A reliable method
Multiply complex expressions using distribution, replace i² with −1, and collect real and imaginary parts separately. Powers of i repeat in a cycle of four.
The common trap
The imaginary part of a + bi is the real coefficient b, not the symbol bi. Read whether the prompt asks for a coefficient, a full number or a magnitude.
Check and transfer
For conjugates a + bi and a − bi, the product is a² + b². This identity is useful for checking multiplication and rationalizing a complex denominator.
Worked comparison: identify the structure
Multiply (2+3i)(4−i): real contributions are 8 and −3i²=3, so the real part is 11. The cross terms are −2i+12i=10i. Treating i² as 1 would change the real part to 5.
Transfer and explain
Multiplication by −i sends a+bi to b−ai. For 2+5i the result is 5−2i, whose imaginary coefficient is −2. This rotation is not simply negating the original imaginary coefficient.
What is the real part of (3 + 3i)(2 − i)?
- Multiply the real terms and the two imaginary terms.
- Since i² = −1, the product of 3i and −i contributes positive 3 to the real part.
- The requested value is 9. Substitute or reverse the operation to check this result.
Compare the other answer choices
Treat i squared as positive one instead of negative one.
Keep the real-real product but omit the imaginary-imaginary contribution.
Incorrectly apply the real coefficient two to the other imaginary term, using 3i times negative 2i and contributing positive six.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
CLEP College Algebra exam overviewSiyavula Grade 10 foundation reference