How this calculation works
Real and imaginary parts are combined separately for addition and subtraction. Multiplication uses i² = −1. Division multiplies by the denominator’s conjugate and divides by its squared magnitude. The result includes exact rational components, a conjugate, approximate magnitude and principal argument.
(a+bi)(c+di) = (ac−bd) + (ad+bc)i; i² = −1A worked example
(1 + 2i)(3 − 4i) = 11 + 2i. The squared magnitude is 125, so the magnitude is approximately 11.18034.
Good to know
Flat Cartesian coordinates with matching units. Each component is between −1,000,000 and 1,000,000; decimal inputs allow up to 6 places and integer fractions have denominators up to 1,000,000. Expressions are not parsed. Exact intermediate fractions have a size limit; reduce unusually complex inputs if that limit is reached. Lengths and angles are decimal approximations.
A component pair represents one number
Enter 1,2 to represent 1 + 2i. The first component is the real part and the second is the coefficient of i. Including the i symbol in the field is unnecessary; the input is deliberately a rational pair rather than a general expression parser.
Complex addition and subtraction combine the corresponding components. Multiplication also uses i² = −1, so the real part includes a subtraction. For (1 + 2i)(3 − 4i), the result is 11 + 2i rather than an entry-by-entry product.
Division is defined for every nonzero complex denominator. Multiplying by its conjugate makes the denominator real and positive, equal to the sum of its squared components. Division by 0 + 0i is rejected.
Magnitude and argument add a geometric view
The magnitude is the Euclidean distance from the origin in the complex plane. The argument gives the direction from the positive real axis. The result uses degrees and a principal angle from above −180° through 180°; directions differing by 360° describe the same ray.
The zero result has magnitude zero but no direction. Returning an argument of 0° for it would suggest a direction it does not possess, so the tool marks that value undefined.
The exact real and imaginary parts can be copied alongside an approximate magnitude and argument. A conjugate changes the sign of the imaginary component while preserving magnitude. Those relationships provide useful independent checks when learning the arithmetic.
A few questions, answered
Can the real or imaginary component be a fraction?
Yes. For example, 1/3,-2/5 represents 1/3 − (2/5)i. Finite decimals are also treated as exact rational inputs.
Does it calculate roots, powers or logarithms?
This page supports the four selected arithmetic operations. It does not parse arbitrary expressions or choose branches for complex roots and logarithms.
Why does the angle use a range instead of every equivalent value?
A principal argument chooses one representative direction. Adding any integer multiple of 360° gives an equivalent angle.