The decision and its conditions
Complex division can be converted into multiplication by a conjugate over a real denominator. The conjugate product is a sum of squares because i² equals negative one. Distribute the numerator fully before collecting real and imaginary coefficients. A reciprocal reverses a multiplication relation, so its sign and scale can be checked by multiplying it by the original number. Quadratic equations with a negative real square on one side have imaginary roots; the coefficient of i is real and should not be confused with the entire root.
Worked example
What is the real part of (3+2i)/(1+2i)? Multiply numerator and denominator by 1−2i. The real denominator is 1²+2²=5. The real numerator is 3+2×2=7; the real part is 7/5.
A contrasting decision
What is the imaginary coefficient of (4+3i)/(1+i)? Multiply both parts of the fraction by 1−i. The imaginary numerator is (3−4)i and the denominator is two. The imaginary coefficient is -1/2; it does not include the symbol i.
Diagnose a competing method
The first example’s tempting answer -1/5 follows this mistake: Treat i² as positive one in the numerator product. Rationalize the whole complex denominator before separating the real component.
Transfer before revealing
What is the imaginary coefficient of 1/(4+i)? Pause and identify the original restrictions. Solution: Multiply by (4−i)/(4−i). The real denominator is 4²+1=17. The numerator coefficient of i is −1, so the answer is -1/17.
Check the boundary of the method
A reciprocal reverses a multiplication relation, so its sign and scale can be checked by multiplying it by the original number. The two cross products contribute to the imaginary numerator with opposite signs. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
What is the real part of (3+2i)/(1+2i)?
- Multiply numerator and denominator by 1−2i.
- The real denominator is 1²+2²=5.
- The real numerator is 3+2×2=7; the real part is 7/5.
Compare the other answer choices
Omit division by the conjugate product denominator.
Treat i² as positive one in the numerator product.
Use 1²−2² in the denominator instead of the sum of squares.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
Show substituted work, state the required domain or geometric condition, and refute a competing method. This extension is not automatically scored.
This draft stays on this page only. Use paper if you want to keep your working.
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 overview