Complex Number Multiplication Calculator — Multiply Complex Numbers Step by Step
Multiply two complex numbers with step-by-step explanation. Learn the FOIL method, apply i² = -1, and understand the geometric interpretation. For related tools, see our complex number addition calculator.
Table of Contents
- 1What Are Complex Numbers?
- 2The Multiplication Formula
- 3How to Multiply Complex Numbers
- 4Step-by-Step Examples
- 5More Multiplication Examples
- 6Geometric Interpretation
- 7Multiplication in Polar Form
- 8Multiplying by the Conjugate
- 9Common Mistakes to Avoid
- 10Real-World Applications
- 11Practice Problems with Solutions
- 12Tips for Mastering Complex Multiplication
- 13Final Thoughts
- 14FAQ: Common Questions
Multiply Two Complex Numbers
Enter two complex numbers in the form a + bi. Click Multiply for step-by-step working.
Complex Number Multiplication – Complete Explanation
Multiplying complex numbers extends the FOIL method to expressions containing the imaginary unit i. The result is always another complex number. Geometrically, complex multiplication represents rotation and scaling on the complex plane.
Complex Multiplication Formula
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Real part: ac - bd | Imaginary part: ad + bc.
1. What Are Complex Numbers?
A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i² = -1.
2. The Multiplication Formula Explained
(a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i
The term bdi² becomes -bd because i² = -1.
3. How to Multiply Complex Numbers
- F: First terms → ac
- O: Outer terms → adi
- I: Inner terms → bci
- L: Last terms → bdi² = -bd
- Combine: (ac - bd) + (ad + bc)i
4. Step-by-Step Examples
Example 1: (3 + 2i)(1 + 4i)
F: 3 × 1 = 3 O: 3 × 4i = 12i I: 2i × 1 = 2i L: 2i × 4i = 8i² = -8 Real: 3 + (-8) = -5 Imaginary: 12i + 2i = 14i Result: -5 + 14i
Example 2: (2 - 3i)(-1 + 5i)
F: 2 × (-1) = -2 O: 2 × 5i = 10i I: (-3i) × (-1) = 3i L: (-3i) × 5i = -15i² = +15 Real: -2 + 15 = 13 Imaginary: 10i + 3i = 13i Result: 13 + 13i
5. More Multiplication Examples
(1 + i)(1 - i) = 2 — imaginary parts cancel
(2i)(3i) = -6 — pure imaginaries product
(4 + i)(4 - i) = 17 — conjugate product
(1 + 2i)² = -3 + 4i
6. Geometric Interpretation
Geometric Rule
Moduli MULTIPLY: |z₁ × z₂| = |z₁| × |z₂| Arguments ADD: arg(z₁ × z₂) = arg(z₁) + arg(z₂)
Multiplying by i rotates 90° counterclockwise: 1 → i → -1 → -i → 1.
7. Multiplication in Polar Form
Polar Multiplication
r₁(cos Īø₁ + i sin Īø₁) × r₂(cos Īø₂ + i sin Īø₂) = r₁r₂[cos(Īø₁+Īø₂) + i sin(Īø₁+Īø₂)]
8. Multiplying by the Conjugate
Conjugate Product
(a + bi)(a - bi) = a² + b² = |z|²
9. Common Mistakes to Avoid
- Forgetting i² = -1.
- Real part is ac - bd, not ac + bd.
- Combining real and imaginary terms.
- Forgetting cross terms.
- Misidentifying pure imaginary numbers.
10. Real-World Applications
- Electrical Engineering: AC circuit analysis.
- Signal Processing: Fourier transforms.
- Quantum Mechanics: Wave functions.
- Computer Graphics: 2D rotations.
- Control Systems: Transfer functions.
11. Practice Problems with Solutions
Problem 1: (2+i)(3+2i) = 4 + 7i
Problem 2: (5-i)(2+3i) = 13 + 13i
Problem 3: (1+i)⁴ = -4
Problem 4: i³ × i⁵ = 1
12. Tips for Mastering Complex Multiplication
- Always use FOIL systematically.
- Remember i² = -1.
- Use the formula (ac-bd) + (ad+bc)i as a shortcut.
- Check geometrically: moduli multiply.
- Conjugate product gives real number.
13. Final Thoughts
Complex multiplication is a gateway to understanding algebra-geometry connections. Use this calculator to verify your work, but practice until FOIL with i² = -1 becomes automatic.
14. Frequently Asked Questions
The formula is: (a + bi)(c + di) = (ac - bd) + (ad + bc)i. The real part is ac - bd (including the -bd from i² = -1), and the imaginary part is ad + bc (the cross terms).
Because i² = -1. When you multiply the imaginary parts (bi × di = bdi²), the i² becomes -1, making this term real. The contribution -bd goes into the real part of the product. This is why imaginary × imaginary = real.
Multiplication scales and rotates. The modulus (distance from origin) of the product equals the product of the moduli: |z₁ × z₂| = |z₁| × |z₂|. The argument (angle) of the product equals the sum of the angles: arg(z₁ × z₂) = arg(z₁) + arg(z₂). Multiplying by i rotates 90° counterclockwise.
Multiplying by the conjugate always gives a real number: (a + bi)(a - bi) = a² + b² = |z|². For example, (3 + 4i)(3 - 4i) = 9 + 16 = 25. This is used in complex division to rationalize denominators.
i² = -1 is the defining property of the imaginary unit. It's crucial in complex multiplication because the "Last" term in FOIL (bi × di = bdi²) must be converted to -bd. Without this step, you'd incorrectly leave an i² term in your answer.
Yes. The product is real when the imaginary part (ad + bc) equals zero. This happens with conjugates: (a + bi)(a - bi) = a² + b². It also happens with pure imaginary numbers: (2i)(3i) = -6.
Yes. The calculator accepts pure imaginary numbers (like "5i" or "-3i") and pure real numbers (like "7" or "-2"). For example, 5i × 3i = -15, and 7 × (2 + i) = 14 + 7i.
Explore Math Topics
Continue your learning journey with our other math guides
Related Calculators
Explore these other math tools that complement complex multiplication:
- š¹ Complex Number Addition — Add complex numbers.
- š¹ Complex Number Division — Divide complex numbers.
- š¹ Complex Conjugate Calculator — Find the conjugate.
- š¹ Complex Modulus Calculator — Find |z|.
- š¹ Complex Polar Form Calculator — Convert to polar form.
- š¹ Complex Number Modulus Calculator — Calculate the modulus.
Conclusion
Complex multiplication reveals the algebra-geometry connection. Use this calculator to verify your work, but practice until FOIL with i² = -1 becomes automatic.
Readers who wish to continue practicing and reinforce what they've learned can explore the companion learning resource for guided exercises.