Complex Number Calculator
Complex numbers combine a real component and an imaginary component, written a + bi where i² = −1. This calculator returns sums, differences, products, quotients and magnitudes for two complex values.
How to use this calculator
- Enter real and imaginary components separately for A and B. A negative imaginary component represents a term such as −4i.
- Check the labels and units before applying the result to a real task.
- Change one input at a time when comparing scenarios.
Formula and method
(a+bi)(c+di)=(ac−bd)+(ad+bc)i; |a+bi|=√(a²+b²)
- (a+bi)(c+di)=(ac−bd)+(ad+bc)i; |a+bi|=√(a²+b²)
- A quotient by zero is undefined. Displayed values are rounded, so use sufficient precision when chaining several operations.
Worked examples
Worked example
Take A = 3 + 2i and B = 1 − 4i.
A+B = 4−2i; A×B = 11−10i; |A| = √13 ≈ 3.606.
A+B = 4−2i; A×B = 11−10i; |A| = √13 ≈ 3.606.
Understanding the result
Complex numbers combine a real component and an imaginary component, written a + bi where i² = −1. This calculator returns sums, differences, products, quotients and magnitudes for two complex values.
A quotient by zero is undefined. Displayed values are rounded, so use sufficient precision when chaining several operations.
Important notes
A quotient by zero is undefined. Displayed values are rounded, so use sufficient precision when chaining several operations.
Displayed values are rounded for readability; retain appropriate precision when continuing a calculation.
Frequently asked questions
What does the imaginary unit i represent?
By definition, i² = −1. A complex number has a real part and an imaginary coefficient, which allows arithmetic beyond the real number line.
Why is division by zero undefined?
No complex number multiplied by zero can produce a non-zero numerator, so a quotient with a zero denominator does not exist.
Is magnitude the same as the real part?
No. Magnitude is the non-negative distance from the origin, √(real² + imaginary²), while the real part is just one component.
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This page provides a calculation tool plus explanatory information so you can understand what the result means and when the underlying method may not fit your situation.