Complex arithmetic
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This document discusses operations on complex numbers including addition, subtraction, multiplication, and division. It provides examples and steps for each operation. For addition and subtraction, complex numbers are treated similarly to polynomials by combining like terms. For multiplication, the distributive property and FOIL method are used depending on whether there is a monomial or binomial factor. Special products include binomial squares and conjugates. Division involves rationalizing the denominator by conjugation for binomial divisors or using reciprocals for monomial divisors.
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Presentation1
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
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Complex arithmetic
- 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