Do arithmetic with a + bi and see the result in rectangular, polar and exponential form.
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Complex numbers a + bi extend the real numbers with i = √−1. They solve every polynomial equation, describe rotations and oscillations, and are essential in electrical engineering (AC circuits), signal processing and quantum physics.
Type numbers like 3+4i, -2i or (1+i)/2 and choose an operation. The result comes with its modulus |z|, argument θ, polar form r(cos θ + i sin θ), exponential form re^(iθ) and conjugate. Powers and roots use De Moivre’s theorem.
| First number z₁ | 3+4i |
| Operation | z₁ × z₂ |
| Second number z₂ | 1-2i |
| Result | (3 + 4i) × (1 − 2i) = 11 − 2i |
| Modulus |z| | 5√5 ≈ 11.18033989 |
| Argument (angle) | −0.1798535 rad = −10.304846° |
| Polar form | 11.18034(cos −10.3048° + i sin −10.3048°) |
| Exponential form | 11.18034 e^(−0.1798535i) |
| Conjugate | 11 + 2i |
(3 + 4i) × (1 − 2i) = 11 − 2i.
| First number z₁ | 3+4i |
| Operation | √z₁ (square root) |
| Result | √(3 + 4i) = 2 + i |
| Modulus |z| | √5 ≈ 2.236067977 |
| Argument (angle) | 0.46364761 rad = 26.565051° |
| Polar form | 2.236068(cos 26.5651° + i sin 26.5651°) |
| Exponential form | 2.236068 e^(0.46364761i) |
| Conjugate | 2 − i |
√(3 + 4i) = 2 + i.
−1. That is the definition of i, and it is how multiplication of complex numbers works.
Multiply the top and bottom by the conjugate of the bottom, which makes the denominator a real number: (1 + i)/(1 − i) = (1 + i)²/2 = i.
5, because √(3² + 4²) = 5. It is the distance from 0 on the complex plane.