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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 & Polar Representation | Mathematics for NDA

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 & Polar Representation | Mathematics for NDA

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 & Polar Representation | Mathematics for NDA . Here we shall see that this definition fits perfectly with the geometrical representation of the complex numbers.

Argand Plane & Polar Representation | Mathematics for NDA
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 =Argand Plane & Polar Representation | Mathematics for NDA

Hence we can say that OP = Argand Plane & Polar Representation | Mathematics for NDASo 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 & Polar Representation | Mathematics for NDA
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 & Polar Representation | Mathematics for NDA
Argument: 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

Q: If z = −2(1+2i)/(3 + i)  where i= \( \sqrt[]{-1} \), then the argument θ(−π < θ ≤ π) of z is:
A) Argand Plane & Polar Representation | Mathematics for NDA
B)   π/4
C)Argand Plane & Polar Representation | Mathematics for NDA
D) Argand Plane & Polar Representation | Mathematics for NDA

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) = -Argand Plane & Polar Representation | Mathematics for NDA
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 = -Argand Plane & Polar Representation | Mathematics for NDA.

The document Argand Plane & Polar Representation | Mathematics for NDA is a part of the NDA Course Mathematics for NDA.
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FAQs on Argand Plane & Polar Representation - Mathematics for NDA

1. What is the Argand plane and how is it related to complex numbers?
The Argand plane is a coordinate system formed by representing complex numbers as points in a two-dimensional plane. In this plane, the real part of a complex number is plotted on the x-axis, while the imaginary part is plotted on the y-axis. The concept of the Argand plane allows us to visualize complex numbers and perform various operations on them geometrically.
2. How is the polar representation of a complex number different from the Cartesian representation?
In the Cartesian representation, a complex number is expressed as a sum of its real and imaginary parts, usually in the form of a + bi, where a represents the real part and b represents the imaginary part. On the other hand, the polar representation of a complex number expresses it in terms of its magnitude (r) and argument (θ), usually in the form of r(cosθ + isinθ). The polar representation provides a different way to express complex numbers, where the magnitude represents the distance from the origin and the argument represents the angle with the positive real axis.
3. How can the polar form of a complex number be converted into the Cartesian form?
To convert a complex number from polar form to Cartesian form, we can use Euler's formula, which states that e^(iθ) = cosθ + isinθ. By substituting the values of r and θ from the polar representation into Euler's formula, we can derive the corresponding Cartesian form. For example, if a complex number is given in polar form as r(cosθ + isinθ), we can rewrite it as r * e^(iθ) and then calculate the real and imaginary parts using Euler's formula.
4. How can the Cartesian form of a complex number be converted into the polar form?
To convert a complex number from Cartesian form to polar form, we can use the magnitude (r) and argument (θ) formulas. The magnitude (r) can be calculated using the Pythagorean theorem as r = √(a^2 + b^2), where a and b are the real and imaginary parts, respectively. The argument (θ) can be determined using trigonometric functions as θ = arctan(b/a), taking into account the signs of a and b. Once the magnitude and argument are determined, the complex number can be expressed in polar form as r(cosθ + isinθ).
5. What are the advantages of using the polar representation of complex numbers?
The polar representation of complex numbers has several advantages. Firstly, it provides a concise and compact way to express complex numbers, especially when dealing with multiplication, division, and exponentiation operations. Secondly, the polar form makes it easier to visualize the geometric properties of complex numbers, such as their magnitude and argument, which can aid in understanding complex analysis and circuit analysis. Additionally, the polar form simplifies certain calculations, such as finding powers and roots of complex numbers.
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