Add, subtract, multiply, or divide two complex numbers. Enter both numbers and pick the operation.
Usage Tip
Multiply complex numbers by distributing (FOIL) and replacing i squared with −1.
i² = −1
They add and subtract component-wise; multiplication and division use i squared equals minus one.
Division requires a non-zero second number.
The modulus of the result is shown.
Complex numbers are added and subtracted by combining the real parts and the imaginary parts separately. Multiplication follows the distributive rule, using the fact that i squared equals minus one, so (a + bi)(c + di) gives (ac – bd) + (ad + bc)i.
Division is the trickiest operation. To divide, multiply the top and bottom by the conjugate of the denominator, which makes the denominator a plain real number. Dividing by zero, meaning a denominator with both real and imaginary parts equal to zero, is undefined, just as with ordinary numbers.
Take A = 3 + 2i and B = 1 – 4i. Their sum is 4 – 2i and their difference is 2 + 6i. For the product, multiply every term: 3 times 1 is 3, 3 times -4i is -12i, 2i times 1 is 2i, and 2i times -4i is -8 times i squared, which equals +8. Adding up gives 11 – 10i.
If you need to confirm a division, multiply the quotient by the divisor and see whether you get the original numerator back. Rounding in the displayed answer can leave tiny differences in the last decimal place.