Mathematics is the study of numbers, sets of points, and various abstract elements, together with relations between them and operations performed on them. Originally mathematics was concerned with the properties of numbers and space, as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or the generalization of these two fields, as in algebra.
Mathematics (from Greek mathema: science, knowledge, learning; mathematikos: fond of learning) is the study of patterns of quantity, structure, change and space. In the modern view, it is the investigation of axiomatically defined abstract structures using formal logic as the common framework. 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, for instance because they realize that the structure provides a unifying generalization for several subfields or a helpful tool in common calculations. Finally, many mathematicians study in the areas that they do for aesthetic reasons - simply because they find the structures they investigate beautiful in and of themselves.
Toward the middle of the 19th century, mathematics increasingly came
to be regarded as the science of relations,
more generally concerned with deductions made in abstract systems.
This latter view encompasses mathematical or symbolic logic, the
science of using symbols to provide an exact theory of logical
deduction and inference based on definitions, axioms, postulates,
and rules for combining and transforming primitive elements into more
complex relations and theorems.
Mathematics is the science of logical reasoning, in which valid conclusions are deduced from a set of basic assumptions, or axioms, It involves a search for truth. It is rigorous and precise.
Mathematics is a tool for problem solving; organizing, simplifying, and interpreting data; and performing calculations that are necessary in subjects such as science, commerce, and industry. The development of modern computers and electronic calculators has enabled mathematicians to solve problems that previously were extremely difficult or impossible to solve.
Mathematics is usually divided into pure mathematics - abstract reasoning based on axioms and rules for maing deductions from them - and applied mathematics, in which mathematical methods are applied to 'real world' problems in engineering, physics, economics, business, navigation, astronomy, chemistry, electronics, computer science, etc. Applied mathematics mathematics concerns itself with the application of mathematical knowledge to other domains.
Some branches of mathematics were developed in order to solve certain physical problems or to explain physical phenomena. In his study of astronomy and astrophysics, Johannes Kepler found it necessary to develop new mathematics.
Mathematical calculations sometimes lead to the discovery of new physical phenomena. Deviations in the motions of Neptune from the predictions of mathematical theory led to the conclusion that an unknown planet existed. Calculations pinpointed the position of this body and led to the discovery of the planet Pluto (1931).
Arithmetic and Algebra
Arithmetic refers generally to the study of the nature and properties of numbers, measurement, and numerical computation (that is, the study of the algorithms of calculation with numbers, the fundamental operations of addition, subtraction, multiplication, and division, as well as raising to powers, and extraction of roots. Algebra is "arithmetic with symbols." Unlike arithmetic, which deals with specific numbers, algebra introduces 'variables' that greatly extend the generality and scope of arithmetic. Algebra may be described as a generalization and extension of arithmetic. | Analysis
Analysis deals with the real and complex numbers and their functions. Analysis is concerned with abstract objects such as sets of numbers, sets of geometric points, or sets of functions that map numbers into numbers or points into points and with the processes, called limit processes, that depend on a measure of closeness between numbers, points, or functions. Analysis uses the concept of a limit. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in general settings. | Calculus
Calculus is concerned with concepts such as the rate of change of one variable quantity with respect to another, the slope of a curve at a prescribed point, the computation of the maximum and minimum values of functions, and the calculation of the area bounded by curves. Evolved from algebra, arithmetic, and geometry, it is the basis of that part of mathematics called analysis. |
Chaos Theory
Established in the 1960s, chaos theory deals with dynamical systems that, while in principle deterministic, have a high sensitivity to initial conditions, because their governing equations are nonlinear. Examples for such systems are the atmosphere, plate tectonics, economics, and population growth. | Fractals
Fractals are a class of complex geometric shapes that commonly exhibit the property of self-similarity. such that a small portion of it can be viewed as a reduced scale replica of the whole. The term fractal is derived from the Latin word fractus ("fragmented," or "broken"). | Game Theory
Game Theory -- the abstract study of games, or the mathematics of competition and cooperation -- analyzes situations in terms of gains and losses of opposing players. Two major theories about modern life have come out of games. The first is probability theory, which was first developed out of games of chance in the 17th century by Blaise Pascal. The strategies used to achieve success on the game board can also be applied in many real-life situations. |
Geometry
Geometry is concerned with the properties of space and of objects in space; e.g. points, lines, surfaces, and solids. In its most elementary form geometry is concerned with such metrical problems as determining the areas and diameters of two-dimensional figures and the surface areas and volumes of solids. The study of plane curves, angles, polygons, and lines is called plane geometry. The study of curves in three-dimensional space such as spheres, cones, cylinders, and polyhedra is called solid geometry. | Graphs
A graph is a two-dimensional representation of data. A typical example would be a graph of some quantity varying with time, e.g. daily temperature. In science, we often use graphs to give us a picture of the relationships between variables. | Infinity
Infinity refers to the concept of limitlessness and unboundedness in size, number or extent. One distinguishes between potential infinity and actual infinity. |
Measurement
Measurement is the determination of the size or magnitude of something. Measurement is not limited to physical quantities, but can extend to quantifying almost any imaginable thing such as degree of uncertainty, consumer confidence, or the rate of increase in the fall in the price of beanie babies. | Logic
Logic is regarded as a branch both of philosophy and of mathematics. A system of logic (or simply "a logic") is a set of rules for reasoning about a given domain. Many different systems of logic have been devised. Such artificial systems of reasoning now find many practical applications in computing. | Number Systems
A number system is any of various sets of symbols and the rules for using them to express quantities as the basis for counting, comparing amounts, performing calculations, determining order, making measurements, representing value, setting limits, abstracting quantities, coding information, and transmitting data. |
