Complex Number Calculator

Complex Number is evaluated from First Number - Real Part, First Number - Imaginary Part and Second Number - Real Part. The calculation reports Sum Real Part, Sum Imaginary Part and Difference Real Part.

Results

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About the Complex Number Calculator

Complex Number is treated here as a quantitative relation between First Number - Real Part, First Number - Imaginary Part, Second Number - Real Part and Second Number - Imaginary Part and Sum Real Part, Sum Imaginary Part, Difference Real Part and Difference Imaginary Part.

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:
Addition/subtraction: combine real and imaginary parts separately. Multiplication: FOIL, using i^2 = - 1. Division: multiply by complex conjugate of denominator to eliminate imaginary denominator. Modulus = Pythagorean distance from origin.

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:

Addition/subtraction: combine real and imaginary parts separately. Multiplication: FOIL, using i^2 = - 1. Division: multiply by complex conjugate of denominator to eliminate imaginary denominator. Modulus = Pythagorean distance from origin.

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: Simple add and multiply: (3+4i) and (1+2i)

Inputs

a_real: 3 a_imag: 4 b_real: 1 b_imag: 2
Sum Real Part: 4. Sum Imaginary Part: 6. Difference Real Part: 2. Difference Imaginary Part: 2. Product Real Part: -5. Product Imaginary Part: 10. Quotient Real Part: 2.2. Quotient Imaginary Part: -0.4. |z₁| Modulus of First Number: 5. |z₂| Modulus of Second Number: 2.236068

With First Number - Real Part = 3, First Number - Imaginary Part = 4, Second Number - Real Part = 1 and Second Number - Imaginary Part = 2 as the stated inputs, the result is Sum Real Part = 4, Sum Imaginary Part = 6 and Difference Real Part = 2. Each value corresponds to the declared output fields.

Example 2: Division: (2+3i) ÷ (1−i)

Inputs

a_real: 2 a_imag: 3 b_real: 1 b_imag: -1
Sum Real Part: 3. Sum Imaginary Part: 2. Difference Real Part: 1. Difference Imaginary Part: 4. Product Real Part: 5. Product Imaginary Part: 1. Quotient Real Part: -0.5. Quotient Imaginary Part: 2.5. |z₁| Modulus of First Number: 3.605551. |z₂| Modulus of Second Number: 1.414214

With First Number - Real Part = 2, First Number - Imaginary Part = 3, Second Number - Real Part = 1 and Second Number - Imaginary Part = -1 as the stated inputs, the result is Sum Real Part = 3, Sum Imaginary Part = 2 and Difference Real Part = 1. Each value corresponds to the declared output fields.

Example 3: Electrical engineering: Z1=(50+30j) Ω parallel with Z2=(100−50j) Ω

Inputs

a_real: 50 a_imag: 30 b_real: 100 b_imag: -50
Sum Real Part: 150. Sum Imaginary Part: -20. Difference Real Part: -50. Difference Imaginary Part: 80. Product Real Part: 6,500. Product Imaginary Part: 500. Quotient Real Part: 0.28. Quotient Imaginary Part: 0.44. |z₁| Modulus of First Number: 58.309519. |z₂| Modulus of Second Number: 111.803399

With First Number - Real Part = 50, First Number - Imaginary Part = 30, Second Number - Real Part = 100 and Second Number - Imaginary Part = -50 as the stated inputs, the result is Sum Real Part = 150, Sum Imaginary Part = -20 and Difference Real Part = -50. Each value corresponds to the declared output fields.

Example 4: Complex roots: solve z² + 1 = 0 → roots i and −i as numbers

Inputs

a_real: 0 a_imag: 1 b_real: 0 b_imag: -1
Sum Real Part: 0. Sum Imaginary Part: 0. Difference Real Part: 0. Difference Imaginary Part: 2. Product Real Part: 1. Product Imaginary Part: 0. Quotient Real Part: -1. Quotient Imaginary Part: 0. |z₁| Modulus of First Number: 1. |z₂| Modulus of Second Number: 1

With First Number - Real Part = 0, First Number - Imaginary Part = 1, Second Number - Real Part = 0 and Second Number - Imaginary Part = -1 as the stated inputs, the result is Sum Real Part = 0, Sum Imaginary Part = 0 and Difference Real Part = 0. Each value corresponds to the declared output fields.

Common Use Cases

  • Add, subtract, multiply, and divide complex numbers
  • Perform complex arithmetic for electrical engineering (impedance)
  • Complex number operations for math homework or course work