Complex Numbers Calculator
Add, subtract, multiply or divide two complex numbers, or convert one between rectangular (a + jb) and polar (magnitude ∠ angle) form.
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Estimate only. Not engineering or safety advice. Check the result against the applicable standard and the equipment data, and have a competent engineer review it before it is relied on. Terms of use.
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About this calculator
This page calculates result magnitude |z|, result angle, result real part and 4 more results from z₁ real part, z₁ imaginary part, operation, z₂ real part and 1 more input. Change any value or its unit and the result updates. The steps below show the working for the default values, so you can check each line against your own method.
Limits of this calculator
For impedance enter R as the real part and X as the imaginary part (X is positive for an inductor and negative for a capacitor). Multiplying two phasors multiplies the magnitudes and adds the angles; dividing divides the magnitudes and subtracts the angles. For a full AC circuit with a supply voltage use the AC Ohm's law calculator.
Cross-check: 70 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review.
Verification status: Engineering reference
Engineering reference. Formula and method documented and tested against separately coded reference calculations. Not yet Verified: the evidence listed on this page is still needed.
What this calculator checks
- The arithmetic of the published formula for the values you entered, in the units shown.
What remains unverified
- That the inputs suit your application. Take ratings, resistances and constants from the manufacturer data sheet or project specification.
Verification register entry
| Calculation author | Rahul Sharma (author) |
|---|---|
| Technical review | No independent technical review has been carried out |
| Individual professional approval | Not recorded |
| Formula reference | not recorded |
| Standard and edition | Not applicable: Pure mathematics or statistics; no electrical standard governs the method |
| Standard review state | Not applicable |
| Scope-screening note | not recorded |
| Cross-check test cases | 70 reference vectors (tests/reference_vectors.json) |
| Review date | not recorded |
| Next review date | not recorded |
| Status | Engineering reference |
Needed before this calculator can be marked Verified: reviewer name, qualification and review date; exact formula reference; formula source (textbook or standard); next review date.
Formula
Worked example
| Input | Value |
|---|---|
| z₁ real part (a) | 3 |
| z₁ imaginary part (b) | 4 |
| Operation | z₁ × z₂ (multiply) |
| z₂ real part (c) | 1 |
| z₂ imaginary part (d) | -2 |
- z₁ = 3 + j4
- z₂ = 1 − j2
- z₁ × z₂ = (ac − bd) + j(ad + bc)
- ac − bd = 3 × 1 − 4 × -2 = 11
- ad + bc = 3 × -2 + 4 × 1 = -2
- Result = 11 − j2
- |z| = √(11² + -2²) = 11.18034 θ = atan2(-2, 11) = -10.3048°
- Polar form = 11.18034 ∠-10.305°
Assumptions
- Angles are in degrees from −180° to 180°, measured from the positive real axis (atan2, so the quadrant is correct)
- j is the square root of −1; engineers write j instead of i
- The values are plain numbers: enter resistance, reactance, volts or amps in one consistent unit
For impedance enter R as the real part and X as the imaginary part (X is positive for an inductor and negative for a capacitor). Multiplying two phasors multiplies the magnitudes and adds the angles; dividing divides the magnitudes and subtracts the angles. For a full AC circuit with a supply voltage use the AC Ohm's law calculator.
Inputs and results
| Input | Unit | Allowed or shown | Note |
|---|---|---|---|
| z₁ real part (a) | (number) | any number | z₁ = a + jb. For a phasor or impedance the real part is the resistance or the in-phase component. |
| z₁ imaginary part (b) | (number) | any number | The coefficient of j. Inductive reactance is positive, capacitive reactance is negative. |
| Operation | (choice) | z₁ × z₂ (multiply); z₁ ÷ z₂ (divide); z₁ + z₂ (add); z₁ − z₂ (subtract); Convert z₁ only (rectangular to polar and back) | |
| z₂ real part (c) | (number) | any number | z₂ = c + jd. Ignored when the operation is "Convert z₁ only". |
| z₂ imaginary part (d) | (number) | any number | The coefficient of j in z₂. Ignored when converting z₁ only. |
| Result | Unit | Allowed or shown | Note |
|---|---|---|---|
| Result magnitude |z| | (number) | Shown to 6 decimal places | |
| Result angle (degrees) | (number) | Shown to 4 decimal places | |
| Result real part | (number) | Shown to 6 decimal places | |
| Result imaginary part (coefficient of j) | (number) | Shown to 6 decimal places | |
| Result angle (radians) | (number) | Shown to 6 decimal places | |
| z₁ magnitude |z₁| | (number) | Shown to 6 decimal places | |
| z₁ angle (degrees) | (number) | Shown to 4 decimal places |
How it is calculated
Each input is converted to the base unit of its quantity, the formula above is applied in the order shown in the worked example, and each result is shown in the unit you choose. Results are held at full precision and rounded only for display, except where a step selects a standard size, as the worked example shows. A value outside its allowed range is refused with a message instead of giving a result.
How the result changes with z₁ real part (a)
| z₁ real part (a) | Result magnitude |z| | Result angle | Result real part |
|---|---|---|---|
| 0.3 | 8.969392 | 22.2759 | 8.3 |
| 0.75 | 9.100137 | 15.9454 | 8.75 |
| 1.5 | 9.552487 | 6.009 | 9.5 |
| 3 | 11.18034 | -10.3048 | 11 |
| 6 | 16.124515 | -29.7449 | 14 |
| 12 | 28.284271 | -45 | 20 |
| 30 | 67.675697 | -55.8403 | 38 |
Other inputs are the same as in the worked example above.
Standards and further reading
Complex numbers, rectangular and polar form and angle conversion are standard mathematics, not the subject of a standard. Standards apply when the result feeds an electrical design, and they set the method there.
Frequently asked questions
What do I need to enter?
Enter z₁ real part (a), z₁ imaginary part (b), operation, z₂ real part (c) and z₂ imaginary part (d). Change the unit of a value with its selector and the result stays the same.
What does this calculator assume?
Angles are in degrees from −180° to 180°, measured from the positive real axis (atan2, so the quadrant is correct). j is the square root of −1; engineers write j instead of i. The values are plain numbers: enter resistance, reactance, volts or amps in one consistent unit.
Which calculators should I use next?
Common next steps are AC Ohm's Law (Phasors), Series RLC Impedance, Inductive and Capacitive Reactance and LC Resonance and Q Factor. The related links at the bottom of this page open them.
How accurate are the results?
Every figure comes from the same tested engine as the Elecmetric calculators and is meant for estimating. 70 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review. Check design values against the standard and the equipment data you work to.
Formula guides and reference
Longer explanations with worked examples and tables, written for the same topic.
- Complex Number Formulas: Polar and Rectangular: Complex number formulas for engineers: rectangular and polar form, conversion with atan2, addition
- AC Circuit Formulas: Reactance and Impedance: AC circuit formulas: RMS and peak, frequency and period, inductive and capacitive reactance
- Resonance Formulas: Frequency, Q, Bandwidth: Resonance formulas for a series or parallel LC circuit: resonant frequency, reactance at resonance
More Engineering Mathematics calculatorsAll engineering calculatorsComplex Numbers (Rectangular and Polar) in Elecmetric