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Complex Arithmetic Calculator

Multiply and divide two complex numbers in rectangular form. Each number uses a real coefficient and an imaginary coefficient, with i²=−1. The quotient always means the first number divided by the second. Read each pair of outputs as real + imaginary×i.

Enter every field in the stated units. Decimal and scientific notation are accepted for non-integer fields; blanks, commas, fractions and non-finite entries are rejected. Zero is accepted only where a field permits it.

Product real part5
Product imaginary part-1
Quotient real part0.5
Quotient imaginary part2.5

How to use this calculator

  1. Identify the quantities required by the model and use the field units.
  2. Replace the example values, then select Calculate or press Enter.
  3. Read the labeled outputs and check the interpretation below.

Formula and interpretation

(a+bi)(c+di) = (ac−bd)+(ad+bc)i; quotient = [(ac+bd)+(bc−ad)i]/(c²+d²).

Calculations run in your browser. The previous results are hidden when an input changes or validation fails. This prevents an old answer from being mistaken for the new calculation. Results are displayed to approximately ten significant digits using floating-point arithmetic; that precision does not imply equal precision in measured inputs.

Worked example

First real part a = 3; First imaginary part b = 2; Second real part c = 1; Second imaginary part d = -1. Product real part = 5; Product imaginary part = -1; Quotient real part = 0.5; Quotient imaginary part = 2.5.

Substitute the example inputs into the stated formula in the same order as the labeled fields. These are the default values shown above, so the explanation and the working interface describe the same case.

Limitations

Uses finite real-valued coefficients and floating-point arithmetic. The second number cannot be zero because this page always calculates the quotient as well as the product. It does not return polar phase or solve complex equations.

This is an educational mathematical model. Check the assumptions and units for your situation. Geometry results do not establish material strength, and probability outputs do not predict an individual outcome.

Frequently asked questions

How do I enter a purely imaginary number?

Set its real part to zero and enter the coefficient of i in the imaginary field.

Why does the calculator reject my entry?

Each field shows its supported range. Integer fields reject fractions; mathematical conditions such as a zero divisor or invalid shape produce a separate explanation. Correct the indicated field or condition and calculate again.

How many digits should I use?

Use enough digits to retain your input accuracy. Round the final answer to the precision justified by the data; tiny residuals around theoretical zero can result from floating-point arithmetic.

Review notes and reference

Reviewed October 3, 2026. The calculator script was tested against the default example and separate boundary cases. Checks covered blank, text, comma-formatted, non-finite and out-of-range entries, plus recovery after errors. Background reference for the mathematical method. Explanations and worked examples are original.

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