Complex Number To Polar Form Calculator

Conversion Formula:

\[ z = x + yi \Rightarrow r(\cos\theta + i\sin\theta) \] \[ \text{where } r = \sqrt{x^2 + y^2}, \theta = \arctan\left(\frac{y}{x}\right) \]

Polar Form:

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1. What is Polar Form?

The polar form represents a complex number in terms of its magnitude (distance from origin) and angle (direction from positive real axis). It's particularly useful for multiplication and division of complex numbers.

2. How Does the Conversion Work?

The conversion from rectangular (x + yi) to polar form uses these formulas:

\[ r = \sqrt{x^2 + y^2} \] \[ \theta = \arctan\left(\frac{y}{x}\right) \]

Where:

\( r \) — Magnitude (modulus) of the complex number
\( \theta \) — Angle (argument) in radians or degrees
\( x \) — Real part of the complex number
\( y \) — Imaginary part of the complex number

Note: The calculator uses atan2(y,x) which correctly determines the quadrant of the angle.

3. Importance of Polar Form

Details: Polar form simplifies complex number operations. Multiplication becomes multiplying magnitudes and adding angles, while division becomes dividing magnitudes and subtracting angles.

4. Using the Calculator

Tips: Enter the real (x) and imaginary (y) parts of your complex number. The calculator will display the polar form with magnitude and angle in degrees.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between rectangular and polar forms?
A: Rectangular form uses x + yi notation while polar form uses magnitude and angle (r∠θ or r(cosθ + i sinθ)).

Q2: How is the angle determined?
A: The angle (θ) is measured from the positive real axis, positive counterclockwise, negative clockwise.

Q3: What's the range for the angle θ?
A: Typically -180° to +180° (or -π to +π radians), with 0° along the positive real axis.

Q4: Can I convert back to rectangular form?
A: Yes, using x = r cosθ and y = r sinθ.

Q5: When is polar form most useful?
A: Particularly valuable for complex number multiplication, division, powers, and roots.