Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Concept of complex numbers

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

Revisit first: Applications of plane vectors

TOPIC 01

Concept of complex numbers

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

What you will be able to explain

  • Connect geometric and algebraic forms of concept of complex numbers.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Algebraic form

Real and imaginary parts are real coefficients in a+bi.

i2=−1i^2=-1

Complex plane

Conjugation reflects across the real axis; modulus is distance from the origin.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in concept of complex numbers changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

z=2+(1)i; |z|=2.2361; argument=0.4636 rad (undefined at zero).

z=2+(1)i; |z|=2.2361; argument=0.4636 rad (undefined at zero).

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find the imaginary part.

z=2−4iz=2-4i
  • Use the domain, units and sampling assumptions stated in the question.
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
The imaginary part is the real coefficient of i.
Worked solution

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

  2. Calculate or simplify this relation.

    Im⁡z=−4\operatorname{Im}z=-4

  3. The imaginary part is not the whole term bi.

The requested value is -4.

Checks and common pitfalls: The imaginary part is not the whole term bi.

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

02 / Standard#Worked example

Find m so the complex number is real.

z=3+(m−2)iz=3+(m-2)i
  • Use the domain, units and sampling assumptions stated in the question.
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
A complex number is real exactly when its imaginary part is zero.
Worked solution

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

  2. Calculate or simplify this relation.

    m−2=0⇒m=2m-2=0\Rightarrow m=2

  3. Its real part need not be nonzero.

The requested value is 2.

Checks and common pitfalls: Its real part need not be nonzero.

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

03 / Transfer#Worked example

Find the sum a+b when the complex number is zero.

(a−2)+(b+2)i=0(a-2)+(b+2)i=0
  • Use the domain, units and sampling assumptions stated in the question.
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
Both real and imaginary parts must vanish.
Worked solution

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

  2. Calculate or simplify this relation.

    a=2,b=−2⇒a+b=0a=2,\quad b=-2\Rightarrow a+b=0

  3. One zero part alone does not make the complex number zero.

The requested value is 0.

Checks and common pitfalls: One zero part alone does not make the complex number zero.

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

04 / Foundation#Your turn

Find the imaginary part.

z=3−5iz=3-5i
  • Use the domain, units and sampling assumptions stated in the question.
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
The imaginary part is the real coefficient of i.
Worked solution

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

  2. Calculate or simplify this relation.

    Im⁡z=−5\operatorname{Im}z=-5

  3. The imaginary part is not the whole term bi.

The requested value is -5.

Checks and common pitfalls: The imaginary part is not the whole term bi.

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

05 / Foundation#Your turn

Find m so the complex number is real.

z=3+(m−3)iz=3+(m-3)i
  • Use the domain, units and sampling assumptions stated in the question.
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
A complex number is real exactly when its imaginary part is zero.
Worked solution

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

  2. Calculate or simplify this relation.

    m−3=0⇒m=3m-3=0\Rightarrow m=3

  3. Its real part need not be nonzero.

The requested value is 3.

Checks and common pitfalls: Its real part need not be nonzero.

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

06 / Foundation#Your turn

Find the modulus.

z=9+12iz=9+12i
  • Use the domain, units and sampling assumptions stated in the question.
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
Interpret z as a point in the complex plane.
Worked solution

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

  2. Calculate or simplify this relation.

    ∣z∣=(9)2+(12)2=15|z|=\sqrt{(9)^2+(12)^2}=15

  3. Modulus is the nonnegative distance from the origin.

The requested value is 15.

Checks and common pitfalls: Modulus is the nonnegative distance from the origin.

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

07 / Foundation#Your turn

Write the conjugate.

z=3−3iz=3-3i
  • Use the domain, units and sampling assumptions stated in the question.
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
Reflect the complex-plane point across the real axis.
Worked solution

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

  2. Calculate or simplify this relation.

    z‾=3+3i\overline z=3+3i

  3. Only the sign of the imaginary term changes.

3+3i.

Checks and common pitfalls: Only the sign of the imaginary term changes.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

08 / Standard#Your turn

State the quadrant of the complex-plane point.

z=−3+iz=-3+i
  • Use the domain, units and sampling assumptions stated in the question.
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

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

  2. Calculate or simplify this relation.

    Re⁡z<0,Im⁡z>0\operatorname{Re}z<0,\quad\operatorname{Im}z>0

  3. 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
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

09 / Standard#Your turn

Solve over the complex numbers.

z2+9=0z^2+9=0
  • Use the domain, units and sampling assumptions stated in the question.
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
Use i²=−1.
Worked solution

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

  2. Calculate or simplify this relation.

    z2=−9;(±3i)2=−9z^2=-9;\quad (±3i)^2=-9

  3. The roots 3i and −3i are distinct because 3>0.

z=±3i.

Checks and common pitfalls: The roots 3i and −3i are distinct because 3>0.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

10 / Standard#Your turn

Find the sum a+b when the complex number is zero.

(a−3)+(b+2)i=0(a-3)+(b+2)i=0
  • Use the domain, units and sampling assumptions stated in the question.
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
Both real and imaginary parts must vanish.
Worked solution

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

  2. Calculate or simplify this relation.

    a=3,b=−2⇒a+b=1a=3,\quad b=-2\Rightarrow a+b=1

  3. One zero part alone does not make the complex number zero.

The requested value is 1.

Checks and common pitfalls: One zero part alone does not make the complex number zero.

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

11 / Standard#Your turn

Find the imaginary part.

z=4−6iz=4-6i
  • Use the domain, units and sampling assumptions stated in the question.
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
The imaginary part is the real coefficient of i.
Worked solution

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

  2. Calculate or simplify this relation.

    Im⁡z=−6\operatorname{Im}z=-6

  3. The imaginary part is not the whole term bi.

The requested value is -6.

Checks and common pitfalls: The imaginary part is not the whole term bi.

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

12 / Transfer#Your turn

Find m so the complex number is real.

z=3+(m−4)iz=3+(m-4)i
  • Use the domain, units and sampling assumptions stated in the question.
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
A complex number is real exactly when its imaginary part is zero.
Worked solution

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

  2. Calculate or simplify this relation.

    m−4=0⇒m=4m-4=0\Rightarrow m=4

  3. Its real part need not be nonzero.

The requested value is 4.

Checks and common pitfalls: Its real part need not be nonzero.

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

13 / Transfer#Your turn

Find the modulus.

z=12+16iz=12+16i
  • Use the domain, units and sampling assumptions stated in the question.
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
Interpret z as a point in the complex plane.
Worked solution

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

  2. Calculate or simplify this relation.

    ∣z∣=(12)2+(16)2=20|z|=\sqrt{(12)^2+(16)^2}=20

  3. Modulus is the nonnegative distance from the origin.

The requested value is 20.

Checks and common pitfalls: Modulus is the nonnegative distance from the origin.

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

Focus on one question

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for concept of complex numbers?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Connect geometric and algebraic forms of concept of complex numbers.
    • Algebraic form: Real and imaginary parts are real coefficients in a+bi.
      i2=−1i^2=-1
    • Complex plane: Conjugation reflects across the real axis; modulus is distance from the origin.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • 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.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

    No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

    Curriculum and source notes ↗