Complex Multiplication CalculatorFree online math calculator — instant results, no sign-up
Multiply two complex numbers in rectangular form.
About the Complex Multiplication Calculator
The Complex Multiplication Calculator multiplies two complex numbers in rectangular form, with separate inputs for each real and imaginary part. It returns the real and imaginary parts of their product using the rule i² = −1. With the default inputs, it multiplies (2 + 3i) by (4 − i) to give 11 + 10i.
- First real part
- The real component a of the first complex number a + bi is unitless and defaults to 2.
- First imaginary part
- The real-valued coefficient b of i in the first complex number is unitless and defaults to 3.
- Second real part
- The real component c of the second complex number c + di is unitless and defaults to 4.
- Second imaginary part
- The real-valued coefficient d of i in the second complex number is unitless and defaults to −1.
Formula and method
(a + bi)(c + di) = (ac − bd) + (ad + bc)i. Real part = ac − bd = 2 × 4 − 3 × (−1) = 11. Imaginary part = ad + bc = 2 × (−1) + 3 × 4 = 10. Product = 11 + 10i.
- aReal part of the first complex number; default 2.
- bImaginary coefficient of the first complex number; default 3.
- cReal part of the second complex number; default 4.
- dImaginary coefficient of the second complex number; default −1.
- iImaginary unit, defined by i² = −1.
Distribute the two factors to obtain ac + adi + bci + bdi². Replace i² with −1, then collect the real terms as ac − bd and the imaginary coefficients as ad + bc. For the defaults, the real calculation is 8 − (−3) = 11 and the imaginary calculation is −2 + 12 = 10.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- First real part2
- First imaginary part3
- Second real part4
- Second imaginary part-1
Results
- Real part11
- Imaginary part10
How to use the Complex Multiplication Calculator
- 1Enter 2 in First real part and 3 in First imaginary part to represent 2 + 3i.
- 2Enter 4 in Second real part and −1 in Second imaginary part to represent 4 − i.
- 3Run the calculation to multiply the two complex numbers.
- 4Read the Real part row as 11 and the Imaginary part row as 10.
- 5Combine the output rows in rectangular form to obtain 11 + 10i.
Reading your result
The Real part row gives the component of the product without i: 11 for the default inputs. The Imaginary part row gives the real-valued coefficient of i: 10, meaning the imaginary term is 10i. Together, the rows describe the product 11 + 10i, not two separate products.
Limitations
This tool assumes numeric real-valued components entered in rectangular form; each imaginary input is a coefficient, not an expression containing i. Polar magnitude-and-angle inputs must be converted to rectangular form first. The calculation does not track physical units. Decimal results may be affected by numerical precision or display rounding, particularly for very large or very small inputs.
Common questions
Should I enter −1 or −i for the second imaginary part in the default example?+
Enter −1. The field takes only the coefficient of i, so a second real part of 4 and imaginary part of −1 represent 4 − i.
Why is the default real part 11 rather than 5?+
The imaginary terms multiply to (3i)(−i) = −3i² = 3. Adding this to the real-term product 2 × 4 = 8 gives 11.
How does the calculator get an imaginary part of 10?+
It adds the cross terms: 2 × (−i) + (3i) × 4 = −2i + 12i = 10i. The output row reports the coefficient 10.
Does swapping the two complex numbers change the product?+
No. Complex multiplication is commutative, so (4 − i)(2 + 3i) also equals 11 + 10i.
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.