Add complex numbers
Type two complex numbers, such as 3 + 2i and 1 − 4i, and get their sum, with the real and imaginary part, the modulus and the argument.
Write as a + bi, for example 3 − 2i, −i or 4i.
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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
Add the real parts and the imaginary parts separately: (3 + 2i) + (1 − 4i) = (3 + 1) + (2 − 4)i = 4 − 2i. On the complex plane, that is adding two arrows head to tail.
Modulus and argument
The modulus |z| is the distance to the origin, √(a² + b²), and the argument is the angle with the positive real axis. For 4 − 2i: |z| ≈ 4.472 and the argument ≈ −26.57°.
Write numbers as a + bi, such as 3 − 2i, −i or 4i. See also subtract, multiply and divide complex numbers.
Worked examples
- (2 + 3i) + (4 − i) = 6 + 2i
- 5 + (−3i) = 5 − 3i, a real number plus a purely imaginary one
- (3 + 2i) + (3 − 2i) = 6, adding a number and its conjugate
Series impedances
In an AC circuit, a resistor adds a real part and a coil or capacitor an imaginary part to the impedance. Two components in series simply add: 3 + 4i Ω and 1 − 2i Ω give 4 + 2i Ω. The modulus √20 ≈ 4.47 Ω is the size of the impedance and the argument ≈ 26.6° is the phase shift between voltage and current.
As arrows
Each complex number is an arrow from the origin. Adding is placing the second arrow at the tip of the first, the same as adding vectors. The sum is the arrow from the start to the end. The modulus is its length, and you can convert the point to polar form with the Cartesian to polar converter.
Frequently asked questions
How do I enter a complex number?
Write it as a + bi, for example 3 + 2i, 3 − 2i, −i or 4i. A real number such as 5 also works and counts as 5 + 0i.
What is i?
The imaginary unit, defined by i² = −1. The imaginary part b of a + bi is the number in front of i, so for 3 + 2i the real part is 3 and the imaginary part is 2.
Can I add more than two complex numbers?
Add two at a time and use the result as the first number of the next addition. Addition is associative, so the order does not matter.
What is the sum of a number and its conjugate?
Always a real number, twice the real part. (3 + 2i) + (3 − 2i) = 6, because the imaginary parts cancel.