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Complex Numbers Operations

Complex Numbers Operations
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2015-01-07 22:48
Coordinate GeometryComplex NumberGeometric Construction
This example demonstrates all operations with complex numbers in polar and rectangular forms. It covers fundamental concepts such as addition, subtraction, multiplication, and division. Using GeoGebra commands like Point, Intersect, Angle, etc., the geometric meaning of complex number operations is visually illustrated. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Point(xAxis) → N_2
  2. Intersect(xAxis, yAxis) → F_2
  3. Angle(N_2, F_2, z_{14}) → β
  4. Segment(z_{14}, F_2) → m_4
  5. Point(Circle[A, 3]) → B
  6. Segment(A, B) → a
  7. Point(Circle[B, 1.5]) → C
  8. Segment(B, C) → b
  9. Point(Circle[C, 1.5]) → D
  10. Segment(C, D) → c
  11. Point(Circle[D, 1.5]) → E
  12. Segment(D, E) → d
  13. Point(Circle[E, 1.5]) → F
  14. Segment(E, F) → e
  15. Point(Circle[F, 1.5]) → G
  16. Segment(F, G) → f
  17. OrthogonalLine(A, a) → g
  18. Circle(A, 0.75) → h
  19. Intersect(h, g, 2) → H
  20. Circle(H, 0.75) → k
  21. Intersect(k, g, 2) → I
  22. OrthogonalLine(H, g) → i
  23. OrthogonalLine(I, g) → j
  24. OrthogonalLine(B, a) → l
  25. OrthogonalLine(C, b) → m
  26. OrthogonalLine(D, c) → n
  27. OrthogonalLine(E, d) → p
  28. OrthogonalLine(F, e) → q
  29. OrthogonalLine(G, f) → r
  30. Circle(I, 0.75) → s
  31. Intersect(s, g, 2) → J
  32. OrthogonalLine(J, g) → t
  33. Intersect(j, l) → K
  34. Intersect(l, t) → L
  35. Intersect(m, t) → M
  36. Intersect(n, t) → N
  37. Intersect(p, t) → O
  38. Intersect(q, t) → P
  39. Intersect(r, t) → Q
  40. Intersect(j, r) → R
  41. Intersect(j, q) → S
  42. Intersect(j, p) → T
  43. Intersect(j, n) → U
  44. Intersect(j, m) → V
  45. Intersect(i, m) → W
  46. Intersect(i, l) → Z
  47. Intersect(i, n) → A_1
  48. Intersect(i, p) → B_1
  49. Intersect(i, q) → C_1
  50. Intersect(i, r) → D_1
  51. Midpoint(M, K) → E_1
  52. Midpoint(N, V) → F_1
  53. Midpoint(O, U) → G_1
  54. Midpoint(P, T) → H_1
  55. Midpoint(Q, S) → I_1
  56. Midpoint(R, C_1) → J_1
  57. Midpoint(D_1, F) → K_1
  58. Midpoint(C_1, E) → L_1
  59. Midpoint(B_1, D) → M_1
  60. Midpoint(T, A_1) → N_1
  61. Midpoint(U, W) → O_1
  62. Midpoint(A_1, C) → P_1
  63. Midpoint(W, B) → Q_1
  64. Midpoint(V, Z) → R_1
  65. Midpoint(K, H) → S_1
  66. Midpoint(Z, A) → T_1
  67. Polygon(R, Q, P, S) → polígono1, r_1, q_1, p_1, s_1
  68. Polygon(S, K, L, P) → polígono2, s_2, k_1, l_1, p_2
  69. Segment(L, K) → a_1
  70. Segment(L, Q) → b_1
  71. Segment(Q, R) → c_1
  72. Segment(R, K) → d_1
  73. Segment(P, S) → e_1
  74. Midpoint(S, B_1) → U_1
  75. Polygon(I, K, Z, H) → polígono3, i_1, k_2, z_1, h_1
  76. Polygon(H, A, B, Z) → polígono4, h_2, a_2, b_2, z_2
  77. Segment(Z, D_1) → f_1
  78. Segment(R, G) → g_1
  79. Segment(S, F) → j_1
  80. Segment(O, E) → m_1
  81. Segment(N, D) → n_1
  82. Segment(M, C) → t_1
  83. Polygon(K, Z, W, V) → polígono5, k_3, z_3, w, v
  84. Polygon(V, W, A_1, U) → polígono6, v_1, w_1, a_3, u
  85. Polygon(U, A_1, B_1, T) → polígono7, u_1, a_4, b_3, t_2
  86. Polygon(T, B_1, C_1, S) → polígono8, t_3, b_4, c_2, s_3
  87. Polygon(C_1, D_1, R, S) → polígono9, c_3, d_2, r_2, s_4
  88. Polygon(C_1, F, G, D_1) → polígono10, c_4, f_2, g_2, d_3
  89. Polygon(F, E, B_1, C_1) → polígono11, f_3, e_2, b_5, c_5
  90. Polygon(E, D, A_1, B_1) → polígono12, e_3, d_4, a_5, b_6
  91. Polygon(D, C, W, A_1) → polígono13, d_5, c_6, w_2, a_6
  92. Polygon(C, B, Z, W) → polígono14, c_7, b_7, z_4, w_3
  93. Mirror(C_1, g_1) → V_1
  94. Mirror(F, g_1) → F'
  95. Mirror(G, g_1) → G'
  96. Mirror(D_1, g_1) → W_1
  97. Polygon(V_1, F', G', W_1) → polígono10', v_2, f', g', w_4
  98. Mirror(C_1, g_1) → Z_1
  99. Mirror(D_1, g_1) → A_2
  100. Mirror(R, g_1) → R'
  101. Mirror(S, g_1) → S'
  102. Polygon(Z_1, A_2, R', S') → polígono9', z_5, a_7, r', s'
  103. Mirror(Z_1, r') → B_2
  104. Mirror(A_2, r') → C_2
  105. Mirror(R', r') → R''
  106. Mirror(S', r') → S''
  107. Polygon(B_2, C_2, R'', S'') → polígono9'', b_8, c_8, r'', s''
  108. Mirror(K_1, g_1) → D_2
  109. Mirror(D_2, w_4) → E_2
  110. Textfield(z_6) → CasillaDeEntrada1
  111. Textfield(z_7) → CasillaDeEntrada2
  112. Segment(F_2, z_{9}) → q_2
  113. Segment(F_2, z_{10}) → g_3
  114. Segment(z_{11}, F_2) → h_3
  115. Textfield(u_2) → CasillaDeEntrada3
  116. Segment(z_{13}, F_2) → i_3
  117. Point(xAxis) → G_2
  118. Angle(G_2, F_2, z_6) → α
  119. Textfield(n_3) → CasillaDeEntrada4
  120. OrthogonalLine(z_6, xAxis) → l_3
  121. OrthogonalLine(z_7, xAxis) → m_3
  122. OrthogonalLine(z_6, yAxis) → p_3
  123. OrthogonalLine(z_7, yAxis) → q_3
  124. Intersect(q_3, yAxis) → H_2
  125. Intersect(p_3, yAxis) → I_2
  126. Intersect(m_3, xAxis) → J_2
  127. Intersect(l_3, xAxis) → K_2
  128. OrthogonalLine(z_8, xAxis) → r_3
  129. OrthogonalLine(z_8, yAxis) → e_4
  130. Intersect(r_3, xAxis) → L_2
  131. Intersect(e_4, yAxis) → M_2
  132. Segment(F_2, K_2) → f_4
  133. Vector(H_2, z_7) → u_3
  134. Vector(K_2, L_2) → v_3
  135. Vector(J_2, z_7) → u_4
  136. Vector(I_2, M_2) → i_2
  137. Segment(F_2, I_2) → j_2
  138. Segment(z_8, L_2) → l_2
  139. Segment(z_8, M_2) → g_4
  140. Segment(z_6, K_2) → h_4
  141. Segment(z_6, I_2) → i_4
  142. Segment(z_7, H_2) → j_4
  143. Segment(z_7, J_2) → k_4
  144. Segment(F_2, z_6) → l_4
  145. Sequence(((l_4^(1 / n_3) * cos(α / n_3 + (360° * i / n_3))), (l_4^(1 / n_3) * sin(α / n_3 + (360° * i / n_3)))), i, 0, n_3 - 1) → lista1
  146. Sequence((m_4^(1 / n_3); β / n_3 + (360° * i / n_3)), i, 0, n_3 - 1) → lista2
  147. Textfield(z_{14}) → CasillaDeEntrada5
  148. Segment(z_{15}, F_2) → n_4
  149. Angle(N_2, F_2, z_{15}) → γ
  150. Sequence(Segment[((l_4^(1 / n_3) * cos(α / n_3 + (360° * i / n_3))), (l_4^(1 / n_3) * sin(α / n_3 + (360° * i / n_3)))), ((l_4^(1 / n_3) * cos(α / n_3 + (360° * (i + 1) / n_3))), (l_4^(1 / n_3) * sin(α / n_3 + (360° * (i + 1) / n_3))))], i, 0, n_3 - 1) → lista3
  151. Textfield(z_{15}) → CasillaDeEntrada6
  152. Segment(O_2, P_2) → p_4
  153. Segment(Q_2, F_2) → q_4
  154. Angle(N_2, F_2, Q_2) → δ
  155. OrthogonalLine(z_{9}, xAxis) → r_4
  156. OrthogonalLine(z_{9}, yAxis) → t_4
  157. Intersect(r_4, xAxis) → R_2
  158. Intersect(t_4, yAxis) → S_2
  159. Segment(z_{9}, R_2) → e_5
  160. Segment(z_{9}, S_2) → f_5
  161. Segment(F_2, z_{19}) → g_5
  162. Angle(N_2, F_2, z_{19}) → ε
  163. Segment(z_{21}, F_2) → h_5
  164. Angle(N_2, F_2, z_{14}) → ζ
  165. Angle(N_2, F_2, z_{15}) → η
  166. Angle(N_2, F_2, z_{14}) → κ
  167. Angle(N_2, F_2, z_{15}) → λ
  168. Segment(z_{23}, F_2) → i_5
  169. Angle(z_{23}, F_2, z_{14}) → μ
  170. Angle(N_2, F_2, z_{23}) → ν
  171. Segment(z_{24}, F_2) → j_5
  172. Angle(N_2, F_2, z_{24}) → ξ
  173. Sequence(Angle[N_2, (0, 0), (m_4 / 2; (β * i))], i, 1, u_2) → lista4
  174. Sequence((m_4; (β * i)), i, 1, u_2) → lista6
  175. Sequence(Segment[(m_4^(1 / n_3); β / n_3 + (360° * i / n_3)), (m_4^(1 / n_3); β / n_3 + (360° * (1 + i) / n_3))], i, 0, n_3 - 1) → lista7
  176. Sequence(CircumcircleArc[(m_4, 0), (m_4; (β * i)), (m_4; (β * (1 + i)))], i, 1, u_2 - 1) → lista5
  177. OrthogonalLine(z_{9}, xAxis) → k_5
  178. Intersect(xAxis, k_5) → T_2
  179. Segment(T_2, z_{9}) → l_5
  180. Angle(z_{14}, F_2, z_{21}) → θ