Complex result and interpretation
The result is shown in rectangular form with separate real and imaginary coefficients. Its magnitude is distance from the origin in the complex plane; it is not a third component.
To find real or complex roots of a quadratic equation, use the Quadratic Equation Calculator and follow its own input conventions.
The formula
Addition and subtraction combine like coefficients. Multiplication applies i² = −1. Division uses the divisor’s conjugate and squared magnitude, rejecting a zero complex divisor.
Verified worked example
Multiplying (3 + 2i) by (1 − 4i) gives 3 − 12i + 2i − 8i² = 11 − 10i. Its magnitude is √221, approximately 14.866069.
How to use this calculator
- Enter the real part and imaginary coefficient for A and B; type coefficients without the letter i.
- Choose the operation, with A as the first operand and B as the second.
- Read the rectangular result, its two coefficients, and its nonnegative magnitude.
Complex Number input conventions
A complex number has a real coordinate and an imaginary coefficient. Enter the coefficient alone, such as −4 for −4i, rather than typing a symbolic expression into the numeric box. Addition and subtraction combine matching coordinates. Multiplication uses the identity i squared equals negative one, which changes the sign of the product of imaginary coefficients.
Interpreting complex result
Division multiplies by the conjugate of the divisor and divides by its squared magnitude. This produces real and imaginary coordinates without leaving an imaginary denominator. The magnitude measures distance from the origin in the complex plane and is nonnegative. Small floating-point residues may appear for decimal inputs; this numerical tool does not simplify exact radicals or preserve rational fractions symbolically.
Assumptions & limitations
What this calculation assumes
- The imaginary unit satisfies i² = −1 and all entered coefficients are real numbers.
What to keep in mind
- Numerical rectangular-form arithmetic only; no symbolic simplification or complex functions.
Common questions
Can one part be zero?
Yes. A purely real or purely imaginary number is valid. Only a divisor with both parts zero is forbidden.
Does this compute complex powers?
No. The operation selector supports four arithmetic operations. Branch choices for complex roots, logarithms, and noninteger powers are outside its scope.
Which number is the divisor?
For A ÷ B, B is the divisor. At least one of B’s real or imaginary coefficients must be nonzero; A itself may be zero.
Sources & further reading
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