How to use
Use this calculator to check arithmetic with two complex numbers A = a + bi and B = c + di. Fill the real and imaginary fields for each number and choose an operation. For Conjugate A, the B fields are not used. The plot marks the two operands and R, or just A and R for conjugation.
Enter each numeric component in its own field. Use a dot or comma as the decimal separator without thousands separators; scientific notation such as 2e-4 is accepted. Do not type i into a numeric field, and do not paste a symbolic expression. This tool evaluates numeric components rather than acting as a symbolic algebra system.
Examples appear in gray and work directly with Calculate. Focusing any numeric field clears all examples together. If you leave the numeric fields without entering anything, the examples return. Once you type or paste in any field, examples will not return, even if you delete that entry. Your own values use the normal dark text. Empty or invalid inputs leave an unfinished diagram with axes and no calculated points. Valid values update the diagram. Calculate confirms the detailed result for copying; editing invalidates that confirmed result. Clear empties the inputs without restoring examples.
Method and interpretation
Addition and subtraction combine matching components. Multiplication uses i² = -1: AB = (ac - bd) + (ad + bc)i. Division multiplies by the conjugate of B, giving [(ac + bd) + (bc - ad)i]/(c² + d²). Here b is A’s imaginary component and d is B’s. The denominator is positive unless B = 0 + 0i. Conjugation changes only the imaginary sign and reflects the point across the real axis.
The plot uses equal scales on the real and imaginary axes, with the origin marked 0. Its range adjusts to the current result. A point can appear at the origin when its components are tiny relative to another point. The legend supplies numeric coordinates independently of marker placement; use those values when screen resolution cannot show the difference.
Examples
(2 + 3i)(1 - 4i) = 14 - 5i. The real part is 2 + 12 and the imaginary part is -8 + 3. Both terms involving i are combined before displaying the answer.
(3 + 2i)/(1 - i) = 0.5 + 2.5i. The conjugate of 3 + 2i is 3 - 2i, and their product is 13 + 0i. This illustrates the identity z·conj(z) = |z|².
Limits and troubleshooting
Components may be zero or have absolute values from 1e-100 through 1e100. Results use floating-point arithmetic, with up to six decimals for this mathematical task and scientific notation for very small or large values. Rounded output is not an exact symbolic answer. Use full internal precision for subsequent work rather than repeatedly copying and rounding intermediate values.
Frequently asked questions
Can the divisor have a zero real part?
Yes. 0 + 2i is a valid divisor. Only a divisor whose real and imaginary components are both zero is invalid.
Do I enter the letter i?
No. Enter the coefficient in the imaginary field; for -i use -1.
Why are A and R on top of one another?
Some operations leave the value unchanged, such as conjugating a real number. The coordinate legend distinguishes overlapping markers.
Does this calculate complex square roots?
No. It provides four arithmetic operations and conjugation for numeric components only.