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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+3𝑖+(3+5𝑖) ADD: =5+8𝑖

7. How do we add complex numbers? Add the real parts. Add the imaginary parts. Express sum in standard form.

8. 2−3𝑖+3+5𝑖 7+2𝑖+(−2+𝑖) 2+6𝑖+7+6𝑖 3−3𝑖+3+3𝑖 6𝑖−3+3+8𝑖 2𝑖+4+(1−3𝑖) 3+−16+(2+−4) Illustrative examples: Add these complex numbers

9. SUBTRACTION

10. (6x + 7y) – (2x – 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+5𝑖−(3+4𝑖) FIND THE DIFFERENCE: =2+𝑖

13. How do we SUBTRACT complex numbers? Change the sign of the subtrahend. Proceed to addition. Express difference in standard form.

14. 2−4𝑖−(3+5𝑖) 4−4𝑖−(1−3𝑖) 3−3𝑖−(8+5𝑖) 12−3𝑖−(4+6𝑖) 2𝑖−14−6−4𝑖 2−−25− (−11+3−49) Illustrative examples: Subtract

15. 2+4𝑖+(3+6𝑖) 4−2𝑖−(7+𝑖) 11−2𝑖+(3−15𝑖) 5−𝑖+(3+5𝑖) 3−10𝑖+16+3𝑖−(2+3𝑖) SEATWORK: Perform the Indicated Operation

16. MULTIPLICATION Monomial Factor Binomial Factors

17. 3(2x + 5) 2x(5 + 3x) 7x(3x – 2y) (3x – 2) (5x + 3) (4x + 5) (3x – 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. 62𝑖+3 𝑖7+2𝑖 2𝑖7−𝑖 2𝑖3−1+5𝑖 -> #1-10 4𝑖26+−25 -> # 11-20 A. MONOMIAL FACTOR Using DPMA or DPMS

20. 1−3𝑖5+2𝑖 3−4𝑖7−2𝑖 6+𝑖2−𝑖-> # (21-30) 5+6𝑖3−2𝑖 # 31-40 B. BINOMIAL FACTORS Using FOIL

21. SPECIAL products 1. Binomial Square 2. Conjugates

22. C. BINOMIAL SQUARE (𝑥+𝑦)2= 𝑥2+2𝑥𝑦+𝑦2

23. C. Binomial Square (𝑎+𝑏𝑖)2= 𝑎2+2𝑎𝑏𝑖−𝑏2 Why?

24. 3+𝑖2 6+2𝑖2 4−3𝑖2 1+𝑖2 5−2𝑖2 7+4𝑖2 Illustrative Examples: Find the Product (TEAM-PAIR-SOLO)

25. C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS 𝑥+𝑦𝑥−𝑦= 𝑥2−𝑦2

26. C. SPECIAL PRODUCT OF THE SUM AND DIFFERENCE OF TWO LIKE TERMS CONJUGATES ? 𝑎+𝑏𝑖𝑎−𝑏𝑖=

27. complex numbers which differ only in the sign of their imaginary part Find the conjugate of: (6+2𝑖) (3−7𝑖) 4+6𝑖 3+7𝑖 5−2𝑖 Conjugates

28. 2+3𝑖2−3𝑖 3+8𝑖3−8𝑖 6+7𝑖6−7𝑖 5+6𝑖5−6𝑖 9+10𝑖9−10𝑖 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 𝑎+𝑏𝑖𝑎−𝑏𝑖= 𝑎2+𝑏2 Why?

31. 3+2𝑖2 6−2𝑖22 123+7𝑖2 5−3𝑖2 4−3𝑖2 6−2𝑖6+2𝑖 (8+3𝑖)(8−3𝑖) (7−6𝑖)(7+6𝑖) 7+11𝑖7−11𝑖 2𝑖+142𝑖−14 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? SET INDUCTION

35. A. MONOMIAL DIVISOR RATIONALIZATION

36. 6𝑖 15𝑖 43𝑖 112𝑖 74𝑖 reciprocal of 𝑖 reciprocal of 2𝑖 Illustrative examples

37. How do we make the denominator a rational number? B. BINOMIAL DIVISOR 12+𝑖

38. B. Binomial Divisor CONJUGATION

39. 12+𝑖 1+𝑖2−𝑖 4+3𝑖1−2𝑖 5+3𝑖1+4𝑖 1+𝑖3−3𝑖 𝑖+3 Illustrative Examples

40. 32𝑖 47𝑖3 23+𝑖2𝑖 5−3𝑖6+2𝑖 5−2𝑖3−8𝑖 3+2𝑖3−2𝑖 Reciprocal of 3−7𝑖 SEATWORK: Simplify the following complex numbers