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Enter your values, then click Calculate result.How this calculator helps
This complex-number calculator works with rectangular values supplied as separate real and imaginary coefficients. It supports the four basic arithmetic operations and gives a polar interpretation of the resulting value. Complex arithmetic is a distinct task from ordinary real-number calculation because products use i² = −1 and division must account for both denominator components. The page makes those components explicit instead of parsing an ambiguous expression string.
How to use it
- 1
Enter each complex number as separate real and imaginary coefficients.
- 2
Select addition, subtraction, multiplication or division.
- 3
Click Calculate result to obtain the rectangular output, modulus and principal argument.
- 4
For division, verify the second complex number is nonzero and check by multiplication.
Formula and methodology
(a+bi)(c+di) = (ac−bd)+(ad+bc)i. Division by c+di uses [(ac+bd)+(bc−ad)i]/(c²+d²). Modulus = √(real²+imaginary²); argument = atan2(imaginary,real).
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
Multiply 3+2i by 1−4i. The real term is 3×1 − 2×(−4) = 11, and the imaginary coefficient is 3×(−4) + 2×1 = −10. The result is 11−10i. Its modulus is √221, about 14.8661, and its principal argument is about −42.2737 degrees. Dividing that result by 1−4i returns 3+2i, providing an independent inverse check of the multiplication.
How to interpret your result
Read the rectangular result first and use modulus and argument as a second representation of the same number. The real and imaginary outputs can be transferred into a later arithmetic check without parsing typography. A principal angle describes direction under a branch convention, so crossing the negative real axis may produce a jump between nearly +180 and −180 degrees without a large physical change in the represented point.
For different inputs or formulas, use Basic Calculator; Matrix Multiplication Calculator; Vector Dot Cross Calculator.
Related questions this calculator covers
- complex number calculator
- complex arithmetic calculator
- complex multiplication calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| Imaginary product | i×i equals −1. |
| Conjugate product | (a+bi)(a−bi) is the real value a²+b². |
| Zero sum | a value plus its negative has modulus zero and undefined argument. |
Common mistakes to avoid
- Dropping the negative sign from i².
- Using only the real component to decide whether the denominator is zero.
- Using atan(imaginary/real) without quadrant handling.
Recompute the worked expansion term by term. Divide the result by the same nonzero second number and recover the first. Check a value times its conjugate has zero imaginary coefficient and real part equal to the squared modulus. Test real and imaginary axes to verify angle conventions, and confirm a zero result has no argument.
Authoritative reference. Reviewed on 6 October 2026 for the specific disclosed method or measurement context. The arithmetic is independently implemented. A linked reference does not certify an individual result or extend the worksheet beyond its stated scope.What can affect the result?
Separate coefficients from the imaginary unit
Enter a and b for the first number a+bi, then c and d for the second. For 3−2i, the real input is 3 and the imaginary input is −2. Do not paste the letter i, a degree symbol or an entire expression into a numeric field. The displayed result handles the sign between terms, so a negative imaginary coefficient appears with a subtraction sign rather than an awkward plus-negative notation.
Addition and subtraction preserve components
Addition combines the real components and the imaginary components separately. Subtraction changes the signs of both components of the second number before combining them. A zero real component does not mean the number is zero if its imaginary component is nonzero. These operations can also be understood as adding or subtracting two-dimensional coordinate vectors, although the complex multiplication rule adds a structure that ordinary coordinate addition alone does not describe.
Multiplication uses the defining relation
Expanding the product gives ac, adi, bci and bdi². Since i² equals −1, the last term contributes −bd to the real part. The remaining imaginary coefficient is ad+bc. This is why multiplying two imaginary coefficients does not simply produce another imaginary coefficient. Multiplication by i rotates a nonzero complex value by a quarter turn in the mathematical plane, while multiplying by a positive real number scales its magnitude.
Division requires a nonzero complex denominator
A denominator is zero only when both its real and imaginary components are zero. A purely imaginary denominator can still be valid. Multiplying numerator and denominator by the denominator’s conjugate produces the real denominator c²+d² used by this worksheet. The calculator rejects division by complex zero. Very small denominators can amplify numerical error and create very large outputs, so check the scale rather than assuming every displayed decimal remains reliable.
Modulus is nonnegative magnitude
The modulus is the Euclidean length of the result in the complex plane. It is zero exactly for the zero result and otherwise positive. Moduli multiply under complex multiplication and divide under valid division. The modulus is not the signed real component or the sum of the absolute coefficients. A number with a negative real part can have a positive modulus, just as a coordinate point left of the origin has positive distance from it.
Principal argument uses both coordinates
The argument is reported in degrees using atan2 so the sign and quadrant are preserved. An ordinary arctangent of imaginary divided by real would lose quadrant information and fail for a zero real component. The reported principal angle follows the browser atan2 branch convention between −180 and 180 degrees. Angles differing by a full turn represent the same nonzero direction. Zero has no defined argument and is explicitly labeled rather than assigned a misleading direction.
Numerical scope and other complex functions
Inputs are finite real coefficients within the disclosed numeric bounds. This calculator does not evaluate complex roots, logarithms, symbolic simplification, branch cuts for multivalued functions or arbitrary precision. The modulus and argument describe the result of the selected arithmetic operation, not a second selected operation. If a problem requires exact fractions or radicals, use the decimal outputs as a check and retain the exact algebra separately. Display precision does not increase the precision of the supplied coefficients.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
Dropping the negative sign from i². Read the rectangular result first and use modulus and argument as a second representation of the same number. The real and imaginary outputs can be transferred into a later arithmetic check without parsing typography. A principal angle describes direction under a branch convention, so crossing the negative real axis may produce a jump between nearly +180 and −180 degrees without a large physical change in the represented point. Entries remain on this device for the calculation and are not submitted to a calculation server. Browser floating-point arithmetic and displayed rounding do not establish real-world measurement certainty.
Sources and review information
Frequently asked questions
How do I enter a negative imaginary part?＋
Enter its signed coefficient, such as −4, without the letter i.
Is a purely imaginary denominator allowed?＋
Yes, if its imaginary coefficient is nonzero.
Why does multiplying i by i give a negative real value?＋
The defining relation is i² = −1.
What angle unit is reported?＋
The principal argument is displayed in degrees.
Does zero have an argument?＋
No. The output marks its argument as undefined.