Mathematics
Mathematics is the study and practice of sometimes rigorous inference about usually abstract structures. It includes many practical results concerning quantity and measure, such as calculations involving numbers, financial accounting (like balancing a checkbook), geometric construction of building, astronomical calculations, calendar dating, telling time, engineering, physics, chemistry, etc. but also, and more often, abstract issues such as establishing the conditions under which certain kinds of equations and formulas have solutions. Applied mathematics concerns the use of mathematical methods for practical purposes. Pure mathematics involves reasoning about abstract structures, and its study is wholly unpractical. The main areas are:
- Algebra. Broadly, speaking algebra concerns 'addition' and 'multiplication', but in the widest possible sense. The objects that are being added or multiplied can be numbers, as in Number theory, but they can also can be more general structures such as matrices, functions, polynomials, vectors or many others. Concentrating on addition and multiplication does not exclude subtraction or division, since subtraction is formally considered to be addition of an additive inverse and division is considered to be multiplication by a multiplicative inverse. That is, subtracting 3 from 2 is rigorously defined to adding the number -2 to 3. Minus two is called the additive inverse of +2. Similarly, dividing 3 by 2 is formally defined in terms of multiplying 3 by (1/2), where 1/2 is the multiplicative inverse of 2. Abstract algebra is the study of algebraic structures such as groups, rings, and fields.
- Analysis. Analysis is concerned with limits and other infinite processes. This subject includes the theory of limits of sequences and series and all forms of calculus, including the calculus of several variables, vector calculus and tensor calculus. Also included is numerical analysis, the study of error propagation in algorithms carried out to finite precision. Additional topics in analysis include real analysis and complex analysis.
- Geometry and Topology. Geometry was defined by Felix Klein as the study of invariants under groups of transformations. For example, the Euclidean transformations are translation, rotation and reflection. The quantities that are not altered by these transformations are things like angles and distances, so these are the subjects of interest in Euclidean geometry. Other types of transformations, such as the affine transformations, define other types of geometry. Topology is concerned with the connectedness of objects, rather than the distance between them. It is sometimes called 'rubber sheet geometry', as it concerns properties (that is, arcs between nodes in networks) of objects that would be preserved even if a diagram of them were to be stretched or shrunk. Topology began with Leonard Euler's consideration of the Königsberg Bridges Problem, which also introduced Graph Theory. Beck's map of the London Underground in 1933 used a topological distortion of the locations of the subway stations in order to produce a more useful and artistic map. Differential geometry is a specialized field of its own.
- Logic and set theory. All of mathematics can be expressed in terms of sets. Sets are defined by a collection of axioms called the Zermelo-Fraenkel axioms. One of the axioms, the Axiom of Choice, has been the subject of much discussion.
Children are often forced to take mathematics in school even when they already know more than enough to get along in the real world. For some students this takes so much time it interferes with aspects of the students life that are more important such as religious study.
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There are many examples of mathematics in daily life. Even when children play with blocks, they are practicing mathematics.
