LEARN · EXPLAIN · REVISE

State the quadrant of the complex-plane point.

Read the idea, work independently, then explain what changed.

高一必修第二册(A版).pdf · 7.1 · PDF 75 / printed page 68

TOPIC 01

Concept of complex numbers

Build understanding of concept of complex numbers through definitions, contrasting cases and justified applications.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

z=-3+i

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.

Hint 2

Treat real and imaginary parts as x and y coordinates.

Worked solution

Use z=a+bi with real a,b and i²=−1; distinguish parts from terms.
Calculate or simplify this relation.
$\operatorname{Re}z<0,\quad\operatorname{Im}z>0$
Complex numbers have positions in the plane but no compatible total order like real numbers.

Second quadrant.

Checks and common pitfalls: Complex numbers have positions in the plane but no compatible total order like real numbers.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Connect geometric and algebraic forms of concept of complex numbers.
Algebraic form: Real and imaginary parts are real coefficients in a+bi.
$i^2=-1$
Complex plane: Conjugation reflects across the real axis; modulus is distance from the origin.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Real and imaginary parts are real coefficients in a+bi.
Expected reasoning: Conjugation reflects across the real axis; modulus is distance from the origin.
Expected correction: The imaginary part of a+bi is b, not bi.