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Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics-but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits.From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics.Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

Mathematics --- Math --- Science --- Study and teaching (Higher) --- Abstract algebra. --- Addition. --- Algebra. --- Algebraic equation. --- Algebraic number. --- Algorithm. --- Arbitrarily large. --- Arithmetic. --- Axiom. --- Binomial coefficient. --- Bolzano–Weierstrass theorem. --- Calculation. --- Cantor's diagonal argument. --- Church–Turing thesis. --- Closure (mathematics). --- Coefficient. --- Combination. --- Combinatorics. --- Commutative property. --- Complex number. --- Computable number. --- Computation. --- Constructible number. --- Continuous function (set theory). --- Continuous function. --- Continuum hypothesis. --- Dedekind cut. --- Dirichlet's approximation theorem. --- Divisibility rule. --- Elementary function. --- Elementary mathematics. --- Equation. --- Euclidean division. --- Euclidean geometry. --- Exponentiation. --- Extended Euclidean algorithm. --- Factorization. --- Fibonacci number. --- Floor and ceiling functions. --- Fundamental theorem of algebra. --- Fundamental theorem. --- Gaussian integer. --- Geometric series. --- Geometry. --- Gödel's incompleteness theorems. --- Halting problem. --- Infimum and supremum. --- Integer factorization. --- Integer. --- Least-upper-bound property. --- Line segment. --- Linear algebra. --- Logic. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Method of exhaustion. --- Modular arithmetic. --- Natural number. --- Non-Euclidean geometry. --- Number theory. --- Pascal's triangle. --- Peano axioms. --- Pigeonhole principle. --- Polynomial. --- Predicate logic. --- Prime factor. --- Prime number. --- Probability theory. --- Probability. --- Projective line. --- Pure mathematics. --- Pythagorean theorem. --- Ramsey theory. --- Ramsey's theorem. --- Rational number. --- Real number. --- Real projective line. --- Rectangle. --- Reverse mathematics. --- Robinson arithmetic. --- Scientific notation. --- Series (mathematics). --- Set theory. --- Sign (mathematics). --- Significant figures. --- Special case. --- Sperner's lemma. --- Subset. --- Successor function. --- Summation. --- Symbolic computation. --- Theorem. --- Time complexity. --- Turing machine. --- Variable (mathematics). --- Vector space. --- Word problem (mathematics). --- Word problem for groups. --- Zermelo–Fraenkel set theory.