1 lesson 7 introduction to complex numbers
- 2. Summary of Topics • Simplify numbers of the form −𝑏, where 𝑏 > 0. • Recognize imaginary complex numbers. • Perform operations on imaginary numbers. • Write complex numbers. • Perform operations on Complex Numbers. • Find powers of i.
- 4. History of Imaginary Number, i READ THE WHOLE PARAGRAPH ON PAGE 47
- 5. Math Box Imaginary Unit i 𝒊 = −𝟏 so 𝒊 𝟐 = −𝟏 For any positive real number x, −𝒙 = 𝒊 𝒙.
- 6. Example: Write each imaginary number in terms of i a. −25 b. −13 c. −12 d. −20 e. − −16
- 7. WARM-UP PRACTICE Answer ACTIVITY A Items 1-10 Page 56
- 8. Power of i
- 9. REVIEW: • −1 raised to an EVEN power is 1. • −1 raised to an ODD power is −1. • 𝒊 𝟐 = −𝟏 FACT:
- 10. Look for a Pattern • 𝒊 𝟏 = ( −𝟏) 𝟏= 𝒊 • 𝒊 𝟐 = ( −𝟏) 𝟐 = −𝟏 • 𝒊 𝟑 = 𝒊 𝟐 ∙ 𝒊 = −𝟏 ∙ 𝒊 = −𝒊 • 𝒊 𝟒 = 𝒊 𝟐 ∙ 𝒊 𝟐 = −𝟏 ∙ −𝟏 = 𝟏 • 𝒊 𝟓 = 𝒊 𝟐 ∙ 𝒊 𝟐 ∙ 𝒊 = −𝟏 ∙ −𝟏 ∙ 𝒊 = 𝒊 • 𝒊 𝟔 = 𝒊 𝟐 ∙ 𝒊 𝟐 ∙ 𝒊 𝟐 = −𝟏 ∙ −𝟏 ∙ −𝟏 = −𝟏 • 𝒊 𝟕 = 𝒊 𝟐 ∙ 𝒊 𝟐 ∙ 𝒊 𝟐 ∙ 𝒊 = −𝟏 ∙ −𝟏 ∙ −𝟏 ∙ 𝒊 = −𝒊 • 𝒊 𝟖 = 𝒊 𝟐 ∙ 𝒊 𝟐 ∙ 𝒊 𝟐 ∙ 𝒊 𝟐 = −𝟏 ∙ −𝟏 ∙ −𝟏 ∙ −𝟏 = 𝟏
- 11. Math Box Power of i If n is a natural number that has a nonzero remainder of r when divided by 4, then 𝒊 𝒏 = 𝒊 𝒓 When n is divisible by 4, then 𝒊 𝒏 = 𝟏.
- 12. Simplify the following: a. 𝒊 𝟐𝟗 b. 𝒊 𝟐𝟎 c. 𝒊 𝟒𝟕 d. 𝒊 𝟓𝟏 e. 𝒊 𝟗𝟗
- 13. Complex Numbers
- 14. COMPLEX NUMBER • Using i, the real numbers, and the operations of addition, subtraction, multiplication and division, we obtain numbers that can be written in the form a + bi, where a and b are real numbers. • A combination of a real number and an imaginary number is called complex number.
- 15. Complex Numbers Example Real Term Coefficient of the Imaginary Term Classification 𝟑 − 𝟒𝒊 𝟔 = 𝟔 + 𝟎𝒊 −𝟕𝒊 = 𝟎 − 𝟕𝒊 𝟎 = 𝟎 + 𝟎𝒊 − 𝟐𝟓 = −𝟓 + 𝟎𝒊 −𝟐𝟓 = 𝟎 + 𝟓𝒊 3 6 0 0 -5 0 -4 0 -7 0 0 5 Imaginary Real Pure Imaginary Real Real Pure Imaginary Two complex numbers are equal if, and only if, both their real terms and their imaginary terms are equal
- 16. Determine a and b such that the complex numbers are equal. • 𝑎 + 𝑏𝑖 = 6 + 11𝑖 𝒂 = 𝟔; 𝒃 = 𝟏𝟏 • −5 + 𝑏𝑖 = 𝑎 + 9𝑖 𝒂 = −𝟓; 𝒃 = 𝟗 • 𝑎 + 3 − 8𝑖 = 4 + 2𝑏𝑖 𝑎 + 3 = 4; 2𝑏 = −8 𝒂 = 𝟏; 𝒃 = −𝟒
- 17. WARM-UP PRACTICE Answer ACTIVITY D Items 1-6 Page 56
- 18. The arithmetic of complex numbers is similar to the arithmetic of binomials Addition of Binomials Addition of Complex Numbers 2𝑥 + 3𝑦 + 4𝑥 + 7𝑦 6𝑥 + 10𝑦 2 + 3𝑖 + 4 + 7𝑖 6 + 10𝑖
- 19. Add or Subtract as indicated and simplify • 8 + 6𝑖 + (3 + 2𝑖) • 4 + 5𝑖 − (6 − 3𝑖)
- 20. WARM-UP PRACTICE Answer ACTIVITY B Items 1-4 Page 56 Answer ACTIVITY E Items 1-6 Page 56
- 21. Multiplication of Complex Numbers • 𝑎 𝑏 = 𝑎𝑏, for all POSITIVE RADICANDS only. • 𝑎 𝑏 ≠ 𝑎𝑏, for all NEGATIVE RADICANDS.
- 22. Example: Find the product of the following radicals. • −2 ∙ −5 • −16 ∙ −25 • −5 ∙ −7 • −3𝑖 ∙ 8𝑖 • 4𝑖 3 − 5𝑖 • (1 + 2𝑖)(1 + 3𝑖)
- 23. Major Mistake Territory! To avoid mistakes, always put complex numbers in the form a + bi before doing any computation.
- 24. WARM-UP PRACTICE Answer ACTIVITY C Items 1-8 Page 56
- 25. Conjugates and Division Conjugate of a Complex Number •The conjugate of a complex number a + bi is a – bi, and the conjugate of a - bi is a + bi.
- 26. Find the conjugate of each complex numbers •5 + 7𝑖 •14 + 3𝑖 •−3 − 9𝑖 •4𝑖 When a complex number is multiplied by its conjugate, we get a real number.
- 27. POWER PLUS Answer ACTIVITY A Items 1-6 Page 57
- 28. Multiply. • 5 + 7𝑖 5 − 7𝑖 • (2 − 3𝑖)(2 + 3𝑖)
- 29. Divide and Simplify to the form a + bi. • −5 + 9𝑖 1 − 2𝑖 • 7 + 3𝑖 5𝑖
- 30. POWER PLUS Answer ACTIVITY B Items 1-6 Page 57