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# Discover the World of Mathematics Through Conic Sections with Conics by Keith Kendig

0$$). This parameter also measures how much the conic section deviates from being a circle, which is the case when$$B^2 - 4AC = -4F$$. Using projective geometry, Kendig shows us how to simplify the study of conic sections by using invariants, which are quantities that do not change under projections or transformations. For example, he shows us how to find the center, the axes, the foci, the directrices and the eccentricity of any conic section using invariants. He also shows us how to construct conic sections using simple tools such as rulers and compasses. One of the most interesting applications of projective geometry to conic sections is the Pascal's theorem, which states that if six points are chosen on any conic section, then the three pairs of opposite sides of the hexagon formed by joining them meet at three points that lie on a straight line. This theorem was discovered by Blaise Pascal when he was only 16 years old, and it is considered one of the most beautiful results in mathematics. Kendig proves this theorem using projective geometry and shows us some of its consequences and generalizations. ## Complex Numbers: The Key to Generalization Another tool that Kendig uses to generalize conic sections is complex numbers. These are numbers that have both a real part and an imaginary part, such as$$a + bi$$, where$$a$$and$$b$$are real numbers and$$i$$is the square root of$$-1$$. Complex numbers allow us to extend the notion of conic sections beyond the real plane and into the complex plane, which is a two-dimensional space where each point represents a complex number. But why do we need complex numbers to study conic sections? Well, it turns out that complex numbers can help us solve some problems that are difficult or impossible to solve using real numbers alone. For example, they can help us find the roots of polynomial equations, which are equations that involve powers of a variable, such as$$x^2 - 5x + 6 = 0$$. Some polynomial equations have no real roots, such as$$x^2 + 1 = 0$$, but they always have complex roots, such as$$x = \pm i$$. Complex numbers can also help us find the intersections of conic sections, which are points where two or more conic sections meet. Some pairs of conic sections have no real intersections, such as two parallel lines or two disjoint circles, but they always have complex intersections, such as$$x = \pm i$$for two parallel lines or$$x = \pm i$$and$$y = \pm i$$for two disjoint circles. of lines. He also shows us how to use complex numbers to find the invariants and symmetries of conic sections, as well as to construct them using conformal mappings, which are mappings that preserve angles and shapes. One of the most fascinating applications of complex numbers to conic sections is the Möbius transformation, which is a mapping that transforms any conic section into any other conic section. This transformation is given by a formula of the form$$z \mapsto \fracaz + bcz + d$$, where$$z$$is a complex variable and$$a, b, c$$and$$d are complex constants. Kendig explains how this transformation works and shows us some of its amazing properties and effects. ## Other Topics: The Key to Enrichment Besides projective geometry and complex numbers, Kendig also explores some other topics related to conic sections that enrich our understanding and appreciation of them. These topics include: - Plane geometry: Kendig revisits some classical results and problems from plane geometry, such as the nine-point circle, the Steiner-Lehmus theorem, the Poncelet's porism and the Napoleon's theorem, and shows how they are related to conic sections. He also introduces some modern concepts and techniques from plane geometry, such as inversion, pedal curves and isoptic curves, and shows how they can be used to study conic sections. - Polynomial equations: Kendig delves deeper into the theory and practice of solving polynomial equations, especially those of degree two or higher. He shows us how to use Vieta's formulas, Descartes' rule of signs, Cardano's formula, Ferrari's method and Galois theory to find the roots of polynomial equations. He also shows us how to use resultants, discriminants and Bezout's theorem to find the intersections of polynomial curves, such as conic sections. - Differential equations: Kendig introduces some basic concepts and methods from differential equations, which are equations that involve derivatives of a function. He shows us how to use separation of variables, integrating factors, linearization and phase portraits to solve differential equations. He also shows us how differential equations can be used to model various phenomena involving conic sections, such as pendulums, harmonic oscillators, planetary motion and electromagnetic fields. ## Conclusion In this article, I have given you a brief overview of Kendig's book Conics, which is a wonderful introduction to the world of mathematics through the lens of conic sections. I have shown you some of the main ideas and features of the book, such as: - How projective geometry can unify conic sections by showing that they are essentially the same curve seen from different perspectives. - How complex numbers can generalize conic sections by extending them beyond the real plane and into the complex plane. - How other topics can enrich conic sections by revealing their connections and applications to various branches of mathematics. I hope you have enjoyed this article and learned something new about conic sections and mathematics. If you want to learn more, I highly recommend you to read Kendig's book, which is full of details, examples, exercises, proofs and illustrations that will challenge and delight you. You can find the book online at or at your local library or bookstore. Here are some FAQs that you might have after reading this article: - Q: Who is Keith Kendig? - A: Keith Kendig is a professor emeritus of mathematics at Case Western Reserve University. He has written several books and articles on mathematics, especially on geometry and algebra. He is also an avid musician and composer. - Q: Who is Mary P. Dolciani? - A: Mary P. Dolciani was a professor of mathematics at Hunter College of the City University of New York. She was an influential educator and author of many textbooks on mathematics. She also established the Dolciani Mathematical Expositions series with a generous gift to the Mathematical Association of America. - Q: What are some other books in the Dolciani Mathematical Expositions series? - A: There are currently 50 books in the series, covering a wide range of topics in mathematics, such as number theory, combinatorics, logic, calculus, topology, geometry, algebra and more. You can find the complete list of books at . - Q: What are some other resources to learn more about conic sections and mathematics? - A: There are many online resources that you can use to learn more about conic sections and mathematics, such as: - Khan Academy: A free online platform that offers video lessons, exercises and quizzes on various topics in mathematics, including conic sections. You can find it at https://www.khanacademy.org/math/geometry/conic-sections. - Wolfram MathWorld: A comprehensive online encyclopedia of mathematics that contains definitions, formulas, examples, proofs and references on various topics in mathematics, including conic sections. You can find it at http://mathworld.wolfram.com/topics/ConicSections.html. - YouTube: A popular online platform that hosts millions of videos on various topics, including mathematics. You can find many videos on conic sections by searching for keywords such as "conic sections", "projective geometry", "complex numbers" and so on.

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