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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 />
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+3๐+(3+5๐)<br />ย <br />ADD:<br />=5+8๐<br />ย <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. 2โ3๐+3+5๐<br />7+2๐+(โ2+๐)<br />2+6๐+7+6๐<br />3โ3๐+3+3๐<br />6๐โ3+3+8๐<br />2๐+4+(1โ3๐)<br />3+โ16+(2+โ4)<br />ย <br />Illustrative examples: Add these complex numbers<br />
9. 9. SUBTRACTION<br />
10. 10. (6x + 7y) โ (2x โ 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+5๐โ(3+4๐)<br />ย <br />FIND THE DIFFERENCE:<br />=2+๐<br />ย <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. 2โ4๐โ(3+5๐)<br />4โ4๐โ(1โ3๐)<br />3โ3๐โ(8+5๐)<br />12โ3๐โ(4+6๐)<br />2๐โ14โ6โ4๐<br />2โโ25โย (โ11+3โ49) <br />ย <br />Illustrative examples: Subtract<br />
15. 15. 2+4๐+(3+6๐)<br />4โ2๐โ(7+๐)<br />11โ2๐+(3โ15๐)<br />5โ๐+(3+5๐)<br />3โ10๐+16+3๐โ(2+3๐)<br />ย <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 โ 2y)<br />(3x โ 2) (5x + 3)<br />(4x + 5) (3x โ 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. 62๐+3<br />๐7+2๐<br />2๐7โ๐<br />2๐3โ1+5๐ -> #1-10<br />4๐26+โ25 -> # 11-20<br />ย <br />A. MONOMIAL FACTOR<br />Using DPMA or DPMS<br />
20. 20. 1โ3๐5+2๐<br />3โ4๐7โ2๐<br />6+๐2โ๐-> # (21-30)<br />5+6๐3โ2๐ย # 31-40<br />ย <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 />(๐ฅ+๐ฆ)2=<br />ย <br />๐ฅ2+2๐ฅ๐ฆ+๐ฆ2<br />ย <br />
23. 23. C. Binomial Square<br />(๐+๐๐)2=<br />ย <br />๐2+2๐๐๐โ๐2<br />ย <br />Why?<br />
24. 24. 3+๐2<br />6+2๐2<br />4โ3๐2<br />1+๐2<br />5โ2๐2<br />7+4๐2<br />ย <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 />๐ฅ+๐ฆ๐ฅโ๐ฆ=<br />ย <br />๐ฅ2โ๐ฆ2<br />ย <br />
26. 26. C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS <br />CONJUGATES<br />?<br />๐+๐๐๐โ๐๐=<br />ย <br />
27. 27. complex numbers which differ only in the sign of their imaginary part<br />Find the conjugate of:<br />(6+2๐)<br />(3โ7๐)<br />4+6๐<br />3+7๐<br />5โ2๐<br />ย <br />Conjugates<br />
28. 28. 2+3๐2โ3๐<br />3+8๐3โ8๐<br />6+7๐6โ7๐<br />5+6๐5โ6๐<br />9+10๐9โ10๐<br />ย <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 />๐+๐๐๐โ๐๐=<br />ย <br />๐2+๐2<br />ย <br />Why?<br />
31. 31. 3+2๐2<br />6โ2๐22<br />123+7๐2<br />5โ3๐2<br />4โ3๐2<br />ย <br />6โ2๐6+2๐<br />(8+3๐)(8โ3๐)<br />(7โ6๐)(7+6๐)<br />7+11๐7โ11๐<br />2๐+142๐โ14<br />ย <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 />ย <br />SET INDUCTION<br />
35. 35. A. MONOMIAL DIVISOR<br />RATIONALIZATION<br />
36. 36. 6๐<br />15๐<br />43๐<br />112๐<br />ย <br />74๐<br />reciprocal of ๐<br />reciprocal of 2๐<br />ย <br />Illustrative examples<br />
37. 37. How do we make the denominator a rational number?<br />B. BINOMIAL DIVISOR<br />12+๐<br />ย <br />
38. 38. B. Binomial Divisor<br />CONJUGATION<br />
39. 39. 12+๐<br />1+๐2โ๐<br />4+3๐1โ2๐<br />ย <br />5+3๐1+4๐<br />1+๐3โ3๐<br />๐+3<br />ย <br />Illustrative Examples<br />
40. 40. 32๐<br />47๐3<br />23+๐2๐<br />5โ3๐6+2๐<br />ย <br />5โ2๐3โ8๐<br />3+2๐3โ2๐<br />Reciprocal ofย 3โ7๐<br />ย <br />SEATWORK: Simplify the following complex numbers<br />