Prepared by:<br />Mr. Raymond B. Canlapan<br />COMPLEX ARITHMETIC<br />
1.4. Operations on Complex Numbers<br /> 	1.4.1. Addition<br />  	1.4.2. Subtraction<br />    	1.4.3. Multiplication<br />...
ADDITION<br />
(2x + 3y) + (x + 2y)<br />(3x + 5y) + (2x + y)<br />(3x + 3y) + (3x + 3y)<br />SET INDUCTION: Review of Adding Polynomials...
Does the method of combining like terms in polynomials also applied in adding complex numbers?<br />What are the steps to ...
2+3𝑖+(3+5𝑖)<br />Β <br />ADD:<br />=5+8𝑖<br />Β <br />
How do we add complex numbers?<br />Add the real parts.<br />Add the imaginary parts.<br />Express sum in standard form.<b...
2βˆ’3𝑖+3+5𝑖<br />7+2𝑖+(βˆ’2+𝑖)<br />2+6𝑖+7+6𝑖<br />3βˆ’3𝑖+3+3𝑖<br />6π‘–βˆ’3+3+8𝑖<br />2𝑖+4+(1βˆ’3𝑖)<br />3+βˆ’16+(2+βˆ’4)<br />Β <br />Ill...
SUBTRACTION<br />
(6x + 7y) – (2x – 5y)<br />Review: Subtracting polynomials<br />= 4x + 12y<br />Change the sign of the subtrahend.<br />Pr...
Does the procedure in subtracting polynomials applied in complex numbers?<br />ESSENTIAL QUESTIONs:<br />
5+5π‘–βˆ’(3+4𝑖)<br />Β <br />FIND THE DIFFERENCE:<br />=2+𝑖<br />Β <br />
How do we SUBTRACT complex numbers?<br />Change the sign of the subtrahend.<br />Proceed to addition.<br />Express differe...
2βˆ’4π‘–βˆ’(3+5𝑖)<br />4βˆ’4π‘–βˆ’(1βˆ’3𝑖)<br />3βˆ’3π‘–βˆ’(8+5𝑖)<br />12βˆ’3π‘–βˆ’(4+6𝑖)<br />2π‘–βˆ’14βˆ’6βˆ’4𝑖<br />2βˆ’βˆ’25βˆ’Β (βˆ’11+3βˆ’49) <br />Β <br />Illust...
2+4𝑖+(3+6𝑖)<br />4βˆ’2π‘–βˆ’(7+𝑖)<br />11βˆ’2𝑖+(3βˆ’15𝑖)<br />5βˆ’π‘–+(3+5𝑖)<br />3βˆ’10𝑖+16+3π‘–βˆ’(2+3𝑖)<br />Β <br />SEATWORK: Perform the I...
MULTIPLICATION<br />Monomial Factor<br />Binomial Factors<br />
3(2x + 5)<br />2x(5 + 3x)<br />7x(3x – 2y)<br />(3x – 2) (5x + 3)<br />(4x + 5) (3x – 7) <br />SET INDUCTION (QUIZ GAME): ...
How do we multiply polynomials with a monomial factor?<br />How do we multiply polynomials with two binomial factors?<br /...
62𝑖+3<br />𝑖7+2𝑖<br />2𝑖7βˆ’π‘–<br />2𝑖3βˆ’1+5𝑖 -> #1-10<br />4𝑖26+βˆ’25 -> # 11-20<br />Β <br />A. MONOMIAL FACTOR<br />Using DPMA...
1βˆ’3𝑖5+2𝑖<br />3βˆ’4𝑖7βˆ’2𝑖<br />6+𝑖2βˆ’π‘–-> # (21-30)<br />5+6𝑖3βˆ’2𝑖 # 31-40<br />Β <br />B. BINOMIAL FACTORS<br />Using FOIL<br />
SPECIAL products<br />1. Binomial Square<br />2. Conjugates<br />
C. BINOMIAL SQUARE<br />(π‘₯+𝑦)2=<br />Β <br />π‘₯2+2π‘₯𝑦+𝑦2<br />Β <br />
C. Binomial Square<br />(π‘Ž+𝑏𝑖)2=<br />Β <br />π‘Ž2+2π‘Žπ‘π‘–βˆ’π‘2<br />Β <br />Why?<br />
3+𝑖2<br />6+2𝑖2<br />4βˆ’3𝑖2<br />1+𝑖2<br />5βˆ’2𝑖2<br />7+4𝑖2<br />Β <br />Illustrative Examples: Find the Product (TEAM-PAIR-...
C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS <br />π‘₯+𝑦π‘₯βˆ’π‘¦=<br />Β <br />π‘₯2βˆ’π‘¦2<br />Β <br />
C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS <br />CONJUGATES<br />?<br />π‘Ž+π‘π‘–π‘Žβˆ’π‘π‘–=<br />Β <br />
complex numbers which differ only in the sign of their imaginary part<br />Find the conjugate of:<br />(6+2𝑖)<br />(3βˆ’7𝑖)<...
2+3𝑖2βˆ’3𝑖<br />3+8𝑖3βˆ’8𝑖<br />6+7𝑖6βˆ’7𝑖<br />5+6𝑖5βˆ’6𝑖<br />9+10𝑖9βˆ’10𝑖<br />Β <br />ACTIVITY: PRODUCT OF CONJUGATES<br />
Tabulate the results:<br />ACTIVITY: PRODUCT OF CONJUGATES<br />
C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS <br />π‘Ž+π‘π‘–π‘Žβˆ’π‘π‘–=<br />Β <br />π‘Ž2+𝑏2<br />Β <br />Why?<br />
3+2𝑖2<br />6βˆ’2𝑖22<br />123+7𝑖2<br />5βˆ’3𝑖2<br />4βˆ’3𝑖2<br />Β <br />6βˆ’2𝑖6+2𝑖<br />(8+3𝑖)(8βˆ’3𝑖)<br />(7βˆ’6𝑖)(7+6𝑖)<br />7+11𝑖7βˆ’...
A. Monomial Divisor<br />B. Binomial Divisor<br />DIVISION<br />
How do we divide complex numbers with monomial divisor?<br />How do we divide complex numbers with binomial divisor?<br />...
How do we simplify 12?<br />Β <br />SET INDUCTION<br />
A. MONOMIAL DIVISOR<br />RATIONALIZATION<br />
6𝑖<br />15𝑖<br />43𝑖<br />112𝑖<br />Β <br />74𝑖<br />reciprocal of 𝑖<br />reciprocal of 2𝑖<br />Β <br />Illustrative example...
How do we make the denominator a rational number?<br />B. BINOMIAL DIVISOR<br />12+𝑖<br />Β <br />
B. Binomial Divisor<br />CONJUGATION<br />
12+𝑖<br />1+𝑖2βˆ’π‘–<br />4+3𝑖1βˆ’2𝑖<br />Β <br />5+3𝑖1+4𝑖<br />1+𝑖3βˆ’3𝑖<br />𝑖+3<br />Β <br />Illustrative Examples<br />
32𝑖<br />47𝑖3<br />23+𝑖2𝑖<br />5βˆ’3𝑖6+2𝑖<br />Β <br />5βˆ’2𝑖3βˆ’8𝑖<br />3+2𝑖3βˆ’2𝑖<br />Reciprocal ofΒ 3βˆ’7𝑖<br />Β <br />SEATWORK: S...
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Complex arithmetic

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Complex arithmetic

  1. 1. Prepared by:<br />Mr. Raymond B. Canlapan<br />COMPLEX ARITHMETIC<br />
  2. 2. 1.4. Operations on Complex Numbers<br /> 1.4.1. Addition<br /> 1.4.2. Subtraction<br /> 1.4.3. Multiplication<br /> 1.4.3.1. Monomial: Distribution <br /> 1.4.3.2. Binomials<br /> 1.4.3.3. Special Products<br /> 1.4.3.3.1. Binomial Square<br /> 1.4.3.3.2. Conjugates<br /> 1.4.4. Division<br /> 1.4.4.1. Monomial Divisor<br /> 1.4.4.2. Binomial Divisor<br />SCOPE<br />
  3. 3. ADDITION<br />
  4. 4. (2x + 3y) + (x + 2y)<br />(3x + 5y) + (2x + y)<br />(3x + 3y) + (3x + 3y)<br />SET INDUCTION: Review of Adding Polynomials<br />To add polynomials, simply combine like terms.<br />
  5. 5. Does the method of combining like terms in polynomials also applied in adding complex numbers?<br />What are the steps to be followed in adding complex numbers?<br />ESSENTIAL QUESTIONs:<br />
  6. 6. 2+3𝑖+(3+5𝑖)<br />Β <br />ADD:<br />=5+8𝑖<br />Β <br />
  7. 7. How do we add complex numbers?<br />Add the real parts.<br />Add the imaginary parts.<br />Express sum in standard form.<br />
  8. 8. 2βˆ’3𝑖+3+5𝑖<br />7+2𝑖+(βˆ’2+𝑖)<br />2+6𝑖+7+6𝑖<br />3βˆ’3𝑖+3+3𝑖<br />6π‘–βˆ’3+3+8𝑖<br />2𝑖+4+(1βˆ’3𝑖)<br />3+βˆ’16+(2+βˆ’4)<br />Β <br />Illustrative examples: Add these complex numbers<br />
  9. 9. SUBTRACTION<br />
  10. 10. (6x + 7y) – (2x – 5y)<br />Review: Subtracting polynomials<br />= 4x + 12y<br />Change the sign of the subtrahend.<br />Proceed to addition. <br />
  11. 11. Does the procedure in subtracting polynomials applied in complex numbers?<br />ESSENTIAL QUESTIONs:<br />
  12. 12. 5+5π‘–βˆ’(3+4𝑖)<br />Β <br />FIND THE DIFFERENCE:<br />=2+𝑖<br />Β <br />
  13. 13. How do we SUBTRACT complex numbers?<br />Change the sign of the subtrahend.<br />Proceed to addition.<br />Express difference in standard form.<br />
  14. 14. 2βˆ’4π‘–βˆ’(3+5𝑖)<br />4βˆ’4π‘–βˆ’(1βˆ’3𝑖)<br />3βˆ’3π‘–βˆ’(8+5𝑖)<br />12βˆ’3π‘–βˆ’(4+6𝑖)<br />2π‘–βˆ’14βˆ’6βˆ’4𝑖<br />2βˆ’βˆ’25βˆ’Β (βˆ’11+3βˆ’49) <br />Β <br />Illustrative examples: Subtract<br />
  15. 15. 2+4𝑖+(3+6𝑖)<br />4βˆ’2π‘–βˆ’(7+𝑖)<br />11βˆ’2𝑖+(3βˆ’15𝑖)<br />5βˆ’π‘–+(3+5𝑖)<br />3βˆ’10𝑖+16+3π‘–βˆ’(2+3𝑖)<br />Β <br />SEATWORK: Perform the Indicated Operation<br />
  16. 16. MULTIPLICATION<br />Monomial Factor<br />Binomial Factors<br />
  17. 17. 3(2x + 5)<br />2x(5 + 3x)<br />7x(3x – 2y)<br />(3x – 2) (5x + 3)<br />(4x + 5) (3x – 7) <br />SET INDUCTION (QUIZ GAME): FIND THE PRODUCT (5 MINUTES)<br />
  18. 18. How do we multiply polynomials with a monomial factor?<br />How do we multiply polynomials with two binomial factors?<br />QUESTIONS:<br />Distribution Property<br />FOIL Method<br />
  19. 19. 62𝑖+3<br />𝑖7+2𝑖<br />2𝑖7βˆ’π‘–<br />2𝑖3βˆ’1+5𝑖 -> #1-10<br />4𝑖26+βˆ’25 -> # 11-20<br />Β <br />A. MONOMIAL FACTOR<br />Using DPMA or DPMS<br />
  20. 20. 1βˆ’3𝑖5+2𝑖<br />3βˆ’4𝑖7βˆ’2𝑖<br />6+𝑖2βˆ’π‘–-> # (21-30)<br />5+6𝑖3βˆ’2𝑖 # 31-40<br />Β <br />B. BINOMIAL FACTORS<br />Using FOIL<br />
  21. 21. SPECIAL products<br />1. Binomial Square<br />2. Conjugates<br />
  22. 22. C. BINOMIAL SQUARE<br />(π‘₯+𝑦)2=<br />Β <br />π‘₯2+2π‘₯𝑦+𝑦2<br />Β <br />
  23. 23. C. Binomial Square<br />(π‘Ž+𝑏𝑖)2=<br />Β <br />π‘Ž2+2π‘Žπ‘π‘–βˆ’π‘2<br />Β <br />Why?<br />
  24. 24. 3+𝑖2<br />6+2𝑖2<br />4βˆ’3𝑖2<br />1+𝑖2<br />5βˆ’2𝑖2<br />7+4𝑖2<br />Β <br />Illustrative Examples: Find the Product (TEAM-PAIR-SOLO)<br />
  25. 25. C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS <br />π‘₯+𝑦π‘₯βˆ’π‘¦=<br />Β <br />π‘₯2βˆ’π‘¦2<br />Β <br />
  26. 26. C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS <br />CONJUGATES<br />?<br />π‘Ž+π‘π‘–π‘Žβˆ’π‘π‘–=<br />Β <br />
  27. 27. complex numbers which differ only in the sign of their imaginary part<br />Find the conjugate of:<br />(6+2𝑖)<br />(3βˆ’7𝑖)<br />4+6𝑖<br />3+7𝑖<br />5βˆ’2𝑖<br />Β <br />Conjugates<br />
  28. 28. 2+3𝑖2βˆ’3𝑖<br />3+8𝑖3βˆ’8𝑖<br />6+7𝑖6βˆ’7𝑖<br />5+6𝑖5βˆ’6𝑖<br />9+10𝑖9βˆ’10𝑖<br />Β <br />ACTIVITY: PRODUCT OF CONJUGATES<br />
  29. 29. Tabulate the results:<br />ACTIVITY: PRODUCT OF CONJUGATES<br />
  30. 30. C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS <br />π‘Ž+π‘π‘–π‘Žβˆ’π‘π‘–=<br />Β <br />π‘Ž2+𝑏2<br />Β <br />Why?<br />
  31. 31. 3+2𝑖2<br />6βˆ’2𝑖22<br />123+7𝑖2<br />5βˆ’3𝑖2<br />4βˆ’3𝑖2<br />Β <br />6βˆ’2𝑖6+2𝑖<br />(8+3𝑖)(8βˆ’3𝑖)<br />(7βˆ’6𝑖)(7+6𝑖)<br />7+11𝑖7βˆ’11𝑖<br />2𝑖+142π‘–βˆ’14<br />Β <br />Seatwork: FIND THE PRODUCT<br />
  32. 32. A. Monomial Divisor<br />B. Binomial Divisor<br />DIVISION<br />
  33. 33. How do we divide complex numbers with monomial divisor?<br />How do we divide complex numbers with binomial divisor?<br />ESSENTIAL QUESTIONS<br />
  34. 34. How do we simplify 12?<br />Β <br />SET INDUCTION<br />
  35. 35. A. MONOMIAL DIVISOR<br />RATIONALIZATION<br />
  36. 36. 6𝑖<br />15𝑖<br />43𝑖<br />112𝑖<br />Β <br />74𝑖<br />reciprocal of 𝑖<br />reciprocal of 2𝑖<br />Β <br />Illustrative examples<br />
  37. 37. How do we make the denominator a rational number?<br />B. BINOMIAL DIVISOR<br />12+𝑖<br />Β <br />
  38. 38. B. Binomial Divisor<br />CONJUGATION<br />
  39. 39. 12+𝑖<br />1+𝑖2βˆ’π‘–<br />4+3𝑖1βˆ’2𝑖<br />Β <br />5+3𝑖1+4𝑖<br />1+𝑖3βˆ’3𝑖<br />𝑖+3<br />Β <br />Illustrative Examples<br />
  40. 40. 32𝑖<br />47𝑖3<br />23+𝑖2𝑖<br />5βˆ’3𝑖6+2𝑖<br />Β <br />5βˆ’2𝑖3βˆ’8𝑖<br />3+2𝑖3βˆ’2𝑖<br />Reciprocal ofΒ 3βˆ’7𝑖<br />Β <br />SEATWORK: Simplify the following complex numbers<br />

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