The late Adrien Douady (1935–2006) created a magnificent presentation of complex numbers, which would later be featured in the film Dimensions, in chapters 5 and 6. The film, produced by Étienne Ghys, Aurélien Alvarez and Jos Leys (2008), is available online.
With nothing but a blackboard, a piece of chalk, a ruler, a set square and a protractor, Adrien Douady, in chapter 5, brings complex numbers to life before our eyes. With him, a child would grasp Argand's brilliant idea, through which the points of the plane became numbers in their own right. They would see clearly who this famous number i is, whose square equals –1, and would not hesitate to add points, to multiply them just as with ordinary numbers. They would come to realize that linking algebra and geometry is, as Adrien Douady puts it, "one of the most beautiful ideas in mathematics". They would understand why these numbers, not really so complex after all, have taken over the scientific world, giving reality to incredible representations such as stereographic projection or to enchanting worlds such as that of fractals.
They would also see, on a grand scale, how the complex plane can be transformed in chapter 6. Adrien Douady, a highly pedagogical narrator, would explain the geometric transformations to them, staging his own image on screen: the dilation that sends z to, for example, z / 2 merely shrinks the images. The mathematician then moves on to the one that sends z to iz, the one that "rotates the picture", then to (1 + i) z, then to z², where "squaring is going to stretch things out", continuing with increasingly elaborate transformations, but ones introduced and detailed step by step, until reaching the complex exponential.
