Show Notes
Joseph Bennish returns to take us beyond the rational numbers we usually use to numbers that have been given names that indicate they're crazy or other-worldly. The Greeks were shocked to discover irrational numbers, violating their geometric view of the world. But later it was proved that any irrational number can be approximated remarkably well by a relatively simple fraction. The transcendental numbers were even more mysterious and were not even proved to exist until the 19th century.
More from The Art of Mathematics
Robin Whitty hosts the website theoremoftheday.org, which presents a concise poster about a different theorem each day, with explanation, graphics, and links. He has been doing this for 21 years. Theorems include one by an avant-garde musician that relates to group theory, and an explanation of the intermediate value theorem applied to pancakes. We also discuss how AI might change the role of mathematicians.
Ted Dintersmith, author of Aftermath: The Life-Changing Math that Schools Won't Teach You, argues that much of what we learn in math can be outsourced to machines. That frees people to understand the mathematical ideas that underlie much of modern life. An example is estimation, such as political polls. Understanding the various metrics that fill the news requires creativity coupled with logic. Learning to ask the right questions is more important than, say, learning to do closed-form integrals.
Josh Cole discussed Zeno's paradox that says motion is not possible since you must go half-way, and then half of the remaining distance, and so on, infinitely. Aristotle and others struggled with this, using ideas like "potential infinity" that hint at ideas that would appear in calculus hundreds of years later. The ancient Greeks were focused on the counting numbers, so they expect things to happen in succession. In particular, the Arrow Paradox asks at what point in time does an arrow move.
Joseph Bennish discusses Euler’s formula, which involves pi, e, the imagery i, 0 and 1, a beautiful formula that unites disparate types of numbers. We can think of e raised to an exponent as compound interest or a function with a remarkable property. We can extend the properties we expect from exponentials to imaginary numbers, which gives us periodicity instead of the usual steadily rising exponential growth.
Aris Winger, Math Professor and Executive Director of the National Association of Mathematicians, has experienced first hand how math can save students' lives by uplifting them. Our education system can move beyond workbooks and help students, all students, think crisper and understand what's happening in the world.