What is the polar representation of complex numbers

The polar representation of a complex number is a way of expressing the number in terms of its magnitude (or modulus) and its angle (or argument) relative to the positive real axis. If you have a complex number \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part, its polar form is given by:

\[
z = r(\cos \theta + i\sin \theta)
\]

Alternatively, this can be written using Euler's formula as:

\[
z = r e^{i\theta}
\]

### Key Components:
- **Magnitude (r):** The magnitude (or modulus) \( r \) is the distance of the complex number from the origin in the complex plane, calculated as:
  \[
  r = \sqrt{x^2 + y^2}
  \]
- **Argument (θ):** The argument \( \theta \) is the angle the complex number makes with the positive real axis, calculated as:
  \[
  \theta = \tan^{-1}\left(\frac{y}{x}\right)
  \]
  The angle \( \theta \) is typically measured in radians.

### Example:
If \( z = 3 + 4i \):
- The magnitude \( r \) would be \( \sqrt{3^2 + 4^2} = 5 \).
- The argument \( \theta \) would be \( \tan^{-1}\left(\frac{4}{3}\right) \), which is approximately \( 0.93 \) radians.

Thus, the polar form of \( z = 3 + 4i \) is:
\[
z = 5 \left(\cos(0.93) + i\sin(0.93)\right)
\]
Or equivalently:
\[
z = 5e^{i0.93}
\]

Would you like more details or have any questions?

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Here are 8 related questions:
1. How do you convert from polar to rectangular form of a complex number?
2. What is the geometric interpretation of a complex number's magnitude and argument?
3. How does Euler's formula relate to the polar form of complex numbers?
4. Can you provide an example where polar representation simplifies multiplication or division of complex numbers?
5. What is the significance of the angle \( \theta \) in the polar form?
6. How does the polar form help in understanding the roots of complex numbers?
7. What are the properties of complex numbers in polar form when raised to a power?
8. How do trigonometric identities relate to the polar form of complex numbers?

**Tip:** When converting complex numbers to polar form, always check the quadrant in which the angle lies to correctly determine the argument \( \theta \).

Explore the polar representation of complex numbers with detailed explanations and examples. Learn how to convert complex numbers into polar form using magnitude and argument calculations. Suitable for high school students and anyone interested in advanced algebra concepts.

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