Complex Number Arithmetic Calculator
Add, subtract and multiply two complex numbers a + bi and c + di and see every intermediate step of the real and imaginary parts.
How it works
Sum = (a+c) + (b+d)i · Difference = (a−c) + (b−d)i · Product = (ac − bd) + (ad + bc)i
Complex numbers add and subtract component-wise: the real parts combine and the imaginary parts combine separately. Multiplication uses the distributive law together with i² = −1, which is what turns the b·d term negative and folds the two cross-terms into the imaginary part. The result of any of these operations is itself a single complex number.
Worked example
For (3 + 2i) and (1 + 4i): the sum is (3+1) + (2+4)i = 4 + 6i. The product is (3·1 − 2·4) + (3·4 + 2·1)i = (3 − 8) + (12 + 2)i = −5 + 14i.
Sources: Complex number (Wikipedia)
Frequently asked questions
Why does multiplying two complex numbers flip a sign?
When you expand (a+bi)(c+di) the term bi·di becomes bd·i², and because i² = −1 that product contributes −bd to the real part. That single substitution is the whole reason complex multiplication differs from ordinary polynomial multiplication.
How do I subtract complex numbers?
Subtract the real parts and the imaginary parts independently: (a+bi) − (c+di) = (a−c) + (b−d)i. This calculator shows the difference alongside the sum and product.
What if one number is purely real or purely imaginary?
Enter 0 for the part you do not need. A purely real number has b = 0; a purely imaginary number has a = 0. The same formulas still apply.
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This calculator is for educational purposes. Double-check important results before acting on them.