This calculator performs arithmetic on complex numbers written in standard a + bi form. Enter two complex numbers, pick add, subtract, multiply or divide, and read the result, along with its modulus (distance from the origin) and argument (angle from the real axis). It is built for engineering, signal-processing and maths coursework where complex arithmetic is routine.
How it works
Each operation uses the standard complex-number rules:
- Add / subtract: combine real and imaginary parts separately —
(a + bi) ± (c + di) = (a ± c) + (b ± d)i. - Multiply:
(a + bi)(c + di) = (ac − bd) + (ad + bc)i. - Divide: multiply top and bottom by the conjugate of the divisor —
((ac + bd) + (bc − ad)i) ÷ (c² + d²).
For the result the tool also reports the modulus |z| = √(re² + im²) and the argument atan2(im, re), converted to degrees.
Example
Multiply (3 + 2i) by (1 − 4i):
real = 3·1 − 2·(−4) = 3 + 8 = 11 imaginary = 3·(−4) + 2·1 = −12 + 2 = −10
The result is 11 − 10i, with modulus √(121 + 100) = √221 ≈ 14.866 and argument ≈ −42.27°. Everything runs locally in your browser.
What the modulus and argument tell you
Beyond getting the result in a + bi form, the modulus and argument give you the polar form of the same number — which matters in several practical contexts.
Modulus is the magnitude of the complex number: its distance from the origin on the complex plane. In electrical engineering, where complex numbers represent impedances and phasors, the modulus is the amplitude — the peak value of the signal. When you multiply two complex numbers their moduli multiply; when you divide, the moduli divide.
Argument (also called the phase or angle) measures how far the number is rotated counterclockwise from the positive real axis, expressed in degrees here (or equivalently in radians as atan2(im, re)). In signal processing, the argument represents the phase shift of a sinusoidal signal. Adding two arguments corresponds to multiplying two phasors — which is why the polar form makes certain computations far simpler than the Cartesian form.
The polar-to-Cartesian relationship is: a = |z| · cos(θ), b = |z| · sin(θ).
Division walk-through
Division trips people up because you cannot simply divide real and imaginary parts separately. The method is to multiply numerator and denominator by the conjugate of the denominator:
For (a + bi) / (c + di), the conjugate of the denominator is (c − di). Multiplying:
numerator: (a + bi)(c − di) = (ac + bd) + (bc − ad)i
denominator: (c + di)(c − di) = c² + d² (always real)The denominator becomes a real number, so you can safely split the result into real and imaginary parts: (ac + bd)/(c² + d²) + (bc − ad)/(c² + d²) · i.
For example, (2 + 3i) / (1 − 2i):
- numerator real:
2·1 + 3·(−2) = 2 − 6 = −4… wait, the conjugate is(1 + 2i):- numerator real:
2·1 − 3·2 = 2 − 6 = −4 - numerator imaginary:
3·1 + 2·2 = 3 + 4 = 7 - denominator:
1² + 2² = 5
- numerator real:
- result:
−4/5 + 7/5 · i = −0.8 + 1.4i
The calculator does this automatically — enter the numbers and read the result.