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Please note: The duration of access to a MyLab is set by your instructor for your specific unit of study. Washington Publisher: ISBN: Category: Page: View: Read Now » This tried-and-true text from Allyn Washington builds on the author's highly regarded approach to technical math, while enhancing its pedagogy with full-colour figures and boxes that warn students of Common Errors.
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Download Product Flyer. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as 'rigor'.
Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions, of which many instances have occurred in the history of the subject.
Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs.
Since large computations are hard to verify, such proofs may not be sufficiently rigorous. Axioms in traditional thought were 'self-evident truths', but that conception is problematic. Nonetheless mathematics is often imagined to be as far as its formal content nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change i. In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory foundations , to the empirical mathematics of the various sciences applied mathematics , and more recently to the rigorous study of uncertainty. While some areas might seem unrelated, the Langlands program has found connections between areas previously thought unconnected, such as Galois groups, Riemann surfaces and number theory.
In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed.
Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects.
Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase 'crisis of foundations' describes the search for a rigorous foundation for mathematics that took place from approximately to The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer—Hilbert controversy.
Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework.
Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science, [ citation needed ] as well as to category theory. In the context of recursion theory, the impossibility of a full axiomatization of number theory can also be formally demonstrated as a consequence of the MRDP theorem. Theoretical computer science includes computability theory, computational complexity theory, and information theory.
Computability theory examines the limitations of various theoretical models of the computer, including the most well-known model — the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware.
The study of quantity starts with numbers, first the familiar natural numbers and integers 'whole numbers' and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. The twin prime conjecture and Goldbach's conjecture are two unsolved problems in number theory. As the number system is further developed, the integers are recognized as a subset of the rational numbers 'fractions'.
These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quaternions and octonions.
Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of 'infinity'. According to the fundamental theorem of algebra all solutions of equations in one unknown with complex coefficients are complex numbers, regardless of degree.
Another area of study is the size of sets, which is described with the cardinal numbers. These include the aleph numbers, which allow meaningful comparison of the size of infinitely large sets. Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set.
Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets or structures exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms.
Thus one can study groups, rings, fields and other abstract systems; together such studies for structures defined by algebraic operations constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory.
Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model relations between points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.
The study of space originates with geometry — in particular, Euclidean geometry, which combines space and numbers, and encompasses the well-known Pythagorean theorem. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries which play a central role in general relativity and topology.
Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Convex and discrete geometry were developed to solve problems in number theory and functional analysis but now are pursued with an eye on applications in optimization and computer science.
Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space.
Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th-century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern-day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proved only with the help of computers.
Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on typically infinite-dimensional spaces of functions.
One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations.
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