Imaginary Numbers: Theory, Mathematics, and Possibility

The very idea of imaginary numbers appears at first glance preposterous, like something out of science fiction or the wildest philosophy. A real or ordinary number multiplied by an “imaginary unit” somehow mathematically produces a “complex” number that has both theoretical and practical applications. Rene Descartes, who coined the “imaginary” terminology, deemed them preposterous. And yet, imaginary numbers are indispensable—not only in geometry, number theory, and physics, but also for engineering and design in everything from aerospace to robotics to video games.

How did imaginary and complex numbers develop from their early roots in ancient classical algebra to their manifold applications today? And what are their implications not only for science and technology, but for our very philosophical conceptions of mathematics, and indeed mathematics’ relationship with physical reality?

In this course we will study the mathematics of imaginary numbers, their historical development, and questions they raise in the philosophy of mathematics. Students will trace their history and explore imaginary and complex numbers from an algebraic and geometric points of view. In particular, we will examine the elegant geometric interpretation of imaginary and complex numbers as forming a two-dimensional “complex plane,” which accelerated the acceptance of complex numbers as legitimate mathematical objects.

We will then investigate how applying calculus within this context gave rise to the field of complex analysis—which has proven to be remarkably useful in physics and engineering – and the field of complex dynamics which led to discovery of beautiful mathematical objects such as the Mandelbrot set—and the recognition of fractal geometry in natural phenomena like clouds, plants, geological formations, and more. 

Finally, we’ll discuss not only our overarching questions but what are the further potential theoretical and practical implications of imaginary and complex numbers.