This book contains simple explanations to some of the most important theorems and identities in mathematics. They are not intended to be rigorous proofs but rather intuitive guides that take the mystery out of some of the most powerful mathematical facts ever discovered. These important facts are often presented to students with no justification. Students are expected to simply memorize them. This book not only presents the facts but also shows why they must be true in a logical way. It is not meant to be used as a textbook but it could be used as supplement to formal coursework or by someone who is simply interested in the development of advanced mathematics. It begins with a proof of the Pythagorean Theorem and continues with derivations of several commonly used geometric identities and the Quadratic Formula. It then moves on to the Fundamental Theorem of Calculus, the Gradient Theorem, the Curl Theorem, the Divergence Theorem, and Taylor's Theorem. The book concludes with Euler's Formula and some number theory. It also includes four appendixes that explain the small angle approximations, a few important facts from Euclidean geometry, trigonometric derivatives, and hyperbolic trigonometry. Although many of the identities and theorems presented in this book are very powerful, the explanations should be easy to follow for anyone with a basic understanding and genuine interest in mathematics.
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