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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+3π+(3+5π)<br />Β <br />ADD:<br />=5+8π<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. 2β3π+3+5π<br />7+2π+(β2+π)<br />2+6π+7+6π<br />3β3π+3+3π<br />6πβ3+3+8π<br />2π+4+(1β3π)<br />3+β16+(2+β4)<br />Β <br />Illustrative examples: Add these complex numbers<br />
9. 9. SUBTRACTION<br />
10. 10. (6x + 7y) β (2x β 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β(3+4π)<br />Β <br />FIND THE DIFFERENCE:<br />=2+π<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. 2β4πβ(3+5π)<br />4β4πβ(1β3π)<br />3β3πβ(8+5π)<br />12β3πβ(4+6π)<br />2πβ14β6β4π<br />2ββ25βΒ (β11+3β49) <br />Β <br />Illustrative examples: Subtract<br />
15. 15. 2+4π+(3+6π)<br />4β2πβ(7+π)<br />11β2π+(3β15π)<br />5βπ+(3+5π)<br />3β10π+16+3πβ(2+3π)<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 β 2y)<br />(3x β 2) (5x + 3)<br />(4x + 5) (3x β 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. 62π+3<br />π7+2π<br />2π7βπ<br />2π3β1+5π -> #1-10<br />4π26+β25 -> # 11-20<br />Β <br />A. MONOMIAL FACTOR<br />Using DPMA or DPMS<br />
20. 20. 1β3π5+2π<br />3β4π7β2π<br />6+π2βπ-> # (21-30)<br />5+6π3β2πΒ # 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₯+π¦)2=<br />Β <br />π₯2+2π₯π¦+π¦2<br />Β <br />
23. 23. C. Binomial Square<br />(π+ππ)2=<br />Β <br />π2+2πππβπ2<br />Β <br />Why?<br />
24. 24. 3+π2<br />6+2π2<br />4β3π2<br />1+π2<br />5β2π2<br />7+4π2<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₯+π¦π₯βπ¦=<br />Β <br />π₯2βπ¦2<br />Β <br />
26. 26. C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS <br />CONJUGATES<br />?<br />π+πππβ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+2π)<br />(3β7π)<br />4+6π<br />3+7π<br />5β2π<br />Β <br />Conjugates<br />
28. 28. 2+3π2β3π<br />3+8π3β8π<br />6+7π6β7π<br />5+6π5β6π<br />9+10π9β10π<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+πππβππ=<br />Β <br />π2+π2<br />Β <br />Why?<br />
31. 31. 3+2π2<br />6β2π22<br />123+7π2<br />5β3π2<br />4β3π2<br />Β <br />6β2π6+2π<br />(8+3π)(8β3π)<br />(7β6π)(7+6π)<br />7+11π7β11π<br />2π+142πβ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. 6π<br />15π<br />43π<br />112π<br />Β <br />74π<br />reciprocal of π<br />reciprocal of 2π<br />Β <br />Illustrative examples<br />
37. 37. How do we make the denominator a rational number?<br />B. BINOMIAL DIVISOR<br />12+π<br />Β <br />
38. 38. B. Binomial Divisor<br />CONJUGATION<br />
39. 39. 12+π<br />1+π2βπ<br />4+3π1β2π<br />Β <br />5+3π1+4π<br />1+π3β3π<br />π+3<br />Β <br />Illustrative Examples<br />
40. 40. 32π<br />47π3<br />23+π2π<br />5β3π6+2π<br />Β <br />5β2π3β8π<br />3+2π3β2π<br />Reciprocal ofΒ 3β7π<br />Β <br />SEATWORK: Simplify the following complex numbers<br />