Explore the fascinating world of hypercomplex numbers in this comprehensive math history lecture. Delve into the 19th-century discovery of quaternions by W. R. Hamilton and octonians by Graves and Cayley, examining how these 4 and 8-dimensional extensions of complex numbers relate to ancient number theory. Investigate the geometric aspects of complex numbers and their higher-dimensional counterparts, uncovering surprising connections to formulas from Diophantus, Fibonacci, and Euler. Learn about the arithmetic and geometric properties of these number systems, their most important characteristics, and their relevance to modern mathematics. Gain insights into the historical development of mathematical concepts and their applications in various fields.