Complex Number Calculator
Add, subtract, multiply or divide two complex numbers. Enter real and imaginary parts separately, then inspect magnitude, argument and an equal-scale complex-plane diagram.
YOUR CALCULATION
Result
—
Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Combine real and imaginary parts for addition and subtraction. For multiplication use i² = −1. For division, remove the imaginary denominator by the conjugate relationship; the numerical implementation uses a scaled division to reduce overflow and underflow.
(a+bi)(c+di) = (ac−bd)+(ad+bc)i; |z| = √(real²+imaginary²); argument = atan2(imaginary, real)
Worked example
For z₁ = 3+2i and z₂ = 1−4i, addition gives 4−2i, multiplication gives 11−10i, and division gives −5/17 + (14/17)i.
Assumptions and input details
Enter numeric components, not a complete a+bi expression. Division by 0+0i is rejected. Arguments use the principal atan2 direction, approximately −180° to +180°; the zero complex number has no argument. Values and calculated components use floating-point arithmetic, with the supported result magnitude limited to 1e100. The diagram uses equal x/y scales but is a visualization, not an additional precision guarantee.
References: OpenStax: complex-number operations · OpenStax: complex-plane magnitude
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
Solve ax² + bx + c = 0, including real, repeated and complex roots. Inspect the discriminant, worked steps and the function graph.
Enter two three-component vectors to calculate their sum, difference, dot product, cross product, magnitudes and angle, with an XY-projection diagram.
Enter an angle in degrees or radians to find six trigonometric functions. Explore a unit-circle diagram and sine/cosine graphs.
Reserved advertising space.