How to Multiply Imaginary Numbers- Complex Number Operations
What Is the Imaginary Unit, Anyway?
The imaginary unit is i, defined by one simple rule: i² = -1. That's it. No real number squared gives you -1, so mathematicians invented i to fill that gap. If that sounds strange, you're not alone. But once you accept the definition, everything else falls into place.
Imaginary numbers take the form bi where b is a real number. So 3i, -7i, and ½i are all imaginary numbers.
Powers of i: The Cycle You Need to Memorize
When you multiply i by itself repeatedly, you get a neat four-number cycle. Here's what happens:
- i¹ = i
- i² = -1 (by definition)
- i³ = i² × i = -1 × i = -i
- i⁴ = i² × i² = -1 × -1 = 1
- i⁵ = i⁴ × i = 1 × i = i
Once you hit i⁴, the cycle restarts. This means you can simplify any power of i by finding its remainder when divided by 4.
The Quick Method
Divide the exponent by 4. Use the remainder:
- Remainder 0 → answer is 1
- Remainder 1 → answer is i
- Remainder 2 → answer is -1
- Remainder 3 → answer is -i
Example: i⁷. 7 ÷ 4 = 1 remainder 3. So i⁷ = -i.
Powers of i Reference Table
| Power | Result | Remainder when n ÷ 4 |
|---|---|---|
| i⁰ | 1 | 0 |
| i¹ | i | 1 |
| i² | -1 | 2 |
| i³ | -i | 3 |
| i⁴ | 1 | 0 |
| i⁵ | i | 1 |
| i⁶ | -1 | 2 |
| i⁷ | -i | 3 |
Multiplying Two Imaginary Numbers
Here's where students mess up. When you multiply two imaginary numbers, the result is negative.
3i × 4i = 12i² = 12 × (-1) = -12
The rule: (ai)(bi) = ab × i² = -ab
You multiply the coefficients normally, then multiply by i², which equals -1. That's why 2i × 5i = -10, not 10.
Watch the Signs
- 3i × (-2i) = -6i² = -6 × (-1) = 6
- (-3i) × (-2i) = 6i² = 6 × (-1) = -6
Two negative signs cancel out in the coefficient multiplication, but i² still kicks in and flips the final sign.
Multiplying Complex Numbers
A complex number has both a real and imaginary part: a + bi. To multiply two complex numbers, use FOIL:
(a + bi)(c + di)
- First: a × c
- Outer: a × di
- Inner: bi × c
- Last: bi × di
Then combine like terms. Remember that i² = -1.
Example: (2 + 3i)(4 - 5i)
First: 2 × 4 = 8
Outer: 2 × (-5i) = -10i
Inner: 3i × 4 = 12i
Last: 3i × (-5i) = -15i² = -15 × (-1) = 15
Combine: 8 + (-10i) + 12i + 15 = 23 + 2i
Getting Started: Step-by-Step
Here's how to multiply any complex numbers, start to finish:
- Write out both numbers in the form (a + bi)(c + di)
- Apply FOIL to get four terms
- Simplify the "Last" term — replace i² with -1
- Combine the real parts (no i)
- Combine the imaginary parts (terms with i)
- Write your answer as real + imaginary
Practice Problem
Multiply: (1 + 2i)(3 + 4i)
FOIL gives you: 3 + 4i + 6i + 8i²
Simplify: 8i² = 8 × (-1) = -8
Combine: 3 - 8 + 4i + 6i = -5 + 10i
Answer: -5 + 10i
Common Mistakes
- Forgetting i² = -1. Always replace it. Don't leave i² in your final answer.
- Dropping the negative sign when simplifying i². It's -1, not 1.
- Forgetting to combine like terms. Real parts stay with real parts. Imaginary parts combine with imaginary parts.
- Overcomplicating the cycle. For high powers of i, just divide by 4 and use the remainder.
Why This Matters
Multiplying imaginary numbers isn't abstract math theater. Complex numbers describe waveforms, electrical circuits, quantum mechanics, and signal processing. If you plan to work in engineering, physics, or computer science, you'll use this.