Argand Plane and Polar Representation

The Argand Plane

A complex number $z = x+iy$ can be represented geometrically as a unique point $P(x,y)$ in a plane called the complex plane or Argand plane. The x-axis is called the real axis, and the y-axis is called the imaginary axis.

Polar Representation

Instead of using Cartesian coordinates $(x,y)$, we can represent the point P using polar coordinates $(r, \theta)$.

• Modulus ($r$): The distance of the point P from the origin, $r = |z| = \sqrt{x^2+y^2}$.
• Argument ($\theta$): The angle that the line segment OP makes with the positive real axis, also known as the amplitude of z. $x = r\cos\theta$ and $y = r\sin\theta$.

This leads to the polar form of a complex number:

$$z = r(\cos\theta + i\sin\theta)$$

• The value of the argument $\theta$ such that $-\pi < \theta \le \pi$ is called the principal argument of z, denoted by arg(z).

Euler's Form

Using Euler's formula, $e^{i\theta} = \cos\theta + i\sin\theta$, we can express a complex number in its exponential form:

$$z = re^{i\theta}$$

This form is extremely useful for multiplication, division, and finding powers and roots of complex numbers.

Properties of Arguments

If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$:

• Multiplication: $z_1z_2 = r_1r_2[\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)]$. Thus, arg($z_1z_2$) = arg($z_1$) + arg($z_2$).
• Division: $\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1-\theta_2) + i\sin(\theta_1-\theta_2)]$. Thus, arg($z_1/z_2$) = arg($z_1$) - arg($z_2$).

Example 1: Converting to Polar Form (Quadrant I)

Question: Represent the complex number $z = 1+i\sqrt{3}$ in the polar form.

Solution: Here, $x=1$ and $y=\sqrt{3}$.

1. Find the modulus (r): $r = |z| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1+3} = \sqrt{4} = 2$.
2. Find the argument ($\theta$): $\cos\theta = x/r = 1/2$ and $\sin\theta = y/r = \sqrt{3}/2$. Since both are positive, $\theta$ is in the first quadrant. The principal argument is $\theta = \pi/3$.

$$z = 2(\cos(\pi/3) + i\sin(\pi/3))$$

Example 2: Converting to Polar Form (Quadrant III)

Question: Convert the complex number $z = -\sqrt{3}-i$ into polar form.

Solution: Here, $x=-\sqrt{3}$ and $y=-1$.

1. Find r: $r = |z| = \sqrt{(-\sqrt{3})^2 + (-1)^2} = \sqrt{3+1} = 2$.
2. Find $\theta$: $\cos\theta = -\sqrt{3}/2$ and $\sin\theta = -1/2$. Since both are negative, $\theta$ is in the third quadrant. The principal argument is $\theta = -2\pi/3$.

$$z = 2(\cos(-2\pi/3) + i\sin(-2\pi/3))$$

Example 3: Multiplication using Polar Form

Question: Multiply $z_1 = 3(\cos(\pi/4) + i\sin(\pi/4))$ by $z_2 = 2(\cos(\pi/12) + i\sin(\pi/12))$.

Solution: Using the multiplication rule, we multiply the moduli and add the arguments:

$$z_1z_2 = (3 \cdot 2) [\cos(\frac{\pi}{4} + \frac{\pi}{12}) + i\sin(\frac{\pi}{4} + \frac{\pi}{12})]$$

$$= 6[\cos(\frac{3\pi+\pi}{12}) + i\sin(\frac{3\pi+\pi}{12})] = 6(\cos(\pi/3) + i\sin(\pi/3))$$

Converting back to Cartesian form: $6(1/2 + i\sqrt{3}/2) = 3+3i\sqrt{3}$.

Example 4: Converting a pure imaginary number

Question: Convert the complex number $z = -2i$ into polar form.

Solution: Here, $x=0$ and $y=-2$. The point is $(0, -2)$ on the negative imaginary axis.

1. Find r: $r = |z| = \sqrt{0^2+(-2)^2}=2$.
2. Find $\theta$: The angle for the negative imaginary axis is $-\pi/2$.

$$z = 2(\cos(-\pi/2) + i\sin(-\pi/2))$$

Example 5: Locus based on Argument

Question: Find the locus of z if $\arg(z-1) = \pi/4$.

Solution: Let $z=x+iy$. Then $z-1 = (x-1)+iy$. The argument of this complex number is $\tan^{-1}(\frac{y}{x-1})$. We are given that this argument is $\pi/4$.

$$\tan^{-1}\left(\frac{y}{x-1}\right) = \frac{\pi}{4}$$

$$\frac{y}{x-1} = \tan(\pi/4) = 1 \implies y = x-1$$

This is the equation of a straight line. However, for the argument to be $\pi/4$, the point $(x-1, y)$ must be in the first quadrant, so $x-1>0$ (i.e., $x>1$) and $y>0$. The locus is a ray originating from the point $(1,0)$ (but not including it) and making an angle of $45^\circ$ with the positive x-axis.