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X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
X2 t01 03 argand diagram (2013)
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X2 t01 03 argand diagram (2013)

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  • 1. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 2 1 -4 -3 -2 -1 -1 -2 -3 1 2 3 4 x (real axis)
  • 2. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 2 1 A -4 -3 -2 -1 1 2 3 4 x (real axis) -1 -2 -3
  • 3. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 2 B = -3i 1 A -4 -3 -2 -1 1 2 3 4 x (real axis) -1 -2 -3 B
  • 4. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 2 B = -3i C = -2 + i C 1 A -4 -3 -2 -1 1 2 3 4 x (real axis) -1 -2 -3 B
  • 5. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 2 B = -3i C = -2 + i C 1 A D=4-i -4 -3 -2 -1 1 2 3 4 x (real axis) -1 D -2 -3 B
  • 6. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 NOTE: Conjugates 2 B = -3i are reflected in the E real (x) axis C = -2 + i C 1 A D=4-i -4 -3 -2 -1 1 2 3 4 x (real axis) E=4+i -1 D -2 -3 B
  • 7. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 NOTE: Conjugates 2 B = -3i are reflected in the E real (x) axis C = -2 + i F C 1 A D=4-i -4 -3 -2 -1 1 2 3 4 x (real axis) E=4+i -1 D F = -(4 - i) -2 = - 4 + i NOTE: Opposites are -3 B rotated 180°
  • 8. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4G NOTE: Multiply 3 A=2 NOTE: Conjugates by i results in 2 B = -3i 90° rotation are reflected in the E real (x) axis C = -2 + i F C 1 A D=4-i -4 -3 -2 -1 1 2 3 4 x (real axis) E=4+i -1 D F = -(4 - i) -2 = - 4 + i NOTE: G = i(4 - i) Opposites are -3 B rotated 180° = 1+4i
  • 9. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4G NOTE: Multiply 3 A=2 NOTE: Conjugates by i results in 2 B = -3i 90° rotation are reflected in the E real (x) axis C = -2 + i F C 1 A D=4-i -4 -3 -2 -1 1 2 3 4 x (real axis) E=4+i -1 D F = -(4 - i) -2 = - 4 + i NOTE: G = i(4 - i) Opposites are -3 B rotated 180° = 1+4i Every complex number can be represented by a unique point on the Argand Diagram.
  • 10. Locus in Terms of Complex Numbers Horizontal and Vertical Lines
  • 11. Locus in Terms of Complex Numbers Horizontal and Vertical Lines y c x Im z   c
  • 12. Locus in Terms of Complex Numbers Horizontal and Vertical Lines y c y x Im z   c k x Re z   k
  • 13. y R Circles zz  R 2 R R R x
  • 14. y R Circles zz  R 2 R R x R e.g.Find and describe the locus of points in the Argand diagram; (i)  z  4  i  z  4  i   49
  • 15. y R Circles zz  R 2 R R x R e.g.Find and describe the locus of points in the Argand diagram; (i)  z  4  i  z  4  i   49 z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49
  • 16. y R Circles zz  R 2 R R x R e.g.Find and describe the locus of points in the Argand diagram; (i)  z  4  i  z  4  i   49 z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49 z z  4( z  z )  iz  iz  (4  i )(4  i )  49 z z  4( z  z )  iz  iz  (4  i )(4  i )  49   z z  4( z  z )  iz  iz  (4  i )(4  i )  49 x 2  y 2  8 x  2 y  16  1  49
  • 17. y R Circles zz  R 2 R R x R e.g.Find and describe the locus of points in the Argand diagram; (i)  z  4  i  z  4  i   49 z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49 z z  4( z  z )  iz  iz  (4  i )(4  i )  49 z z  4( z  z )  iz  iz  (4  i )(4  i )  49   z z  4( z  z )  iz  iz  (4  i )(4  i )  49 x 2  y 2  8 x  2 y  16  1  49 x 2  8 x  16  y 2  2 y  1  49
  • 18. y R Circles zz  R 2 R R x R e.g.Find and describe the locus of points in the Argand diagram; (i)  z  4  i  z  4  i   49 z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49 x 2  8 x  16  y 2  2 y  1  49 z z  4( z  z )  iz  iz  (4  i )(4  i )  49  x  4    y  1 z z  4( z  z )  iz  iz  (4  i )(4  i )  49   z z  4( z  z )  iz  iz  (4  i )(4  i )  49 x 2  y 2  8 x  2 y  16  1  49 2 2  49
  • 19. y R Circles zz  R 2 R R x R e.g.Find and describe the locus of points in the Argand diagram; (i)  z  4  i  z  4  i   49 z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49 x 2  8 x  16  y 2  2 y  1  49 z z  4( z  z )  iz  iz  (4  i )(4  i )  49  x  4    y  1 z z  4( z  z )  iz  iz  (4  i )(4  i )  49   z z  4( z  z )  iz  iz  (4  i )(4  i )  49 x 2  y 2  8 x  2 y  16  1  49 2 2 Locus is a circle centre :  4 ,  1 radius : 7 units  49
  • 20. y R Circles  z    z     R 2 zz  R 2 R R x Locus is a circle centre ω radius R R e.g.Find and describe the locus of points in the Argand diagram; (i)  z  4  i  z  4  i   49 z z  (4  i ) z  (4  i ) z  (4  i )(4  i )  49 x 2  8 x  16  y 2  2 y  1  49 z z  4( z  z )  iz  iz  (4  i )(4  i )  49  x  4    y  1 z z  4( z  z )  iz  iz  (4  i )(4  i )  49   z z  4( z  z )  iz  iz  (4  i )(4  i )  49 x 2  y 2  8 x  2 y  16  1  49 2 2 Locus is a circle centre :  4 ,  1 radius : 7 units  49
  • 21. 1 1 (ii )   1 z z
  • 22. 1 1 (ii )   1 z z z  z  zz
  • 23. 1 1 (ii )   1 z z z  z  zz 2x  x 2  y 2
  • 24. 1 1 (ii )   1 z z z  z  zz 2x  x 2  y 2 x2  2 x  y 2  0 ( x  1) 2  y 2  1
  • 25. 1 1 (ii )   1 z z z  z  zz 2x  x 2  y 2 x2  2 x  y 2  0 ( x  1) 2  y 2  1 Locus is a circle centre: 1,0  radius: 1 unit
  • 26. 1 1 (ii )   1 z z z  z  zz Locus is a circle centre: 1,0  radius: 1 unit 2x  x 2  y 2 x2  2 x  y 2  0 y ( x  1) 2  y 2  1 1 2 x
  • 27. 1 1 (ii )   1 z z z  z  zz Locus is a circle centre: 1,0  radius: 1 unit 2x  x 2  y 2 x2  2 x  y 2  0 y ( x  1) 2  y 2  1 excluding the point (0,0) 1 2 x
  • 28. 1 1 (ii )   1 z z z  z  zz Locus is a circle centre: 1,0  radius: 1 unit 2x  x 2  y 2 x2  2 x  y 2  0 y ( x  1) 2  y 2  1 excluding the point (0,0) 1 2 x Cambridge: Exercise 1C; 1 ace, 2 bdf, 3, 4 ace, 5 bdfh, 6, 10 to 15

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