Divide complex numbers
Type two complex numbers and get the quotient, with the real and imaginary part, the modulus and the argument.
Write as a + bi, for example 3 − 2i, −i or 4i.
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
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Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
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Instant answer
The answer appears as you type, with the most important intermediate values.
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See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
How it works
Multiply the numerator and the denominator by the complex conjugate of the denominator (change the sign of its imaginary part). The denominator then becomes a real number:
(3 + 2i) / (1 − 4i) = (3 + 2i)(1 + 4i) / ((1 − 4i)(1 + 4i)) = (−5 + 14i) / 17 ≈ −0.294 + 0.824i.
When dividing, the moduli divide and the arguments subtract. Dividing by 0 is not possible. See also multiply complex numbers.
Another example
(4 + 2i) ÷ (1 + i): multiply top and bottom by 1 − i. The top becomes (4 + 2i)(1 − i) = 4 − 4i + 2i − 2i² = 6 − 2i, the bottom (1 + i)(1 − i) = 2. The result is 3 − i. Check by multiplying back: (3 − i)(1 + i) = 3 + 3i − i − i² = 4 + 2i.
Dividing in polar form
Because moduli divide and arguments subtract, polar form is quicker by hand: a number with modulus 6 and angle 90° divided by one with modulus 3 and angle 60° has modulus 2 and angle 30°. Convert with the Cartesian to polar converter if your numbers are in a + bi form.
Check your answer
Multiply the quotient by the divisor and you must get the dividend back. The multiply complex numbers calculator does that in one step, which catches sign slips in the conjugate.
Frequently asked questions
Why multiply by the conjugate?
Because z · z̄ = |z|² is a real number. Multiplying top and bottom by the conjugate of the denominator makes the denominator real, so you can split the result into a real and an imaginary part.
What is 1 divided by i?
−i. Multiply by the conjugate: 1 ÷ i = (−i) ÷ (i · −i) = −i ÷ 1 = −i. Check: i × (−i) = −i² = 1.
Does the result come as a fraction or a decimal?
As decimals, for example −0.2941 + 0.8235i, which is exactly (−5 + 14i) ÷ 17. The calculator also shows the modulus and argument.
What if the divisor is 0?
Division by 0 + 0i is not possible, and the calculator tells you so instead of showing a result.