השורשים של מספר מרוכב
Author: המרכז הארצי למורים למתמטיקה בחינוך העל יסודי

Drag the BLUE point to change the value of the complex number. Drag the blue slider to select the number of roots to display. Roots are shown as RED points.

שורשי המשוואה 

$ z^{n}=0 $

הצג חזקות של השורש

$ z_0^{k} $

חבר בין החזקות של השורש

חבר בין השורשים

הצג שורשים

z^{n}=0

i = (0, 1)
z = (-1, 3)
n = 2 (slider from 2 to 9, step 1)
α = Angle(z)
a = Length(z)
z_0: a^(1 / n) * (cos(α / n) + sin(α / n) * i)
z_1: a^(1 / n) * (cos((α + 360°) / n) + sin((α + 360°) / n) * i)
z_2: a^(1 / n) * (cos((α + (2 * 360)°) / n) + sin((α + (2 * 360)°) / n) * i)
z_3: a^(1 / n) * (cos((α + (3 * 360)°) / n) + sin((α + (3 * 360)°) / n) * i)
z_4: a^(1 / n) * (cos((α + (4 * 360)°) / n) + sin((α + (4 * 360)°) / n) * i)
z_5: a^(1 / n) * (cos((α + (5 * 360)°) / n) + sin((α + (5 * 360)°) / n) * i)
z_6: a^(1 / n) * (cos((α + (6 * 360)°) / n) + sin((α + (6 * 360)°) / n) * i)
z_7: a^(1 / n) * (cos((α + (7 * 360)°) / n) + sin((α + (7 * 360)°) / n) * i)
z_8: a^(1 / n) * (cos((α + (8 * 360)°) / n) + sin((α + (8 * 360)°) / n) * i)
z_{15}: z_0^(2)
z_9: z_0^(3)
z_{10}: z_0^(4)
A = Intersect(xAxis, yAxis)
b = Segment(A, z_{15})
c = Segment(A, z_9)
d = Segment(A, z)
e = Segment(A, z_0)
z_{12}: z_0^(6)
z_{14}: z_0^(8)
f = Segment(z_0, z_{15})
h = Segment(z_9, z_{10})
p = Segment(z_{14}, z)
g = Segment(z_{15}, z_9)
q = Segment(z_{10}, A)
s = Segment(z_{12}, A)
a_1 = Segment(z_{14}, A)
poly1, z0_1, z1_1, z2_1, z3_1, z4_1, z5_1, z6_1, z7_1, z8_1 = Polygon(z_0, z_1, z_2, z_3, z_4, z_5, z_6, z_7, z_8)
o = false
u = false
v = false
text4: "(" + (LaTeX[z]) + ")^{" + (LaTeX[n]) + "}"
text6: "z_1=" + z_1 + ""
text5: "z_0=" + z_0 + ""
text7: "z_2=" + z_2 + ""
text8: "z_3=" + z_3 + ""
text9: "z_4=" + z_4 + ""
text10: "z_5=" + z_5 + ""
text11: "z_6=" + z_6 + ""
text12: "z_7=" + z_7 + ""
text13: "z_8=" + z_8 + ""
roots = false
i_1 = (0, 1)
z_{11} = (-5.528386468878475, 8.297949909851585)
n_1 = 9 (slider from 2 to 9, step 1)
α_1 = Angle(z_{11})
a_2 = Length(z_{11})
z_{13}: a_2^(1 / n_1) * (cos(α_1 / n_1) + sin(α_1 / n_1) * i_1)
z_{23}: a_2^(1 / n_1) * (cos((α_1 + 360°) / n_1) + sin((α_1 + 360°) / n_1) * i_1)
z_{24}: a_2^(1 / n_1) * (cos((α_1 + (2 * 360)°) / n_1) + sin((α_1 + (2 * 360)°) / n_1) * i_1)
z_{25}: a_2^(1 / n_1) * (cos((α_1 + (3 * 360)°) / n_1) + sin((α_1 + (3 * 360)°) / n_1) * i_1)
z_{26}: a_2^(1 / n_1) * (cos((α_1 + (4 * 360)°) / n_1) + sin((α_1 + (4 * 360)°) / n_1) * i_1)
z_{27}: a_2^(1 / n_1) * (cos((α_1 + (5 * 360)°) / n_1) + sin((α_1 + (5 * 360)°) / n_1) * i_1)
z_{28}: a_2^(1 / n_1) * (cos((α_1 + (6 * 360)°) / n_1) + sin((α_1 + (6 * 360)°) / n_1) * i_1)
z_{29}: a_2^(1 / n_1) * (cos((α_1 + (7 * 360)°) / n_1) + sin((α_1 + (7 * 360)°) / n_1) * i_1)
z_{30}: a_2^(1 / n_1) * (cos((α_1 + (8 * 360)°) / n_1) + sin((α_1 + (8 * 360)°) / n_1) * i_1)
z_{16}: z_{13}^(2)
z_{17}: z_{13}^(3)
z_{18}: z_{13}^(4)
z_{19}: z_{13}^(5)
z_{20}: z_{13}^(6)
z_{21}: z_{13}^(7)
z_{22}: z_{13}^(8)