# 1
z1 = 3 - 2*I
z2 = -1 + 2*I
z3 = sqrt(2) - I
z4 = expand(z3*conjugate(z1-z2))
z5 = expand(conjugate(z1+z3)*(z2-z3))
# Befehlsinfo
pretty_print("komplexe Zahl: " + str(z1))
pretty_print("Realteil: " + str(z1.real_part()))
pretty_print("Imaginärteil: " + str(z1.imag_part()))
pretty_print("Komplex Konjugiert: " + str(z1.conjugate()))
pretty_print("Betrag: " + str(z1.abs()))
pretty_print("Argument: " + str(arg(z1)))

# a)
print((z4).real_part())
print((z4).imag_part())
print((z5).real_part())
print((z5).imag_part())
print("")
# b)
print(abs(z4).full_simplify())
print(arg(z4).full_simplify())
print(abs(z5).full_simplify())
print(arg(z5).full_simplify())
print("")

z1 = 3*(cos(225/180*pi())+I*sin(225/180*pi()))
z2 = 2*(cos(150/180*pi())-I*sin(150/180*pi()))
z3 = 6*exp(2/3*pi()*I)
z4 = -4*exp(-5/4*pi()*I)
z5 = 3*(cos(45/180*pi())+I*sin(30/180*pi()))
z6 = 3*(cos(60/180*pi())-I*sin(150/180*pi()))
list_plot([z1, 0], plotjoined=True)+list_plot([z2, 0], plotjoined=True, color='red')+list_plot([z3, 0], plotjoined=True, color='green')+list_plot([z4, 0], plotjoined=True, color='orange')+list_plot([z5, 0], plotjoined=True, color='black')+list_plot([z6, 0], plotjoined=True, color='magenta')+text("z1",(real_part(z1),imag_part(z1)))+text("z2",(real_part(z2),imag_part(z2)))+text("z3",(real_part(z3),imag_part(z3)))+text("z4",(real_part(z4),imag_part(z4)))+text("z5",(real_part(z5),imag_part(z5)))+text("z6",(real_part(z6),imag_part(z6)))

z1 = -3/sqrt(2)-I*3/sqrt(2)
z2 = -sqrt(3)-I
z3 = -3 + I*3*sqrt(3)
z4 = 2*sqrt(2)-I*2*sqrt(2)
z5 = 3/sqrt(2) + I*3/2
z6 = 3/2 - I*3/2
list_plot([z1, 0], plotjoined=True)+list_plot([z2, 0], plotjoined=True, color='red')+list_plot([z3, 0], plotjoined=True, color='green')+list_plot([z4, 0], plotjoined=True, color='orange')+list_plot([z5, 0], plotjoined=True, color='black')+list_plot([z6, 0], plotjoined=True, color='magenta')+text("z1",(real_part(z1),imag_part(z1)))+text("z2",(real_part(z2),imag_part(z2)))+text("z3",(real_part(z3),imag_part(z3)))+text("z4",(real_part(z4),imag_part(z4)))+text("z5",(real_part(z5),imag_part(z5)))+text("z6",(real_part(z6),imag_part(z6)))

plot(arctan(x),x,-10,10,color="red")+plot(arctan(x)+pi(),x,-10,10,color="green")+plot(+pi()/2,x,-10,10,linestyle="dashed",color="black")+plot(+3*pi()/2,x,-10,10,linestyle="dashed",color="black")+plot(arctan(x)+2*pi(),x,-10,0,color="blue",ticks=[1,pi/2], tick_formatter=[1,pi], axes_labels=['$x$','$\\arctan(x)$'])

z10 = 3*cos(pi()/3)-I*3*sin(5*pi()/6) #(1-I)/sqrt(2)
print((z10).real_part())
print((z10).imag_part())
print("")
print(abs(z10).full_simplify())
print(arg(z10).full_simplify())

z11 = -4*sqrt(2) - 14 + I*6
list_plot([z11,0],plotjoined=true)