X2 T01 01 definitions (2011)

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<ul><li> 1. Complex NumbersSolving Quadratics</li></ul> <p> 2. Complex NumbersSolving Quadraticsx2 1 0 3. Complex NumbersSolving Quadratics x2 1 0x 2 1 no real solutions 4. Complex NumbersSolving Quadraticsx2 1 0 x 2 1no real solutionsIn order to solve this equation we define a new number 5. Complex NumbersSolving Quadraticsx2 1 0 x 2 1no real solutionsIn order to solve this equation we define a new numberi 1or i 2 1 i is an imaginary number 6. Complex NumbersSolving Quadraticsx2 1 0 x 2 1no real solutionsIn order to solve this equation we define a new numberi 1ori 2 1 i is an imaginary number x 2 1x 1x i 7. e.g . x 2 3 x 7 0 8. e.g . x 2 3 x 7 0 3 9 28 x2 3 19 2 9. e.g . x 2 3 x 7 0 3 9 28 x2 3 19 2 3 19i 2 10. e.g . x 2 3 x 7 0 3 9 28x2 3 19 2 3 19i 2 3 19i 3 19ixor x 2 2 11. e.g . x 2 3 x 7 0x 2 3x 7 0 3 9 28xOR2 3 19 2 3 19i 2 3 19i 3 19ixor x 2 2 12. e.g . x 2 3 x 7 0x 2 3x 7 0 2 x3 19 3 9 28 0xOR 2 42 3 19 2 3 19i 2 3 19i 3 19ixor x 2 2 13. e.g . x 2 3 x 7 0x 2 3x 7 0 2 x3 19 3 9 28 0xOR 2 422 3 19 3 19 2 x i 0 2 2 4 3 19i 2 3 19i 3 19ixor x 2 2 14. e.g . x 2 3 x 7 0 x 2 3x 7 02x3 19 3 9 28 0xOR2 42 2 3 193 19 2 x i 0 22 4 3 19i319 319 x i x i 0 2 22 22 3 19i 3 19ixor x 2 2 15. e.g . x 2 3 x 7 0 x 2 3x 7 02x3 19 3 9 28 0xOR2 42 2 3 193 19 2 x i 0 22 4 3 19i319 319 x i x i 0 2 22 22 3 19i 3 19ixor x 319 3192 2x i or x i2222 16. All complex numbers (z) can be written as; z = x + iy 17. All complex numbers (z) can be written as; z = x + iyDefinitions; 18. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part 19. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part Re z x Im z y 20. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part Re z x Im z ye.g . z 3 5i 21. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part Re z x Im z ye.g . z 3 5iRe z 3 22. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part Re z x Im z ye.g . z 3 5iRe z 3 Im z 5 23. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary partRe z xIm z y e.g . z 3 5i Re z 3 Im z 5(2) If Re(z) = 0, then z is a pure imaginary number 24. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary partRe z xIm z y e.g . z 3 5i Re z 3 Im z 5(2) If Re(z) = 0, then z is a pure imaginary number e.g . 3i,6i 25. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part Re z x Im z y e.g . z 3 5i Re z 3 Im z 5(2) If Re(z) = 0, then z is a pure imaginary number e.g . 3i,6i(3) If Im(z) = 0, then z is a real number 26. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part Re z x Im z y e.g . z 3 5i Re z 3 Im z 5(2) If Re(z) = 0, then z is a pure imaginary number e.g . 3i,6i(3) If Im(z) = 0, then z is a real number 3e.g . , , e,4 4 27. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary partRe z xIm z y e.g . z 3 5i Re z 3 Im z 5(2) If Re(z) = 0, then z is a pure imaginary number e.g . 3i,6i(3) If Im(z) = 0, then z is a real number 3e.g . , , e,4 4(4) Every complex number z = x + iy, has a complex conjugate 28. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary partRe z xIm z y e.g . z 3 5i Re z 3 Im z 5(2) If Re(z) = 0, then z is a pure imaginary number e.g . 3i,6i(3) If Im(z) = 0, then z is a real number 3e.g . , , e,4 4(4) Every complex number z = x + iy, has a complex conjugate z x iy 29. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary partRe z xIm z y e.g . z 3 5i Re z 3 Im z 5(2) If Re(z) = 0, then z is a pure imaginary number e.g . 3i,6i(3) If Im(z) = 0, then z is a real number3e.g . , , e,44(4) Every complex number z = x + iy, has a complex conjugate z x iye.g . z 2 7i 30. All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary partRe z xIm z y e.g . z 3 5i Re z 3 Im z 5(2) If Re(z) = 0, then z is a pure imaginary number e.g . 3i,6i(3) If Im(z) = 0, then z is a real number3e.g . , , e,44(4) Every complex number z = x + iy, has a complex conjugate z x iye.g . z 2 7i z 2 7i 31. Complex Numbers x + iy 32. Complex Numbers x + iy Real Numbersy=0 33. Complex Numbers x + iyImaginary NumbersReal Numbersy0 y=0 34. Complex Numbers x + iyImaginary NumbersReal Numbersy0 y=0 Rational Numbers 35. Complex Numbers x + iyImaginary NumbersReal Numbersy0 y=0 Rational Numbers Irrational Numbers 36. Complex Numbers x + iyImaginary Numbers Real Numbersy0y=0 Rational NumbersFractions Irrational Numbers 37. Complex Numbers x + iyImaginary Numbers Real Numbersy0y=0 Rational NumbersFractionsIntegers Irrational Numbers 38. Complex Numbers x + iyImaginary Numbers Real Numbersy0y=0 Rational NumbersFractionsIntegers Naturals Irrational Numbers 39. Complex Numbers x + iyImaginary Numbers Real Numbersy0y=0 Rational NumbersFractionsIntegers Naturals Zero Irrational Numbers 40. Complex Numbers x + iyImaginary Numbers Real Numbersy0y=0 Rational NumbersFractionsIntegers Naturals Zero Negatives Irrational Numbers 41. Complex Numbers x + iyImaginary Numbers Real Numbersy0y=0 Rational Numbers Pure FractionsIntegersImaginary NumbersNaturalsx 0, y 0 Zero Negatives Irrational Numbers 42. Complex Numbersx + iy Imaginary NumbersReal Numbers y0 y=0 Rational NumbersPureFractionsIntegers Imaginary Numbers Naturals x 0, y 0 Zero NegativesNOTE: Imaginary numbers cannot be ordered Irrational Numbers 43. Basic Operations 44. Basic OperationsAs i a surd, the operations with complex numbers are the same as surds 45. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition 46. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition4 3i 8 2i 47. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition4 3i 8 2i 4 i 48. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 i 49. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 50. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 12 5i 51. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 12 5iMultiplication 52. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 12 5iMultiplication 4 3i 8 2i 53. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 12 5iMultiplication 4 3i 8 2i 32 8i 24i 6i 2 54. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 12 5iMultiplication 4 3i 8 2i 32 8i 24i 6i 2 32 32i 6 55. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 12 5iMultiplication 4 3i 8 2i 32 8i 24i 6i 2 32 32i 6 26 32i 56. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 12 5iMultiplicationDivision (Realising The Denominator) 4 3i 8 2i 32 8i 24i 6i 2 32 32i 6 26 32i 57. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 12 5iMultiplicationDivision (Realising The Denominator) 4 3i 8 2i 4 3i 8 2i 32 8i 24i 6i 2 8 2i 8 2i 32 32i 6 26 32i 58. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 12 5iMultiplicationDivision (Realising The Denominator) 4 3i 8 2i 4 3i 8 2i 32 8i 24i 6i 2 8 2i 8 2i 32 32i 6 32 8i 24i 6 26 32i 64 4 59. Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition Subtraction4 3i 8 2i 4 3i 8 2i 4 i 12 5iMultiplicationDivision (Realising The Denominator) 4 3i 8 2i 4 3i 8 2i 32 8i 24i 6i 2 8 2i 8 2i 32 32i 6 32 8i 24i 6 26 32i 64 4 38 16i68 19 8 i 34 34 60. Exercise 4A; 1 to 16 evens</p>