Multiplying Complex Numbers Calculator
Multiply two complex numbers and see the product in standard form with every step shown: enter a + bi and c + di to get the FOIL expansion with the i² = −1 substitution, plus the polar form (modulus and argument) of both inputs and the result, verifying that multiplication scales the moduli and adds the angles. This page works with complex numbers like 2 + 3i. If you meant ordinary fractions like 2/3, that is a different tool: the multiply fractions calculator.
You entered: –
For coursework: the FOIL steps match what you would write by hand. Angles are in degrees, normalized to (−180°, 180°]. The “You entered” line above updates as you type, so you can check the tool read your numbers the way you meant them.
Quick Answer
For (2 + 3i)(1 − 4i), expand with FOIL: 2 − 8i + 3i − 12i². Since i² = −1, −12i² becomes +12, so the product is 14 − 5i. The polar check agrees: 3.6056 × 4.1231 ≈ 14.8666 for the modulus, and 56.31° + (−75.96°) = −19.65° for the argument. Enter your own numbers in the calculator to get the FOIL steps and the polar check for them.
Key takeaways
- Complex multiplication is ordinary binomial multiplication with one extra rule: i² = −1 turns the “Last” FOIL term into a real number.
- The compact formula is (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The calculator shows this formula filled in with your numbers.
- Geometrically, multiplication scales the moduli and adds the arguments, so multiplying by i rotates a number 90° counterclockwise.
- Angles are reported in degrees, normalized to (−180°, 180°], and the polar check spells out the modulus product and angle sum for your exact inputs.
- Quick Answer
- How to use the complex numbers calculator
- The formula, and why FOIL works here
- Worked example: (2 + 3i)(1 − 4i), checked against this exact tool
- The geometric view: multiply the sizes, add the angles
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the complex numbers calculator
Four boxes, one button, and a live echo that shows what the tool thinks you typed. Fill in the two numbers, click Calculate, and read the product in standard form, the separate real and imaginary parts, the FOIL steps and the polar check. The results stay hidden until you calculate.
- Enter the first number: its real part a in the first box and its imaginary part b in the second. 2 + 3i is pre-filled as the worked example.
- Enter the second number: real part c and imaginary part d the same way. All four boxes accept decimals and negatives; the “You entered” line above the boxes updates as you type so you can confirm the parsing.
- Read four answers: the product in standard form (large type), the real and imaginary parts separately, the FOIL expansion line by line, and the polar form check comparing moduli and arguments.
- Copy or share: the two buttons under the results copy the full working to your clipboard, or hand it to your device’s share sheet.
If a box is left blank, the calculator names exactly which part is missing, for example “Enter the imaginary part of the second number,” instead of guessing.
The formula, and why FOIL works here
Multiplying complex numbers is ordinary binomial multiplication: distribute every term of the first number over every term of the second (First, Outer, Inner, Last). The only new rule is i² = −1, which turns the “Last” term bdi² into the real number −bd. Grouping real with real and imaginary with imaginary gives (ac − bd) + (ad + bc)i. Paul’s Online Math Notes at Lamar University works through this exact distribution in its complex numbers notes, and OpenStax states the same product formula in the complex numbers chapter of Algebra and Trigonometry.
because i² = −1: (a + bi)(c + di) = ac + adi + bci + bdi² = ac + adi + bci − bd
The single most common mistake is forgetting the i² substitution or getting its sign wrong: bdi² equals −bd, not +bd. The calculator’s middle FOIL line keeps the i² term visible and labels the substitution explicitly, so you can see exactly where the sign flips.
Worked example: (2 + 3i)(1 − 4i), checked against this exact tool
The calculator is pre-filled with a = 2, b = 3, c = 1, d = −4. Running it gives the product 14 − 5i, and the FOIL panel shows the same lines you would write by hand:
= 2(1) − 8i + 3i − 12i²
= 2 − 8i + 3i + 12 (i² = −1)
= 14 − 5i
The polar panel then verifies the geometry for these exact inputs: |2+3i| = 3.6056 at 56.31°, |1−4i| = 4.1231 at −75.96°, and the product shows modulus 14.8666 ≈ 3.6056 × 4.1231 (rounded) with argument −19.65° = 56.31° + (−75.96°). The “(rounded)” label is honest: the comparison multiplies the two displayed four-decimal moduli, so it can differ in the last shown digit from the product’s own modulus.
FOIL looks trivial until the signs start biting, so the four stress cases below were each computed in this calculator to show where the arithmetic turns. One has a real part that flips negative, one is the definition i² = −1 itself (the whole product collapsing to a pure real −1), and one throws all-negative inputs at the sign rules. Type the four numbers of any row into the boxes above and every figure reproduces exactly.
| Problem | Product | Why it is interesting |
|---|---|---|
| (1 + 2i)(4 + 5i) | −6 + 13i | Real part goes negative: 4 − 10 = −6 |
| (i)(i) | −1 | The definition itself: i² = −1, pure real result |
| (−3 − 4i)(−2 − 5i) | −14 + 23i | All-negative inputs: 6 − 20 = −14, 15 + 8 = 23 |
| (1.5 − 0.5i)(2.25 + 3.75i) | 5.25 + 4.5i | Decimals work: 3.375 + 1.875 = 5.25 real |
The geometric view: multiply the sizes, add the angles
Every complex number has a polar form: a modulus r = √(a² + b²), its distance from the origin, and an argument θ, its angle. Multiplying two complex numbers multiplies their moduli and adds their arguments: r = r1 × r2, θ = θ1 + θ2. That is why multiplying by i rotates a number 90° counterclockwise, and it is the property the tool’s polar panel checks for your numbers. Khan Academy’s complex numbers unit and Math is Fun’s complex numbers page both develop this picture.
Three special products are worth memorizing, because they come up constantly:
| Multiply z by | Geometric effect | Example on (2 + 3i) |
|---|---|---|
| i | Rotate 90° counterclockwise | (2 + 3i)i = −3 + 2i |
| −1 | Rotate 180° (point through the origin) | −(2 + 3i) = −2 − 3i |
| its conjugate (a − bi) | Angles cancel: a pure real number, |z|² | (2 + 3i)(2 − 3i) = 13 |
The conjugate row is the workhorse of complex division: multiplying top and bottom by the conjugate clears the imaginary part from a denominator. Purplemath’s complex numbers module shows that application alongside the multiplication mechanics.
One more connection for students going further: the complex eigenvalue calculator and the complex eigenvector calculator both lean on complex arithmetic, and both are separate tools for the linear-algebra side of the topic.
What competing calculators miss
Complex-number multiplication tools cluster into a few patterns, and each leaves a gap:
- General complex calculators (learnmathclass, calculator-cloud): add, subtract, multiply and divide with steps and FAQs. The product is correct, but the geometry stays in the prose: they describe “scale the modulus, add the arguments” without showing the modulus product and angle sum for your numbers.
- Textbook-style pages (Pearson): correct formulas and clean worked examples with De Moivre, but the examples are fixed. You cannot check your own homework numbers.
- Software documentation (RedCrab): excellent on the distributive-law alternative and special cases, but again a fixed example, no interactive tool for your inputs.
- Solved-problem pages (mathportal, gauthmath): one-shot FOIL tables for a specific problem. Useful if your problem happens to match, useless otherwise.
- Tutorials (Sophia): good teaching of the FOIL sign flip, no calculator at all.
The gap this page fills is the combination: FOIL computed line by line for your exact inputs, the product in standard form, and the polar form of both inputs and the product with the modulus-product and angle-sum check spelled out in numbers, not just words.
Frequently asked questions
Use FOIL: multiply (a + bi)(c + di) out to ac + adi + bci + bdi², then replace i² with −1 and group the real and imaginary parts. The compact result is (ac − bd) + (ad + bc)i. For (2 + 3i)(1 − 4i): 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i.
First, Outer, Inner, Last: the four products you get when you distribute two binomials. First: a × c. Outer: a × di. Inner: bi × c. Last: bi × di. It is a memory aid for the distributive property, not a separate rule: with complex numbers the only extra step is substituting i² = −1 in the Last term.
That is the defining property of the imaginary unit i. No real number squared gives −1, so mathematicians introduced i as the square root of −1, and everything else in complex arithmetic follows from i² = −1. Forgetting the substitution, or getting its sign wrong, is the most common error in complex multiplication.
It verifies the geometry of the product for your numbers. It reports the modulus and argument of both inputs and the product, then shows that the product’s modulus equals the two input moduli multiplied and its argument equals the two input arguments added. Angles are in degrees, normalized to (−180°, 180°].
Yes. All four boxes accept decimals and negatives: (1.5 − 0.5i)(2.25 + 3.75i) = 5.25 + 4.5i. The product is displayed rounded to four decimal places, with −0 normalized to 0.
It echoes your inputs back in standard form, updating as you type, before you calculate. If it shows something different from what you meant, for example a blank box reading as 0, fix the input first. It never runs the calculation by itself.
That is correct when the imaginary parts cancel. The classic cases are i × i = −1 and a number times its conjugate: (2 + 3i)(2 − 3i) = 4 + 9 = 13, which is |2 + 3i|². The calculator shows a plain real number and an imaginary part of 0.
The calculator names exactly which part is missing, for example “Enter the imaginary part of the second number,” and shows no result. Fill in all four parts to calculate.
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References
- Paul Dawkins, Lamar University. Complex Numbers (Paul’s Online Math Notes). Accessed October 23, 2026.
- OpenStax. Complex Numbers (Algebra and Trigonometry 2e, Section 2.4). Accessed October 23, 2026.
- Khan Academy. Complex numbers. Accessed October 23, 2026.
- Math is Fun. Complex Numbers. Accessed October 23, 2026.
- Purplemath. Complex Numbers. Accessed October 23, 2026.