The history of mathematics extends back millennia. The needs of trade, taxation, and time-keeping drove the development of principles of arithmetic, algebra, and geometry, which had already acquired some sophistication by 5,000 years ago. Perhaps most fundamental to the development of mathematics has been discoveries on the nature of numbers themselves, or what mathematicians refer to as Number Theory. Today’s topic is the history and development of Number Theory, viewed through the lens of numbers and number systems. Our guide to Number Theory is Bryden Cais, professor of mathematics at the University of Arizona and the Director of the Southwest Center for Arithmetic Geometry.  Bryden completed a BA in mathematics at Harvard University in 2002 and a PhD also in mathematics at the University of Michigan in 2007. He was a postdoctoral fellow at McGill University, a visiting scholar at Universität Bielefeld, and a professor at the University of Wisconsin, Madison before joining the faculty at the University of Arizona in 2011. We explore the nature and history of different number systems, highlight the obstacles that mathematicians and civilizations faced with new concepts of number, and touch on some unsolved problems in modern number theory. A study guide for this episode is available in PDF form HERE, or as LaTeX HERE.