Modulus and Argument Calculator

Modulus and Argument is evaluated from Real Part and Imaginary Part. The calculation reports |z| Modulus, |z|^2 = a^2 + b^2 and arg in Radians.

Results

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About the Modulus and Argument Calculator

Modulus and Argument is treated here as a quantitative relation between Real Part and Imaginary Part and |z| Modulus, |z|^2 = a^2 + b^2, arg in Radians and arg in Degrees.

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 = Euclidean distance from origin in complex plane. Argument = angle from positive real axis (atan2 handles all quadrants). Conjugate: negate imaginary part. Inverse: conjugate divided by modulus squared. z x z̄ = |z|^2 is key identity.

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 = Euclidean distance from origin in complex plane. Argument = angle from positive real axis (atan2 handles all quadrants). Conjugate: negate imaginary part. Inverse: conjugate divided by modulus squared. z x z̄ = |z|^2 is key identity.

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: z = −3 + 4i (Q2 complex number)

Inputs

real_part: -3 imag_part: 4

|z| Modulus: 5. |z|^2 = a^2 + b^2: 25. arg in Radians: 2.214297 rad. arg in Degrees: 126.8699 deg. Conjugate z̄ - Real Part: -3. Conjugate z̄ - Imaginary Part: -4. Inverse 1/z - Real Part: -0.12. Inverse 1/z - Imaginary Part: -0.16. Principal Argument Range: Positive angle: 0 to 180 (Q1 or Q2)

With Real Part = -3 and Imaginary Part = 4 as the stated inputs, the result is |z| Modulus = 5, |z|^2 = a^2 + b^2 = 25 and arg in Radians = 2.214297 rad. Each value corresponds to the declared output fields.

Example 2: Unit complex number: z = cos(45°) + i×sin(45°) ≈ 0.707 + 0.707i

Inputs

real_part: 0.7071 imag_part: 0.7071

|z| Modulus: 0.99999. |z|^2 = a^2 + b^2: 0.999981. arg in Radians: 0.785398 rad. arg in Degrees: 45 deg. Conjugate z̄ - Real Part: 0.7071. Conjugate z̄ - Imaginary Part: -0.7071. Inverse 1/z - Real Part: 0.707114. Inverse 1/z - Imaginary Part: -0.707114. Principal Argument Range: Positive angle: 0 to 180 (Q1 or Q2)

With Real Part = 0.7071 and Imaginary Part = 0.7071 as the stated inputs, the result is |z| Modulus = 0.99999, |z|^2 = a^2 + b^2 = 0.999981 and arg in Radians = 0.785398 rad. Each value corresponds to the declared output fields.

Example 3: Reciprocal of impedance: z = 2 + 3i (admittance calculation)

Inputs

real_part: 2 imag_part: 3

|z| Modulus: 3.605551. |z|^2 = a^2 + b^2: 13. arg in Radians: 0.982794 rad. arg in Degrees: 56.3099 deg. Conjugate z̄ - Real Part: 2. Conjugate z̄ - Imaginary Part: -3. Inverse 1/z - Real Part: 0.153846. Inverse 1/z - Imaginary Part: -0.230769. Principal Argument Range: Positive angle: 0 to 180 (Q1 or Q2)

With Real Part = 2 and Imaginary Part = 3 as the stated inputs, the result is |z| Modulus = 3.605551, |z|^2 = a^2 + b^2 = 13 and arg in Radians = 0.982794 rad. Each value corresponds to the declared output fields.

Example 4: High-magnitude complex: z = 12 + 5i (Pythagorean pair)

Inputs

real_part: 12 imag_part: 5

|z| Modulus: 13. |z|^2 = a^2 + b^2: 169. arg in Radians: 0.394791 rad. arg in Degrees: 22.6199 deg. Conjugate z̄ - Real Part: 12. Conjugate z̄ - Imaginary Part: -5. Inverse 1/z - Real Part: 0.071006. Inverse 1/z - Imaginary Part: -0.029586. Principal Argument Range: Positive angle: 0 to 180 (Q1 or Q2)

With Real Part = 12 and Imaginary Part = 5 as the stated inputs, the result is |z| Modulus = 13, |z|^2 = a^2 + b^2 = 169 and arg in Radians = 0.394791 rad. Each value corresponds to the declared output fields.

Common Use Cases

Find modulus and argument of complex numbers for exam problems
Calculate phase angle of impedance in AC circuit analysis
Verify complex number properties like triangle inequality