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Complex arithmetic

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  • 1. Prepared by:
    Mr. Raymond B. Canlapan
    COMPLEX ARITHMETIC
  • 2. 1.4. Operations on Complex Numbers
    1.4.1. Addition
    1.4.2. Subtraction
    1.4.3. Multiplication
    1.4.3.1. Monomial: Distribution
    1.4.3.2. Binomials
    1.4.3.3. Special Products
    1.4.3.3.1. Binomial Square
    1.4.3.3.2. Conjugates
    1.4.4. Division
    1.4.4.1. Monomial Divisor
    1.4.4.2. Binomial Divisor
    SCOPE
  • 3. ADDITION
  • 4. (2x + 3y) + (x + 2y)
    (3x + 5y) + (2x + y)
    (3x + 3y) + (3x + 3y)
    SET INDUCTION: Review of Adding Polynomials
    To add polynomials, simply combine like terms.
  • 5. Does the method of combining like terms in polynomials also applied in adding complex numbers?
    What are the steps to be followed in adding complex numbers?
    ESSENTIAL QUESTIONs:
  • 6. 2+3๐‘–+(3+5๐‘–)
    ย 
    ADD:
    =5+8๐‘–
    ย 
  • 7. How do we add complex numbers?
    Add the real parts.
    Add the imaginary parts.
    Express sum in standard form.
  • 8. 2โˆ’3๐‘–+3+5๐‘–
    7+2๐‘–+(โˆ’2+๐‘–)
    2+6๐‘–+7+6๐‘–
    3โˆ’3๐‘–+3+3๐‘–
    6๐‘–โˆ’3+3+8๐‘–
    2๐‘–+4+(1โˆ’3๐‘–)
    3+โˆ’16+(2+โˆ’4)
    ย 
    Illustrative examples: Add these complex numbers
  • 9. SUBTRACTION
  • 10. (6x + 7y) โ€“ (2x โ€“ 5y)
    Review: Subtracting polynomials
    = 4x + 12y
    Change the sign of the subtrahend.
    Proceed to addition.
  • 11. Does the procedure in subtracting polynomials applied in complex numbers?
    ESSENTIAL QUESTIONs:
  • 12. 5+5๐‘–โˆ’(3+4๐‘–)
    ย 
    FIND THE DIFFERENCE:
    =2+๐‘–
    ย 
  • 13. How do we SUBTRACT complex numbers?
    Change the sign of the subtrahend.
    Proceed to addition.
    Express difference in standard form.
  • 14. 2โˆ’4๐‘–โˆ’(3+5๐‘–)
    4โˆ’4๐‘–โˆ’(1โˆ’3๐‘–)
    3โˆ’3๐‘–โˆ’(8+5๐‘–)
    12โˆ’3๐‘–โˆ’(4+6๐‘–)
    2๐‘–โˆ’14โˆ’6โˆ’4๐‘–
    2โˆ’โˆ’25โˆ’ย (โˆ’11+3โˆ’49)
    ย 
    Illustrative examples: Subtract
  • 15. 2+4๐‘–+(3+6๐‘–)
    4โˆ’2๐‘–โˆ’(7+๐‘–)
    11โˆ’2๐‘–+(3โˆ’15๐‘–)
    5โˆ’๐‘–+(3+5๐‘–)
    3โˆ’10๐‘–+16+3๐‘–โˆ’(2+3๐‘–)
    ย 
    SEATWORK: Perform the Indicated Operation
  • 16. MULTIPLICATION
    Monomial Factor
    Binomial Factors
  • 17. 3(2x + 5)
    2x(5 + 3x)
    7x(3x โ€“ 2y)
    (3x โ€“ 2) (5x + 3)
    (4x + 5) (3x โ€“ 7)
    SET INDUCTION (QUIZ GAME): FIND THE PRODUCT (5 MINUTES)
  • 18. How do we multiply polynomials with a monomial factor?
    How do we multiply polynomials with two binomial factors?
    QUESTIONS:
    Distribution Property
    FOIL Method
  • 19. 62๐‘–+3
    ๐‘–7+2๐‘–
    2๐‘–7โˆ’๐‘–
    2๐‘–3โˆ’1+5๐‘– -> #1-10
    4๐‘–26+โˆ’25 -> # 11-20
    ย 
    A. MONOMIAL FACTOR
    Using DPMA or DPMS
  • 20. 1โˆ’3๐‘–5+2๐‘–
    3โˆ’4๐‘–7โˆ’2๐‘–
    6+๐‘–2โˆ’๐‘–-> # (21-30)
    5+6๐‘–3โˆ’2๐‘–ย # 31-40
    ย 
    B. BINOMIAL FACTORS
    Using FOIL
  • 21. SPECIAL products
    1. Binomial Square
    2. Conjugates
  • 22. C. BINOMIAL SQUARE
    (๐‘ฅ+๐‘ฆ)2=
    ย 
    ๐‘ฅ2+2๐‘ฅ๐‘ฆ+๐‘ฆ2
    ย 
  • 23. C. Binomial Square
    (๐‘Ž+๐‘๐‘–)2=
    ย 
    ๐‘Ž2+2๐‘Ž๐‘๐‘–โˆ’๐‘2
    ย 
    Why?
  • 24. 3+๐‘–2
    6+2๐‘–2
    4โˆ’3๐‘–2
    1+๐‘–2
    5โˆ’2๐‘–2
    7+4๐‘–2
    ย 
    Illustrative Examples: Find the Product (TEAM-PAIR-SOLO)
  • 25. C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS
    ๐‘ฅ+๐‘ฆ๐‘ฅโˆ’๐‘ฆ=
    ย 
    ๐‘ฅ2โˆ’๐‘ฆ2
    ย 
  • 26. C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS
    CONJUGATES
    ?
    ๐‘Ž+๐‘๐‘–๐‘Žโˆ’๐‘๐‘–=
    ย 
  • 27. complex numbers which differ only in the sign of their imaginary part
    Find the conjugate of:
    (6+2๐‘–)
    (3โˆ’7๐‘–)
    4+6๐‘–
    3+7๐‘–
    5โˆ’2๐‘–
    ย 
    Conjugates
  • 28. 2+3๐‘–2โˆ’3๐‘–
    3+8๐‘–3โˆ’8๐‘–
    6+7๐‘–6โˆ’7๐‘–
    5+6๐‘–5โˆ’6๐‘–
    9+10๐‘–9โˆ’10๐‘–
    ย 
    ACTIVITY: PRODUCT OF CONJUGATES
  • 29. Tabulate the results:
    ACTIVITY: PRODUCT OF CONJUGATES
  • 30. C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS
    ๐‘Ž+๐‘๐‘–๐‘Žโˆ’๐‘๐‘–=
    ย 
    ๐‘Ž2+๐‘2
    ย 
    Why?
  • 31. 3+2๐‘–2
    6โˆ’2๐‘–22
    123+7๐‘–2
    5โˆ’3๐‘–2
    4โˆ’3๐‘–2
    ย 
    6โˆ’2๐‘–6+2๐‘–
    (8+3๐‘–)(8โˆ’3๐‘–)
    (7โˆ’6๐‘–)(7+6๐‘–)
    7+11๐‘–7โˆ’11๐‘–
    2๐‘–+142๐‘–โˆ’14
    ย 
    Seatwork: FIND THE PRODUCT
  • 32. A. Monomial Divisor
    B. Binomial Divisor
    DIVISION
  • 33. How do we divide complex numbers with monomial divisor?
    How do we divide complex numbers with binomial divisor?
    ESSENTIAL QUESTIONS
  • 34. How do we simplify 12?
    ย 
    SET INDUCTION
  • 35. A. MONOMIAL DIVISOR
    RATIONALIZATION
  • 36. 6๐‘–
    15๐‘–
    43๐‘–
    112๐‘–
    ย 
    74๐‘–
    reciprocal of ๐‘–
    reciprocal of 2๐‘–
    ย 
    Illustrative examples
  • 37. How do we make the denominator a rational number?
    B. BINOMIAL DIVISOR
    12+๐‘–
    ย 
  • 38. B. Binomial Divisor
    CONJUGATION
  • 39. 12+๐‘–
    1+๐‘–2โˆ’๐‘–
    4+3๐‘–1โˆ’2๐‘–
    ย 
    5+3๐‘–1+4๐‘–
    1+๐‘–3โˆ’3๐‘–
    ๐‘–+3
    ย 
    Illustrative Examples
  • 40. 32๐‘–
    47๐‘–3
    23+๐‘–2๐‘–
    5โˆ’3๐‘–6+2๐‘–
    ย 
    5โˆ’2๐‘–3โˆ’8๐‘–
    3+2๐‘–3โˆ’2๐‘–
    Reciprocal ofย 3โˆ’7๐‘–
    ย 
    SEATWORK: Simplify the following complex numbers