Explore the fundamental distinction between pure and applied geometry in this 39-minute lecture from the Universal Hyperbolic Geometry series. Delve into the relationship between rational and decimal numbers, challenging the notion that real numbers accurately model the continuum. Examine the logical shortcomings of common mathematical concepts like the square root of 2, pi, and e. Discover a more sensible approach to understanding the hierarchy of continuums, based on rational numbers and concrete constructions. Learn about the differences between exact and approximate solutions in geometry, using examples of circles, lines, and polynomial curves. Gain insights into the continuum problem and its significance in mathematics.