Subtract complex numbers
Type two complex numbers and get the difference, 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
Subtract the real parts and the imaginary parts separately: (3 + 2i) − (1 − 4i) = (3 − 1) + (2 + 4)i = 2 + 6i. Watch the signs: subtracting −4i adds 4i.
Write the numbers as a + bi, such as 3 − 2i, −i or 4i. See also add complex numbers.
Worked examples
- (5 + i) − (2 + 3i) = 3 − 2i
- (3 + 2i) − (3 + 2i) = 0, any number minus itself
- 10 − (4 + 3i) = 6 − 3i
The distance between two points
On the complex plane, the distance between z₁ and z₂ is |z₁ − z₂|. The points 1 + 2i and 4 + 6i are 3 across and 4 up from each other, so the distance is √(9 + 16) = 5. This also gives the equation of a circle: |z − (2 + i)| = 3 is every point 3 away from 2 + i.
From difference to polar form
The modulus and argument of the result are shown, so you can read direction and distance straight away. To combine numbers instead, use add complex numbers; to see the polar form of any point, the Cartesian to polar converter.
Frequently asked questions
Why do signs cause so many mistakes?
Because subtracting −4i adds 4i. Put the second number in brackets and change both signs: (3 + 2i) − (1 − 4i) = 3 + 2i − 1 + 4i = 2 + 6i.
How do I find the distance between two complex numbers?
Subtract them and take the modulus of the difference. For 1 + 2i and 4 + 6i the difference is −3 − 4i, and |−3 − 4i| = 5 is the distance.
Is z₁ − z₂ the same as z₂ − z₁?
No. Swapping them flips both signs, so the modulus stays the same but the argument turns by 180°.
How do I subtract a complex number from a real number?
Enter the real number as the first number. 10 − (4 + 3i) = 6 − 3i, because the imaginary part has nothing to subtract from and just changes sign.