Everyday Tools

Complex Number Multiplication Calculator

Multiply two complex numbers in standard a + bi form.

Updated April 17, 2026
Free online tool
Planning and research use

Why this page exists

Complex multiplication is easier to check when the real and imaginary pieces are expanded cleanly instead of being FOILed by hand each time. This calculator helps users multiply two complex numbers in standard form and clearly shows the resulting complex number with the original inputs used.

Interactive tool

Run the estimate

Enter your numbers and read the result first, then use the sections below to understand what affects the outcome.

Calculator

Complex number multiplication calculator

Multiply two complex numbers in a + bi form using the standard complex-product rule.

Result

11 - 10i

Calculated complex-number multiplication with the standard (a + bi)(c + di) rule.

Resulting complex number11 - 10i

First complex number used3 + 2i

Second complex number used1 - 4i

3 + 2i times 1 - 4i gives 11 - 10i after combining the real and imaginary parts with the standard complex-product formula.
Complex multiplication mixes both numbers together, so the real result comes from ac - bd and the imaginary result comes from ad + bc.
Use the result as a quick check when complex-number multiplication is one step inside a longer algebra or engineering problem.

This is standard complex-number multiplication for numbers already written in real-and-imaginary-part form.

Planning note

Last updated April 17, 2026. Use this tool to compare scenarios and plan ahead, then confirm important details with the lender, employer, insurer, contractor, or other qualified provider involved in the final decision.

How it works

What the calculator is doing

Enter the real and imaginary parts of both complex numbers.

The calculator applies the standard complex multiplication rule (a + bi)(c + di) = (ac - bd) + (ad + bc)i.

It shows the resulting complex number in standard a + bi form.

Understanding your result

This is standard complex-number multiplication. The final real part comes from ac - bd and the final imaginary part comes from ad + bc.

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Examples

Ways people use this tool

Example scenarios help turn a quick estimate into a more useful comparison or planning step.

Check a FOIL-style complex multiplication step

A quick product can help verify that the real and imaginary pieces were combined correctly.

See how signs change the result

Negative imaginary parts can shift both the real and imaginary outcomes, so a direct calculator check can prevent common mistakes.

Use it with related complex tools

Complex multiplication becomes more useful when reviewed beside addition, subtraction, and modulus tools.

When to use it

Good times to run this calculator

Use this when you want a quick product for two complex numbers already written in standard form.

It is especially useful for algebra, electrical, and engineering-style problems where complex multiplication appears repeatedly.

Assumptions and limitations

The calculator assumes both complex numbers are entered in rectangular form using real and imaginary parts.

It handles multiplication only and does not convert the result into polar or exponential form.

Common mistakes

Avoid the usual input mistakes

Forgetting that i squared becomes -1 is one of the most common reasons hand-worked complex products go wrong.

Mixing the cross terms can change the imaginary part quickly even when the real part looks correct.

Practical tips

Use the calculator as a FOIL checker when complex multiplication appears inside a longer expression.

Pair the result with subtraction, addition, or modulus tools if the complex product is only one stage of a larger workflow.

Worked example

Walk through a realistic scenario

A worked example shows how the estimate behaves when the inputs resemble a real planning decision.

Multiply two complex numbers in standard form

A student wants to multiply 3 + 2i by 1 - 4i and check the resulting real and imaginary parts.

1. Enter the real and imaginary parts for both numbers.

2. Apply the standard complex multiplication rule.

3. Read the final result in a + bi form.

Takeaway: The result provides a clean complex-product check without rebuilding every term manually.

FAQ

Common questions

How are complex numbers multiplied here?

The calculator uses the standard formula (a + bi)(c + di) = (ac - bd) + (ad + bc)i.

Why does the real part use subtraction?

Because multiplying the two imaginary terms brings in i squared, and i squared equals -1.

Why keep the result in a + bi form?

Standard form keeps the real and imaginary parts clear and makes the result easier to reuse in later algebra steps.

Related tools

Keep comparing

Addition, subtraction, modulus, and logarithm tools help connect complex multiplication to the broader algebra and engineering workflow.

Square-root and quadratic-formula tools add context when complex products appear inside longer symbolic or equation-solving problems.

Everyday ToolsUpdated April 17, 2026