Complex Division CalculatorFree online math calculator — instant results, no sign-up
Divide two complex numbers with a nonzero divisor.
About the Complex Division Calculator
The Complex Division Calculator divides a complex dividend by a nonzero complex divisor, with each number entered as separate real and imaginary parts. It reports the real and imaginary components of the quotient. With the default inputs, it evaluates (2 + 3i) / (4 − i) and returns approximately 0.2941176471 + 0.8235294118i.
- Dividend real part
- The real component a of the number being divided, defaulting to 2 and entered as a unitless value.
- Dividend imaginary part
- The real coefficient b of i in the number being divided, defaulting to 3 and entered as a unitless value.
- Divisor real part
- The real component c of the number you divide by, defaulting to 4 and entered as a unitless value.
- Divisor imaginary part
- The real coefficient d of i in the number you divide by, defaulting to −1 and entered as a unitless value.
Formula and method
For real a, b, c, and d with c² + d² > 0: (a + bi)/(c + di) = (ac + bd)/(c² + d²) + [(bc − ad)/(c² + d²)]i. Real part = [2×4 + 3×(−1)]/[4² + (−1)²] = 5/17 ≈ 0.2941176471. Imaginary part = [3×4 − 2×(−1)]/[4² + (−1)²] = 14/17 ≈ 0.8235294118.
- aDividend real part; the default is 2.
- bDividend imaginary coefficient; the default is 3.
- cDivisor real part; the default is 4.
- dDivisor imaginary coefficient; the default is −1.
- iThe imaginary unit, defined by i² = −1.
Multiply the numerator and denominator by the divisor’s complex conjugate, c − di. The denominator becomes the real quantity c² + d², while the numerator becomes (ac + bd) + (bc − ad)i. Divide each numerator component by that denominator. For the default inputs, the conjugate is 4 + i, the denominator is 17, and the numerator is 5 + 14i.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Dividend real part2
- Dividend imaginary part3
- Divisor real part4
- Divisor imaginary part-1
Results
- Real part0.2941176471
- Imaginary part0.8235294118
How to use the Complex Division Calculator
- 1Enter 2 in Dividend real part and 3 in Dividend imaginary part to represent 2 + 3i.
- 2Enter 4 in Divisor real part and −1 in Divisor imaginary part to represent 4 − i.
- 3Check that the divisor’s real and imaginary parts are not both zero.
- 4Run the calculation to divide the dividend by the divisor.
- 5Read Real part = 0.2941176471 and Imaginary part = 0.8235294118, then combine them as approximately 0.2941176471 + 0.8235294118i.
Reading your result
The Real part row gives the quotient’s component without i. The Imaginary part row gives the real coefficient multiplying i, not a separate complex number. Combine the rows as Real part + (Imaginary part)i, preserving the sign of the imaginary coefficient. For the default inputs, the exact quotient is 5/17 + (14/17)i; the displayed decimals are rounded approximations.
Limitations
The divisor cannot have both components equal to zero because division by zero is undefined. Inputs are assumed to be finite, unitless real numbers representing the components of two complex numbers. Decimal results may be rounded, and finite-precision arithmetic can lose accuracy for extremely large or small inputs, particularly when squaring the divisor components or subtracting nearly equal terms.
Common questions
Should I enter −i or −1 for the imaginary part of the divisor 4 − i?+
Enter −1. The imaginary input accepts the coefficient of i, so the default divisor uses real part 4 and imaginary part −1.
Why does complex division use the divisor’s conjugate?+
Multiplying c + di by c − di produces c² + d², a real denominator. For the default divisor 4 − i, multiplying by its conjugate 4 + i produces 17.
Can the divisor have a zero real part or a zero imaginary part?+
Yes. Either component may be zero individually; only a divisor with both components zero is invalid. The denominator c² + d² remains positive whenever at least one component is nonzero.
How can I verify the quotient for the default inputs?+
Multiply the exact quotient by the divisor: [5/17 + (14/17)i](4 − i) = 2 + 3i, which recovers the dividend. Using the displayed rounded decimals instead may introduce a small rounding difference.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.