This applet shows the iteration of complex maps of type [math]f(z)=z^2+c[/math], were [math]z,c\in\mathbb{C}[/math], as a generalization of the concepts discussed in the previous chapters.[br][br]Complex numbers [math]z=x+iy\in\mathbb{C}[/math], for [math]x,y\in\mathbb{R}[/math], may be introduced as numbers needed to solve equations of degree 2, [math]ax^2+bx+c=0[/math], when their discriminant [math]\Delta=b^2-4ac[/math] is negative: there exist no real number whose square is negative, so we "invent" a new number, [math]i[/math], such that its square equals [math]-1[/math], i.e. [math]i^2=-1[/math]. Using this definition and algebraic properties, it is easy to find rules for mathematical operations on complex numbers (conjugation and modulus are used to define the reciprocal number and, therefore, division):[br][list][br][*] Addition: [math]z_1+z_2=(x_1+iy_1)+(x_2+iy_2)=(x_1+x_2)+i(y_1+y_2)[/math][br][*] Subtraction: [math]z_1-z_2=(x_1+iy_1)-(x_2+iy_2)=(x_1-x_2)+i(y_1-y_2)[/math][br][*] Multiplication: [math]z_1\cdot z_2=(x_1+iy_1)\cdot(x_2+iy_2)=(x_1\cdot x_2-y_1\cdot y_2)+i(x_1\cdot y_2+x_2\cdot y_1)[/math][br][*] Conjugation: [math]\overline{z}=\overline{(x+iy)}=x-iy[/math][br][*] Modulus: [math]\vert z\vert=\vert x+iy\vert=\sqrt{z\cdot\overline{z}}=\sqrt{x^2+y^2}[/math][br][*] Reciprocal number: [math]z^{-1}=\frac{1}{z}=\frac{1}{x+iy}=\frac{\overline{z}}{\vert z\vert^2}=\frac{x-iy}{x^2+y^2}=\frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}[/math][br][*] Division: [math]\frac{z_1}{z_2}=\frac{x_1+iy_1}{x_2+iy_2}=z_1\cdot\frac{1}{z_2}=(x_1+iy_1)\cdot\left(\frac{x_2}{x_2^2+y_2^2}-i\frac{y_2}{x_2^2+y_2^2}\right)=\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2}+i\frac{x_2y_1-x_1y_2}{x_2^2+y_2^2}[/math][br][/list][br]Since a complex number [math]z=x+iy[/math] is defined in terms of two real numbers, it can be represented by a point on the plane: its abscissa [math]x[/math] is the [i]real part[/i] of [math]z[/math] and its ordinate [math]y[/math] is the [i]immaginary coefficient[/i] (the [i]immaginary part[/i] being [math]iy[/math]).