Polar form is a way of representing complex numbers in terms of their distance from the origin (modulus) and the angle they make with the positive real axis (argument or phase). This form is especially useful for operations like multiplication, division, and exponentiation of complex numbers.

Polar Form of a Complex Number:

A complex number in rectangular form is written as: $z = x + iy$ where:

• $x$ is the real part,
• $y$ is the imaginary part, and
• $i$ is the imaginary unit ($i^2 = -1$).

In polar form, the same complex number is represented as: $z = r (\cos \theta + i \sin \theta)$ where:

• $r = \sqrt{x^2 + y^2}$ is the modulus (the distance from the origin),
• $\theta = \tan^{-1} \left(\frac{y}{x}\right)$ is the argument (the angle with the positive real axis).

This is often written using Euler's formula: $z = r e^{i \theta}$ where $e^{i \theta} = \cos \theta + i \sin \theta$.

Steps to Convert a Complex Number from Rectangular to Polar Form:

1. Find the modulus: $r = \sqrt{x^2 + y^2}$
2. Find the argument: $\theta = \tan^{-1} \left(\frac{y}{x}\right)$
   • Make sure the angle is in the correct quadrant based on the signs of $x$ and $y$.
3. Express the complex number as $z = r (\cos \theta + i \sin \theta)$ or $z = r e^{i \theta}$.

Example:

For the complex number $z = 3 + 4i$:

1. The modulus is: $r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
2. The argument is: $\theta = \tan^{-1} \left(\frac{4}{3}\right) \approx 0.93 \, \text{radians}$
3. Thus, the polar form is: $z = 5 \left( \cos 0.93 + i \sin 0.93 \right) \quad \text{or} \quad z = 5 e^{i 0.93}$

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Would you like more details or further explanations on how to convert between polar and rectangular forms?

Here are some related questions:

1. How do you multiply complex numbers in polar form?
2. How do you divide complex numbers in polar form?
3. How do you convert a complex number from polar to rectangular form?
4. What is Euler’s formula and how is it used with complex numbers?
5. How do you raise complex numbers to a power in polar form?

Tip: In polar form, multiplying complex numbers becomes much simpler. You just multiply their moduli and add their arguments!