1.4 complex numbers
- 2. Complex Numbers Because the square of any real number can't be negative, the equation x2 = –1 does not have any real solution.
- 3. Complex Numbers Because the square of any real number can't be negative, the equation x2 = –1 does not have any real solution. We make up a new number called an imaginary number –1 to be a solution of this equation
- 4. Complex Numbers Because the square of any real number can't be negative, the equation x2 = –1 does not have any real solution. We make up a new number called an imaginary number –1 ↔ i to be a solution of this equation and we name it “ i ”,
- 5. Complex Numbers Because the square of any real number can't be negative, the equation x2 = –1 does not have any real solution. We make up a new number called an imaginary number –1 ↔ i to be a solution of this equation and we name it “ i ”, i.e. (±i)2 = –1
- 6. Complex Numbers Using i, the “solutions” of the equations of the form Because the square of any real number can't be negative, the equation x2 = –1 does not have any real solution. We make up a new number called an imaginary number –1 ↔ i to be a solution of this equation and we name it “ i ”, i.e. (±i)2 = –1 x2 = –r
- 7. Complex Numbers Using i, the “solutions” of the equations of the form Because the square of any real number can't be negative, the equation x2 = –1 does not have any real solution. We make up a new number called an imaginary number –1 ↔ i to be a solution of this equation and we name it “ i ”, i.e. (±i)2 = –1 x2 = –r are x = ± ir
- 8. Complex Numbers Using i, the “solutions” of the equations of the form Example A. Solve x2 + 49 = 0 using imaginary numbers. Because the square of any real number can't be negative, the equation x2 = –1 does not have any real solution. We make up a new number called an imaginary number –1 ↔ i to be a solution of this equation and we name it “ i ”, i.e. (±i)2 = –1 x2 = –r are x = ± ir
- 9. Complex Numbers Using i, the “solutions” of the equations of the form Example A. Solve x2 + 49 = 0 using imaginary numbers. Using the square-root method: x2 + 49 = 0 → x2 = –49 Because the square of any real number can't be negative, the equation x2 = –1 does not have any real solution. We make up a new number called an imaginary number –1 ↔ i to be a solution of this equation and we name it “ i ”, i.e. (±i)2 = –1 x2 = –r are x = ± ir
- 10. Complex Numbers Using i, the “solutions” of the equations of the form Example A. Solve x2 + 49 = 0 using imaginary numbers. Using the square-root method: x2 + 49 = 0 → x2 = –49 so x = ±–49 Because the square of any real number can't be negative, the equation x2 = –1 does not have any real solution. We make up a new number called an imaginary number –1 ↔ i to be a solution of this equation and we name it “ i ”, i.e. (±i)2 = –1 x2 = –r are x = ± ir
- 11. Complex Numbers Using i, the “solutions” of the equations of the form Example A. Solve x2 + 49 = 0 using imaginary numbers. Using the square-root method: x2 + 49 = 0 → x2 = –49 so x = ±–49 x = ±49–1 x = ±7i Because the square of any real number can't be negative, the equation x2 = –1 does not have any real solution. We make up a new number called an imaginary number –1 ↔ i to be a solution of this equation and we name it “ i ”, i.e. (±i)2 = –1 x2 = –r are x = ± ir
- 12. A complex number is a number of the form a + bi where a and b are real numbers, Complex Numbers
- 13. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part Complex Numbers
- 14. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number. Complex Numbers
- 15. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number. Complex Numbers Example B. 5 – 3i, 6i, –17 are complex numbers.
- 16. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number. Complex Numbers Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i.
- 17. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number. Complex Numbers Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.
- 18. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number. Complex Numbers Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0. Any real number a is also complex because a = a + 0i hence –17 = –17 + 0i.
- 19. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0. Any real number a is also complex because a = a + 0i hence –17 = –17 + 0i. Complex Numbers (Addition and subtraction of complex numbers) Treat the "i" as a variable when adding or subtracting complex numbers.
- 20. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number Complex Numbers (Addition and subtraction of complex numbers) Treat the "i" as a variable when adding or subtracting complex numbers. Example C. (7 + 4i) + (5 – 3i) Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0. Any real number a is also complex because a = a + 0i hence –17 = –17 + 0i.
- 21. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number Complex Numbers (Addition and subtraction of complex numbers) Treat the "i" as a variable when adding or subtracting complex numbers. Example C. (7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0. Any real number a is also complex because a = a + 0i hence –17 = –17 + 0i.
- 22. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number Complex Numbers (Addition and subtraction of complex numbers) Treat the "i" as a variable when adding or subtracting complex numbers. Example C. (7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0. Any real number a is also complex because a = a + 0i hence –17 = –17 + 0i.
- 23. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number Complex Numbers (Addition and subtraction of complex numbers) Treat the "i" as a variable when adding or subtracting complex numbers. Example C. (7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i (7 + 4i) – (5 – 3i) Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0. Any real number a is also complex because a = a + 0i hence –17 = –17 + 0i.
- 24. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number Complex Numbers (Addition and subtraction of complex numbers) Treat the "i" as a variable when adding or subtracting complex numbers. Example C. (7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i (7 + 4i) – (5 – 3i) = 7 + 4i – 5 + 3i Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0. Any real number a is also complex because a = a + 0i hence –17 = –17 + 0i.
- 25. A complex number is a number of the form a + bi where a and b are real numbers, a is called the real part and bi is called the imaginary part of the complex number Complex Numbers (Addition and subtraction of complex numbers) Treat the "i" as a variable when adding or subtracting complex numbers. Example C. (7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i (7 + 4i) – (5 – 3i) = 7 + 4i – 5 + 3i = 2 + 7i Example B. 5 – 3i, 6i, –17 are complex numbers. The imaginary part of 5 – 3i is –3i. The real part of 6i is 0. Any real number a is also complex because a = a + 0i hence –17 = –17 + 0i.
- 26. (Multiplication of complex numbers) Complex Numbers
- 27. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers
- 28. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i)
- 29. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2
- 30. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21
- 31. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i
- 32. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i The conjugate of (a + bi) is (a – bi) and vice–versa.
- 33. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i The conjugate of (a + bi) is (a – bi) and vice–versa. The most important complex number multiplication formula is the product of a pair of conjugate numbers.
- 34. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i (Conjugate Multiplication) The conjugate of (a + bi) is (a – bi) and vice–versa. The most important complex number multiplication formula is the product of a pair of conjugate numbers.
- 35. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i (Conjugate Multiplication) The nonzero conjugate product is (a + bi)(a – bi) = a2 + b2 which is always positive. The conjugate of (a + bi) is (a – bi) and vice–versa. The most important complex number multiplication formula is the product of a pair of conjugate numbers.
- 36. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i Example E. (4 – 3i)(4 + 3i) The conjugate of (a + bi) is (a – bi) and vice–versa. The most important complex number multiplication formula is the product of a pair of conjugate numbers. (Conjugate Multiplication) The nonzero conjugate product is (a + bi)(a – bi) = a2 + b2 which is always positive.
- 37. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i Example E. (4 – 3i)(4 + 3i) = 42 + 32 = 25 The conjugate of (a + bi) is (a – bi) and vice–versa. The most important complex number multiplication formula is the product of a pair of conjugate numbers. (Conjugate Multiplication) The nonzero conjugate product is (a + bi)(a – bi) = a2 + b2 which is always positive.
- 38. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i Example E. (4 – 3i)(4 + 3i) = 42 + 32 = 25 (5 – 7i)(5 + 7i) The conjugate of (a + bi) is (a – bi) and vice–versa. The most important complex number multiplication formula is the product of a pair of conjugate numbers. (Conjugate Multiplication) The nonzero conjugate product is (a + bi)(a – bi) = a2 + b2 which is always positive.
- 39. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i Example E. (4 – 3i)(4 + 3i) = 42 + 32 = 25 (5 – 7i)(5 + 7i) = (5)2 + 72 The conjugate of (a + bi) is (a – bi) and vice–versa. The most important complex number multiplication formula is the product of a pair of conjugate numbers. (Conjugate Multiplication) The nonzero conjugate product is (a + bi)(a – bi) = a2 + b2 which is always positive.
- 40. (Multiplication of complex numbers) To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result. Complex Numbers Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i Example E. (4 – 3i)(4 + 3i) = 42 + 32 = 25 (5 – 7i)(5 + 7i) = (5)2 + 72 = 54 The conjugate of (a + bi) is (a – bi) and vice–versa. The most important complex number multiplication formula is the product of a pair of conjugate numbers. (Conjugate Multiplication) The nonzero conjugate product is (a + bi)(a – bi) = a2 + b2 which is always positive.
- 41. Complex Numbers (Division of Complex Numbers)
- 42. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.
- 43. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify
- 44. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i)
- 45. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)*
- 46. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32
- 47. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2iExample F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 4 + 3i
- 48. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 12 – 8i – 9i + 6i2
- 49. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 12 – 8i – 9i + 6i2 –6
- 50. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 12 – 8i – 9i + 6i2 –6 6 – 17i = 25 6 25 17i –
- 51. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 12 – 8i – 9i + 6i2 –6 6 – 17i = 25 6 25 17i – Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.
- 52. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 12 – 8i – 9i + 6i2 –6 6 – 17i = 25 6 25 17i – Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers. Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.
- 53. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 12 – 8i – 9i + 6i2 –6 6 – 17i = 25 6 25 17i – Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers. To find b2 – 4ac first: a = 2, b = –2, c = 3, Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.
- 54. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 12 – 8i – 9i + 6i2 –6 6 – 17i = 25 6 25 17i – Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers. To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20. Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions.
- 55. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 12 – 8i – 9i + 6i2 –6 6 – 17i = 25 6 25 17i – Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers. To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20. x = 2 ± –20 4 Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions. Hence
- 56. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 12 – 8i – 9i + 6i2 –6 6 – 17i = 25 6 25 17i – Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers. To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20. x = 2 ± –20 4 = 2 ± 2–5 4 Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions. Hence
- 57. Complex Numbers (Division of Complex Numbers) To divide complex numbers, we write the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator. 3 – 2i 4 + 3i Example F. Simplify Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom. (3 – 2i) (4 + 3i) = (4 – 3i) (4 – 3i)* 42 + 32 = 25 12 – 8i – 9i + 6i2 –6 6 – 17i = 25 6 25 17i – Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers. To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20. x = 2 ± –20 4 = 2 ± 2–5 4 = 2(1 ± i5) 4 = 1 ± i5 2 Using the quadratic formula, we can solve all 2nd degree equations and obtain their complex number solutions. Hence
- 58. Powers of i The powers of i go in a cycle as shown below:
- 59. The powers of i go in a cycle as shown below: i -1 = i2 Powers of i
- 60. The powers of i go in a cycle as shown below: i -1 = i2 -i = i3 Powers of i
- 61. The powers of i go in a cycle as shown below: i -1 = i2 -i = i3 1 = i4 Powers of i
- 62. The powers of i go in a cycle as shown below: i = i5 -1 = i2 -i = i3 1 = i4 Powers of i
- 63. Complex Numbers Exercise D. Divide by rationalizing the denominators. 2 + 3i i24. 3 – 4i i25. 3 – 4i i26. 1 + i 1 – i27. 2 – i 3 – i28. 3 – 2i 2 + i29. 2 + 3i 2 – 3i 30. 3 – 4i 3 – 2i 31. 3 – 4i 2 + 5i 32. 33. Is there a difference between √4i and 2i?
- 64. The powers of i go in a cycle as shown below: i = i5 -1 = i2 = i6 .. -i = i3 1 = i4 Powers of i
- 65. The powers of i go in a cycle as shown below: i = i5 -1 = i2 = i6 .. -i = i3 = i7 .. 1 = i4 Powers of i
- 66. The powers of i go in a cycle as shown below: i = i5 -1 = i2 = i6 .. -i = i3 = i7 .. 1 = i4 = i8 .. Powers of i
- 67. The powers of i go in a cycle as shown below: i = i5 = i9 .. -1 = i2 = i6 .. -i = i3 = i7 .. 1 = i4 = i8 .. Powers of i
- 68. The powers of i go in a cycle as shown below: i = i5 = i9 .. -1 = i2 = i6 .. -i = i3 = i7 .. 1 = i4 = i8 .. Example H. Simplify i59 Powers of i
- 69. The powers of i go in a cycle as shown below: i = i5 = i9 .. -1 = i2 = i6 .. -i = i3 = i7 .. 1 = i4 = i8 .. Example H. Simplify i59 59 = 4*14 + 3, Powers of i
- 70. The powers of i go in a cycle as shown below: i = i5 = i9 .. -1 = i2 = i6 .. -i = i3 = i7 .. 1 = i4 = i8 .. Example H. Simplify i59 59 = 4*14 + 3, hence i59 = i4*14+3 Powers of i
- 71. The powers of i go in a cycle as shown below: i = i5 = i9 .. -1 = i2 = i6 .. -i = i3 = i7 .. 1 = i4 = i8 .. Example H. Simplify i59 59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3 Powers of i
- 72. The powers of i go in a cycle as shown below: i = i5 = i9 .. -1 = i2 = i6 .. -i = i3 = i7 .. 1 = i4 = i8 .. Example H. Simplify i59 59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3 Powers of i
- 73. The powers of i go in a cycle as shown below: i = i5 = i9 .. -1 = i2 = i6 .. -i = i3 = i7 .. 1 = i4 = i8 .. Example H. Simplify i59 59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3 = 114 i3 Powers of i
- 74. The powers of i go in a cycle as shown below: i = i5 = i9 .. -1 = i2 = i6 .. -i = i3 = i7 .. 1 = i4 = i8 .. Example H. Simplify i59 59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3 = 114 i3 = i3 = -i Powers of i
- 75. Quadratic Formula and Complex Numbers and From example G, the solutions of 2x2 – 2x + 3 = 0 are x = 1 + i5 2 x = 1 – i5 2 because b2 – 4ac = –20 < 0.
- 76. Quadratic Formula and Complex Numbers and From example G, the solutions of 2x2 – 2x + 3 = 0 are x = 1 + i5 2 x = 1 – i5 2 because b2 – 4ac = –20 < 0. Therefore we have a complex conjugate pair as solutions.
- 77. Quadratic Formula and Complex Numbers and From example G, the solutions of 2x2 – 2x + 3 = 0 are x = 1 + i5 2 x = 1 – i5 2 In general, for ax2 + bx + c = 0 with a, b, and c real numbers, and b2 – 4ac < 0, because b2 – 4ac = –20 < 0. Therefore we have a complex conjugate pair as solutions.
- 78. Quadratic Formula and Complex Numbers x = –b +b2 – 4ac 2a x = –b –b2 – 4ac 2a and From example G, the solutions of 2x2 – 2x + 3 = 0 are x = 1 + i5 2 x = 1 – i5 2 and In general, for ax2 + bx + c = 0 with a, b, and c real numbers, and b2 – 4ac < 0, then its two roots: because b2 – 4ac = –20 < 0. are of the form A + Bi and A – Bi, a conjugate pair. Therefore we have a complex conjugate pair as solutions.
- 79. Quadratic Formula and Complex Numbers x = –b +b2 – 4ac 2a x = –b –b2 – 4ac 2a and If a, b, and c are real numbers, then the complex roots* for ax2 + bx + c = 0 are a complex conjugates pair. ( * if b2 – 4ac < 0). From example G, the solutions of 2x2 – 2x + 3 = 0 are x = 1 + i5 2 x = 1 – i5 2 and In general, for ax2 + bx + c = 0 with a, b, and c real numbers, and b2 – 4ac < 0, then its two roots: because b2 – 4ac = –20 < 0. are of the form A + Bi and A – Bi, a conjugate pair. Therefore we have a complex conjugate pair as solutions.
- 80. Quadratic Formula and Complex Numbers x = –b +b2 – 4ac 2a x = –b –b2 – 4ac 2a and If a, b, and c are real numbers, then the complex roots* for ax2 + bx + c = 0 are a complex conjugates pair. ( * if b2 – 4ac < 0). From example G, the solutions of 2x2 – 2x + 3 = 0 are x = 1 + i5 2 For example, if x = i is a solution of #x2 + # x + # = 0, then automatically x = – i is the other root (# real numbers.) x = 1 – i5 2 and In general, for ax2 + bx + c = 0 with a, b, and c real numbers, and b2 – 4ac < 0, then its two roots: because b2 – 4ac = –20 < 0. are of the form A + Bi and A – Bi, a conjugate pair. Therefore we have a complex conjugate pair as solutions.
- 81. Complex Numbers In what sense are the complex numbers, numbers? Real numbers are physically measurable quantities (or the lack of such quantities in the case of the negative numbers). Theoretically, we can forever improve upon the construction of a stick with length exactly 2. But how do we make a stick of length 3i, or a cookie that weighs 3i oz? Well, we can’t. Imaginary numbers and complex numbers in general are not physically measurable in the traditional sense. Only the real numbers, which are a part of the complex numbers, are tangible in the traditional sense. Complex numbers are directional measurements. They keep track of measurements and directions, i.e. how much and in what direction (hence the two– component form of the complex numbers). Google the terms “complex numbers, 2D vectors” for further information.
- 82. Complex Numbers Exercise A. Write the complex numbers in i’s. Combine the following expressions. 1. 2 – 2i + 3 + √–4 2. 4 – 5i – (4 – √–9) 3. 3 + 2i + (4 – i√5) 4. 4 – 2i + (–6 + i√3) 5. 4 – √–25 – (9 – √–16) 6. 11 – 9i + (–7 + i√12) 7. ½ – (√–49)/3 – (3/4 – √–16) Exercise B. Do by inspection. 8. (1 – 2i)(1 + 2i) 9. (1 + 3i)(1 – 3i) 10. (2 + 3i)(2 – 3i) 11. (3 – 4i)(3 + 4i) 12. (9 + i√3)(9 – √3i) 13. (7 – i√5)(7 + i√5) 14. (9 + i√3) (7 – i√5)(9 – i√3) (7 + i√5) 15. (√3 + i√3) (√7 – i√5)(√3 – i√3)(√7 + i√5) Exercise C. Expand and simplify. 16. (1 – 3i)(1 + 2i) 17. (2 + 3i)(1 – 3i) 18. (2 + 3i)(3 – 2i) 19. (4 – 3i)(3 – 4i) 20. (5 + 3i)(5 + 3i) 21. (1 – i)2 22. (2 + 3i)2 23. (5 + 2i)2
- 83. Complex Numbers Exercise D. Divide by rationalizing the denominators. 2 + 3i i24. 3 – 4i i25. 3 + 4i i26. 1 + i 1 – i27. 2 – i 3 – i28. 3 – 2i 2 + i29. 2 + 3i 2 – 3i 30. 3 – 4i 3 – 2i 31. 3 – 4i 2 + 5i 32. Simplify 33. i92 38. Find a and b if (a + bi) 2 = i. 34. i –25 36. i 205 37. i –102 39. Is there a difference between √4i and 2i?
- 84. Complex Numbers (Answers to odd problems) Exercise A. 1. 5 3. 7 + (2 – √5) i Exercise B. 9. 10 11. 5 Exercise C. 17. 11 – 3i 19. – 25i 5. –5 – i 7. –1/4 + 5/3 i 13. 54 15. 72 21. – 2i 23. 21 + 20i Exercise D. – 4 – 3i25. 27. 33. 1 i 29. 1 5 (4 – 7i) 31. 1 13 (17 – 6i) 37. – 1 39. There is no difference