The polar form of a complex number is a way to represent complex numbers using their magnitude and angle rather than their Cartesian coordinates (real and imaginary parts). This representation emphasizes the geometric properties of complex numbers in the complex plane.

Key Elements of Polar Form:

1. Magnitude (r):
   The distance from the origin (0,0) to the point representing the complex number. It is the modulus of the complex number and is given by: $r = |z| = \sqrt{x^2 + y^2}$ where $x$ is the real part and $y$ is the imaginary part.

2. Argument (θ):
   The angle (in radians or degrees) measured counterclockwise from the positive real axis to the line representing the complex number. This angle is called the argument of the complex number and is denoted as $\theta$. It can be calculated as: $\theta = \arg(z) = \tan^{-1} \left( \frac{y}{x} \right)$

3. Polar Form Representation:
   The complex number $z = x + iy$ can be expressed in polar form as: $z = r (\cos \theta + i \sin \theta)$ This is often abbreviated as: $z = r \text{cis} \theta$ Alternatively, using Euler's formula, the polar form can also be written as: $z = r e^{i \theta}$

Conversion from Cartesian to Polar:

If $z = x + iy$, you can convert it to polar form using the following steps:

1. Compute the magnitude: $r = \sqrt{x^2 + y^2}$
2. Compute the argument: $\theta = \tan^{-1} \left( \frac{y}{x} \right)$

Advantages of Polar Form:

• Multiplication and Division: In polar form, multiplication and division of complex numbers become much simpler. The magnitudes multiply/divide and the arguments add/subtract.

  • For multiplication: $z_1 \cdot z_2 = r_1 r_2 \text{cis} (\theta_1 + \theta_2)$
  • For division: $\frac{z_1}{z_2} = \frac{r_1}{r_2} \text{cis} (\theta_1 - \theta_2)$

• Exponential Powers: Raising a complex number to a power or finding roots is easier in polar form using De Moivre’s theorem.

Example:

Let’s express $z = 3 + 4i$ in polar form.

• Magnitude: $r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$
• Argument: $\theta = \tan^{-1} \left( \frac{4}{3} \right) \approx 53.13^\circ$ or $0.93$ radians

Thus, the polar form is: $z = 5 \text{cis} 53.13^\circ \quad \text{or} \quad z = 5 e^{i 0.93}$

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Would you like further details or clarification on this? Here are 5 related questions to expand your understanding:

1. How does Euler's formula relate to polar form?
2. What is De Moivre's theorem and how is it applied?
3. How do you perform addition and subtraction in polar form?
4. How is the polar form used in solving complex equations?
5. What is the geometrical interpretation of polar form in the complex plane?

Tip: When converting between forms, always check whether the angle is in degrees or radians to avoid confusion.