Complex Number Polar Converter

Complex Number Polar Converter is evaluated from Real Part - from Rectangular Form and Imaginary Part - from a + bi. The calculation reports Modulus r = |z|, Argument θ and Argument θ.

Results

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About the Complex Number Polar Converter

Complex Number Polar Converter is treated here as a quantitative relation between Real Part - from Rectangular Form and Imaginary Part - from a + bi and Modulus r = |z|, Argument θ, Argument θ and Back to Rectangular - Real.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
Modulus = distance from origin (Pythagorean theorem). Argument = angle from positive real axis (use atan2 for correct quadrant). Polar multiplication: multiply moduli, add angles. Polar power: raise modulus to power, multiply angle.

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

Modulus = distance from origin (Pythagorean theorem). Argument = angle from positive real axis (use atan2 for correct quadrant). Polar multiplication: multiply moduli, add angles. Polar power: raise modulus to power, multiply angle.

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Classic 3-4-5 complex: 3 + 4i

Inputs

real_part: 3 imag_part: 4
Modulus r = |z|: 5. Argument θ: 53.1301 deg. Argument θ: 0.927295 rad. Back to Rectangular - Real: 3. Back to Rectangular - Imag: 4. Quadrant of Complex Plane: Q1 (0 to 90)

With Real Part - from Rectangular Form = 3 and Imaginary Part - from a + bi = 4 as the stated inputs, the result is Modulus r = |z| = 5, Argument θ = 53.1301 deg and Argument θ = 0.927295 rad. Each value corresponds to the declared output fields.

Example 2: Pure imaginary: 0 + 5i (pointing straight up)

Inputs

real_part: 0 imag_part: 5
Modulus r = |z|: 5. Argument θ: 90 deg. Argument θ: 1.570796 rad. Back to Rectangular - Real: 0. Back to Rectangular - Imag: 5. Quadrant of Complex Plane: Q2 (90 to 180)

With Real Part - from Rectangular Form = 0 and Imaginary Part - from a + bi = 5 as the stated inputs, the result is Modulus r = |z| = 5, Argument θ = 90 deg and Argument θ = 1.570796 rad. Each value corresponds to the declared output fields.

Example 3: Negative real: −7 + 0i (pointing left)

Inputs

real_part: -7 imag_part: 0
Modulus r = |z|: 7. Argument θ: 180 deg. Argument θ: 3.141593 rad. Back to Rectangular - Real: -7. Back to Rectangular - Imag: 0. Quadrant of Complex Plane: Q3 (-180 to -90)

With Real Part - from Rectangular Form = -7 and Imaginary Part - from a + bi = 0 as the stated inputs, the result is Modulus r = |z| = 7, Argument θ = 180 deg and Argument θ = 3.141593 rad. Each value corresponds to the declared output fields.

Example 4: Fourth quadrant phasor: 5 − 5i (−45°)

Inputs

real_part: 5 imag_part: -5
Modulus r = |z|: 7.071068. Argument θ: -45 deg. Argument θ: -0.785398 rad. Back to Rectangular - Real: 5. Back to Rectangular - Imag: -5. Quadrant of Complex Plane: Q4 (-90 to 0)

With Real Part - from Rectangular Form = 5 and Imaginary Part - from a + bi = -5 as the stated inputs, the result is Modulus r = |z| = 7.071068, Argument θ = -45 deg and Argument θ = -0.785398 rad. Each value corresponds to the declared output fields.

Common Use Cases

  • Convert a + bi to polar form r∠θ for multiplication/division
  • Convert polar form back to rectangular for addition/subtraction
  • Find modulus and argument for AC circuit phasor analysis