Kenneth ChengMathematical modeling
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Arithmetic of complex numbers

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

高一必修 第二册(A版).pdf · 7.2 · PDF 82 / printed page 75

Revisit first: Concept of complex numbers

TOPIC 01

Arithmetic of complex numbers

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

What you will be able to explain

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

Multiplication

Expand algebraically and replace i² with −1.

Division

A conjugate turns a nonzero complex denominator into its positive squared modulus.

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 arithmetic 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 real part of the product.

(2+i)(2−i)(2+i)(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
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
The product i(−i) equals +1.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    (2+i)(2−i)=5+0i(2+i)(2-i)=5+0i

  3. Separate the real terms from the coefficient of i.

The requested value is 5.

Checks and common pitfalls: Separate the real terms from the coefficient of i.

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

02 / Standard#Worked example

Simplify the quotient.

2+2i1+i\frac{2+2i}{1+i}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
Factor the numerator.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    2(1+i)1+i=2\frac{2(1+i)}{1+i}=2

  3. The factor 1+i is nonzero, so cancellation is legal.

2.

Checks and common pitfalls: The factor 1+i is nonzero, so cancellation is legal.

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.

03 / Transfer#Worked example

Simplify the sum and explain the cancellation.

1+i+i2+i31+i+i^2+i^3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
Replace i² and i³ by their elementary values.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    1+i−1−i=01+i-1-i=0

  3. The four points cancel in opposite pairs on the unit circle.

The requested value is 0.

Checks and common pitfalls: The four points cancel in opposite pairs on the unit circle.

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

04 / Foundation#Your turn

Find the real part of the product.

(3+i)(2−i)(3+i)(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
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
The product i(−i) equals +1.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    (3+i)(2−i)=7+−1i(3+i)(2-i)=7+-1i

  3. Separate the real terms from the coefficient of i.

The requested value is 7.

Checks and common pitfalls: Separate the real terms from the coefficient of i.

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

05 / Foundation#Your turn

Simplify the quotient.

3+3i1+i\frac{3+3i}{1+i}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
Factor the numerator.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    3(1+i)1+i=3\frac{3(1+i)}{1+i}=3

  3. The factor 1+i is nonzero, so cancellation is legal.

3.

Checks and common pitfalls: The factor 1+i is nonzero, so cancellation is legal.

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.

06 / Foundation#Your turn

Calculate z times its conjugate.

z=3+2iz=3+2i
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
Use the difference-of-squares identity.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    zz‾=(3+2i)(3−2i)=9+4z\overline z=(3+2i)(3-2i)=9+4

  3. The product equals |z|² and is real nonnegative.

The requested value is 13.

Checks and common pitfalls: The product equals |z|² and is real nonnegative.

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

07 / Foundation#Your turn

Evaluate the power.

i14i^{14}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
Powers of i repeat every four exponents.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    i14=(i4)3i2=−1i^{14}=(i^4)^{3}i^2=-1

  3. Reduce the exponent modulo four.

The requested value is -1.

Checks and common pitfalls: Reduce the exponent modulo four.

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

08 / Standard#Your turn

Solve the quadratic over the complex numbers.

z2−6z+10=0z^2-6z+10=0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
Complete the square.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    (z−3)2=−1⇒z=3±i(z-3)^2=-1\Rightarrow z=3±i

  3. Substituting either root gives zero in the original polynomial.

z=3±i.

Checks and common pitfalls: Substituting either root gives zero in the original polynomial.

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

Find the imaginary part of the reciprocal.

13+i\frac1{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
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
Multiply numerator and denominator by the conjugate.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    13+i=3−i10\frac1{3+i}=\frac{3-i}{10}

  3. The positive denominator is the squared modulus.

The requested value is -0.1.

Checks and common pitfalls: The positive denominator is the squared modulus.

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

10 / Standard#Your turn

Explain why the conjugate of a complex root is also a root of a polynomial with real coefficients.

p(z)=0,p∈R[z]p(z)=0,\quad p\in\mathbb R[z]
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Conjugation preserves sums and products.
Hint 2
Real coefficients are unchanged by conjugation.
Worked solution

  1. Conjugation preserves sums and products.

  2. Calculate or simplify this relation.

    p(z‾)=p(z)‾=0p(\overline z)=\overline{p(z)}=0

  3. The real-coefficient assumption is essential.

Conjugate every term: p(conjugate z)=conjugate p(z)=0.

Checks and common pitfalls: The real-coefficient assumption is essential.

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.

11 / Standard#Your turn

Find the real part of the product.

(4+i)(2−i)(4+i)(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
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
The product i(−i) equals +1.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    (4+i)(2−i)=9+−2i(4+i)(2-i)=9+-2i

  3. Separate the real terms from the coefficient of i.

The requested value is 9.

Checks and common pitfalls: Separate the real terms from the coefficient of i.

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

12 / Transfer#Your turn

Simplify the quotient.

4+4i1+i\frac{4+4i}{1+i}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
Factor the numerator.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    4(1+i)1+i=4\frac{4(1+i)}{1+i}=4

  3. The factor 1+i is nonzero, so cancellation is legal.

4.

Checks and common pitfalls: The factor 1+i is nonzero, so cancellation is legal.

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.

13 / Transfer#Your turn

Calculate z times its conjugate.

z=4+2iz=4+2i
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Expand with i²=−1 and use a conjugate to remove a complex denominator.
Hint 2
Use the difference-of-squares identity.
Worked solution

  1. Expand with i²=−1 and use a conjugate to remove a complex denominator.

  2. Calculate or simplify this relation.

    zz‾=(4+2i)(4−2i)=16+4z\overline z=(4+2i)(4-2i)=16+4

  3. The product equals |z|² and is real nonnegative.

The requested value is 20.

Checks and common pitfalls: The product equals |z|² and is real nonnegative.

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

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for arithmetic 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 arithmetic of complex numbers.
    • Multiplication: Expand algebraically and replace i² with −1.
    • Division: A conjugate turns a nonzero complex denominator into its positive squared modulus.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Expand algebraically and replace i² with −1.
    • Expected reasoning: A conjugate turns a nonzero complex denominator into its positive squared modulus.
    • Expected correction: The sign from i² must not be lost.

    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.

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