides a more concise and intuitive representation of complex numbers. 2. It simplifies multiplication and division of complex numbers by using the properties of exponents and trigonometric functions. 3. It allows for easier visualization of complex numbers on the Argand Plane, as the magnitude and argument provide geometric information about the number. 4. It facilitates the calculation of powers and roots of complex numbers by using the properties of exponents and trigonometric functions.
5. Can we convert a complex number from polar representation back to the Argand Plane?
Ans. Yes, we can convert a complex number from polar representation back to the Argand Plane. Using the formula: x = r*cosθ y = r*sinθ where r is the magnitude and θ is the argument of the complex number, we can calculate the real part (x) and the imaginary part (y) of the number. These values can then be used to plot the complex number on the Argand Plane.
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