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# X2 t01 01 arithmetic of complex numbers (2013)

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• No - I'm wrong. Conjugate of (z1-z2) does in fact equal conjugate z1 - conjugate z2. Should perhaps point this out in this lesson.

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• Should point out that in z=x+iy the coefficients x and y are REAL numbers.
Also, should point out that conjugate z1 MINUS conjugate z2 is not always the same as conjugate(z1 minus z2), unlike sums, multiplications and quotients.

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### X2 t01 01 arithmetic of complex numbers (2013)

1. 1. Complex Numbers Solving Quadratics x2 1  0
2. 2. Complex Numbers Solving Quadratics x2 1  0 x 2  1 no real solutions
3. 3. Complex Numbers Solving Quadratics x2 1  0 x 2  1 no real solutions In order to solve this equation we define a new number
4. 4. Complex Numbers Solving Quadratics x2 1  0 x 2  1 no real solutions In order to solve this equation we define a new number i  1 or i 2  1 i is an imaginary number
5. 5. Complex Numbers Solving Quadratics x2 1  0 x 2  1 no real solutions In order to solve this equation we define a new number i  1 or i 2  1 i is an imaginary number x 2  1 x   1 x  i
6. 6. All complex numbers (z) can be written as; z = x + iy
7. 7. All complex numbers (z) can be written as; z = x + iy Definitions; (1) All complex numbers contain a real and an imaginary part
8. 8. All complex numbers (z) can be written as; z = x + iy Definitions; (1) All complex numbers contain a real and an imaginary part Re z   x Im z   y
9. 9. All complex numbers (z) can be written as; z = x + iy Definitions; (1) All complex numbers contain a real and an imaginary part Re z   x Im z   y e.g . z  3  5i
10. 10. All complex numbers (z) can be written as; z = x + iy Definitions; (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
11. 11. All complex numbers (z) can be written as; z = x + iy Definitions; (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
12. 12. All complex numbers (z) can be written as; z = x + iy Definitions; (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 3 e.g . ,  , e,4 4
13. 13. All complex numbers (z) can be written as; z = x + iy Definitions; (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 3 e.g . ,  , e,4 4 (4) Every complex number z = x + iy, has a complex conjugate z  x  iy
14. 14. All complex numbers (z) can be written as; z = x + iy Definitions; (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 3 e.g . ,  , e,4 4 (4) Every complex number z = x + iy, has a complex conjugate z  x  iy z  2  7i e.g . z  2  7i
15. 15. Complex Numbers x + iy
16. 16. Complex Numbers x + iy Real Numbers y=0
17. 17. Complex Numbers x + iy Imaginary Numbers y0 Real Numbers y=0
18. 18. Complex Numbers x + iy Imaginary Numbers y0 Real Numbers y=0 Rational Numbers Irrational Numbers
19. 19. Complex Numbers x + iy Imaginary Numbers y0 Real Numbers y=0 Rational Numbers Fractions Integers Irrational Numbers
20. 20. Complex Numbers x + iy Imaginary Numbers y0 Real Numbers y=0 Rational Numbers Fractions Integers Naturals Zero Irrational Numbers
21. 21. Complex Numbers x + iy Imaginary Numbers y0 Real Numbers y=0 Rational Numbers Fractions Integers Naturals Zero Negatives Irrational Numbers
22. 22. Complex Numbers x + iy Imaginary Numbers y0 Real Numbers y=0 Rational Numbers Pure Imaginary Numbers x  0, y  0 Fractions Integers Naturals Zero Negatives Irrational Numbers
23. 23. Complex Numbers x + iy Imaginary Numbers y0 Real Numbers y=0 Rational Numbers Pure Imaginary Numbers x  0, y  0 Fractions Integers Naturals Zero Negatives NOTE: Imaginary numbers cannot be ordered Irrational Numbers
24. 24. Basic Operations As i a surd, the operations with complex numbers are the same as surds
25. 25. Basic Operations As i a surd, the operations with complex numbers are the same as surds Addition 4  3i    8  2i   4  i
26. 26. Basic Operations As i a surd, the operations with complex numbers are the same as surds Addition Subtraction 4  3i    8  2i  4  3i    8  2i   4  i  12  5i
27. 27. Basic Operations As i a surd, the operations with complex numbers are the same as surds Addition Subtraction 4  3i    8  2i  4  3i    8  2i   4  i  12  5i Multiplication 4  3i  8  2i 
28. 28. Basic Operations As i a surd, the operations with complex numbers are the same as surds Addition Subtraction 4  3i    8  2i  4  3i    8  2i   4  i  12  5i Multiplication 4  3i  8  2i   32  8i  24i  6i 2
29. 29. Basic Operations As i a surd, the operations with complex numbers are the same as surds Addition Subtraction 4  3i    8  2i  4  3i    8  2i   4  i  12  5i Multiplication 4  3i  8  2i   32  8i  24i  6i 2  32  32i  6  26  32i
30. 30. Basic Operations As i a surd, the operations with complex numbers are the same as surds Addition Subtraction 4  3i    8  2i  4  3i    8  2i   4  i  12  5i Multiplication 4  3i  8  2i   32  8i  24i  6i 2  32  32i  6  26  32i Division (Realising The Denominator) 4  3i   8  2i 
31. 31. Basic Operations As i a surd, the operations with complex numbers are the same as surds Addition Subtraction 4  3i    8  2i  4  3i    8  2i   4  i  12  5i Multiplication 4  3i  8  2i   32  8i  24i  6i 2  32  32i  6  26  32i Division (Realising The Denominator) 4  3i   8  2i    8  2i   8  2i   32  8i  24i  6  64  4
32. 32. Basic Operations As i a surd, the operations with complex numbers are the same as surds Addition Subtraction 4  3i    8  2i  4  3i    8  2i   4  i  12  5i Multiplication 4  3i  8  2i   32  8i  24i  6i 2  32  32i  6  26  32i Division (Realising The Denominator) 4  3i   8  2i    8  2i   8  2i   32  8i  24i  6  64  4  38  16i  68  19 8   i 34 34
33. 33. Conjugate Basics 1 z1  z2  z1  z2
34. 34. Conjugate Basics 1 z1  z2  z1  z2  2 z1 z2  z1  z2  z1  z1  3     z2  z2
35. 35. Conjugate Basics 1 z1  z2  z1  z2  2 z1 z2  z1  z2  z1  z1  3     z2  z2  4  zz  x 2  y 2
36. 36. Conjugate Basics 1 z1  z2  z1  z2  4  zz  x 2  y 2  2 z1 z2  z1  z2 1 z  5  z zz  z1  z1  3     z2  z2
37. 37. Conjugate Basics 1 z1  z2  z1  z2  4  zz  x 2  y 2  2 z1 z2  z1  z2 1 z  5  z zz  z1  z1  3     z2  z2 1 e.g.  4  3i 
38. 38. Conjugate Basics 1 z1  z2  z1  z2  4  zz  x 2  y 2  2 z1 z2  z1  z2 1 z  5  z zz  z1  z1  3     z2  z2 1 4  3i  e.g.  4  3i  16  9 4 3   i 25 25
39. 39. Complex Equations If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal i.e. If a1  b1i  a2  b2i then a1  a2 b1  b2
40. 40. Complex Equations If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal i.e. If a1  b1i  a2  b2i then a1  a2 b1  b2
41. 41. Complex Equations If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal i.e. If a1  b1i  a2  b2i then a1  a2 b1  b2 e.g . (i)  2  3i  x  iy    4  2i 
42. 42. Complex Equations If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal i.e. If a1  b1i  a2  b2i then a1  a2 b1  b2 e.g . (i)  2  3i  x  iy    4  2i  2 x  2iy  3ix  3 y  4  2i
43. 43. Complex Equations If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal i.e. If a1  b1i  a2  b2i then a1  a2 b1  b2 e.g . (i)  2  3i  x  iy    4  2i  2 x  2iy  3ix  3 y  4  2i  2 x  3 y  4 and 3 x  2 y  2
44. 44. Complex Equations If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal i.e. If a1  b1i  a2  b2i then a1  a2 b1  b2 e.g . (i)  2  3i  x  iy    4  2i  2 x  2iy  3ix  3 y  4  2i  2 x  3 y  4 and 3 x  2 y  2 4x  6 y  8  9 x  6 y  6
45. 45. Complex Equations If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal i.e. If a1  b1i  a2  b2i then a1  a2 b1  b2 e.g . (i)  2  3i  x  iy    4  2i  2 x  2iy  3ix  3 y  4  2i  2 x  3 y  4 and 3 x  2 y  2 4x  6 y  8  9 x  6 y  6 5x  14 14 x 5
46. 46. Complex Equations If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal i.e. If a1  b1i  a2  b2i then a1  a2 b1  b2 e.g . (i)  2  3i  x  iy    4  2i  2 x  2iy  3ix  3 y  4  2i  2 x  3 y  4 and 3 x  2 y  2 14 16 , y x   5 5 4x  6 y  8  9 x  6 y  6 5x  14 14 x 5
47. 47. ii  z  2iw  4  3i 2 z  iw  3  4i
48. 48. ii  z  2iw  4  3i 2 z  iw  3  4i  z  2iw  4  3i 4 z  2iw  6  8i
49. 49. ii  z  2iw  4  3i 2 z  iw  3  4i  z  2iw  4  3i 4 z  2iw  6  8i  2  5i 3z 2 5 z  i 3 3
50. 50. ii  z  2iw  4  3i 2 z  iw  3  4i 2 5   i  2iw  4  3i 3 3 10 4 2iw   i 3 3 10 4 w  6i 6 2 5   i 3 3  z  2iw  4  3i 4 z  2iw  6  8i  2  5i 3z 2 5 z  i 3 3
51. 51. ii  z  2iw  4  3i 2 z  iw  3  4i  2 5   i  2iw  4  3i 3 3 10 4 2iw   i 3 3 10 4 w  6i 6 2 5   i 3 3 2 5 2 5 w  i , z  i 3 3 3 3 z  2iw  4  3i 4 z  2iw  6  8i  2  5i 3z 2 5 z  i 3 3
52. 52. Cambridge: Exercise 1A; 1 to 10 ace etc, 11bd, 13 to 20 Patel: Exercise 4A; 28 ac