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MathematicsMathematics (often abbreviated to maths or, in American English, math) is commonly defined as the study of patterns of structure, change, and space. In the modern view, it is the investigation of axiomatically defined abstract structures using formal logic as the common framework, although some contest that this is necessary.The specific structures investigated often have their origin in the natural sciences, most commonly in physics, but mathematicians also define and investigate structures for reasons purely internal to mathematics, because the structures may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations. Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science. The word "mathematics" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning". Historically, the major disciplines within mathematics arose out of the need to do calculations in commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space and change. The study of structure starts with numbers, firstly the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by the familiar numbers. The physically important concept of vector, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space. The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space, but later also generalized to non-Euclidean geometries which play a central role in general relativity. Several long standing questions about ruler and compass constructions were finally settled by Galois theory. The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of coordinate system, smoothness and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations. Group theory investigates the concept of symmetry abstractly and provides a link between the studies of space and structure. Topology connects the study of space and the study of change by focusing on the concept of continuity. Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for doing just that. The central concept used to describe a changing variable is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods to solve these are studied in the field of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. For several reasons, it is convenient to generalise to the complex numbers which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for quantum mechanics among many other things. Many phenomena in nature can be described by dynamical systems and chaos theory deals with the fact that many of these systems exhibit unpredictable yet deterministic behavior. In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic and model theory were developed. When computers were first conceived, several essential theoretical concepts were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, information theory and algorithmic information theory. Many of these questions are now investigated in theoretical computer science. Discrete mathematics is the common name for those fields of mathematics useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and is used in all sciences. Numerical analysis investigates the methods of efficiently solving various mathematical problems numerically on computers and takes rounding errors into account.
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An alphabetical list of mathematical topics is available; together with the "Watch links" feature, this list is useful to track changes in mathematics articles. The following list of subfields and topics reflects one organizational view of mathematics.
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a number of young men came for their vacations to their homes in
the Grammar School private pupils, called openly at the rectory on
by Flora and Hilda, ruined the walks precisely as Messrs. Chalton
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The reason for this secrecy was never explained to Rosalie and
a thing she was not able to tell to her mother, and Rosalie was in
also the form of mysterious and vaguely alarming perils during the
with people. One of the party would often cry, "Look! Who's this?"
they were walking by themselves. Sometimes they would break right
sent on ahead to a turning and told to come back at once if anybody
was. Sometimes she was posted to keep watch while the girls and
She got to know by sometimes rushing in with warnings that Flora
Then when they got home, they would rush up to their room and wash
young men took their leave at the end of a walk there would be long
"Well, say you were--" or "Look here, we'll say we were--" and
to."
It all had the effect of making Rosalie feel unhappy and rather
mother?" She was always told, and only told, "Because father doesn't
given, and Rosalie, ceaselessly disturbed by the concealment, could
that she could imagine; and she was thus immensely taken aback when
"The girls have no chance of ever meeting men in this infernal
pertained to Ibbotsfield. Some question of expenses had started.
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