Mathematics is the rigorous analysis of abstract structures, including numeric and logical systems. The earliest known beginning of this topic is about 2400 B.C., the date of the oldest extant mathematical tablets.
Mathematics includes many practical results concerning quantity and measure, such as calculations involving numbers, financial accounting, geometric construction of building, astronomical calculations, calendar dating, telling time, engineering, physics, chemistry, etc. but also more abstract issues such as establishing the conditions under which certain kinds of equations and formulas have solutions.
Symbols, Equations, and Theories
Mathematics is expressed with symbols. Some of the most commonly used are the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are symbols used to express our intuitive notion of quantity. Other symbol used in elementary mathematics are the equality ( = ), addition ( + ), subtraction ( - ), multiplication ( x ), less than ( < ), greater than ( > ), etc. More advanced branches of mathematics have their own symbols.
An equation is a mathematical statement that asserts the equality of two expressions. Some equations, like (x + 2 = 5), express the equality of two quantities. Other equations, called differential equations, express the equality of two functions.
A mathematical theory is expressed as a set of sentences, called axioms. These axioms should be self consistent, that is, they must not contradict with each other. From these axioms, new results can be derived adhering strictly to mathematical logic. These derived results are called theorems. It is important to note that, according to the Godel's Incompleteness Theorems, it is impossible to state a self consistent set of axioms from which the whole mathematics can be derived.
Pure and Applied Mathematics
Applied mathematics has its emphasis in applications, and is used extensively in the sciences such as physics, chemistry, medicine, and biology, as well as engineering, mechanics and technology. Economics and information theory also uses applied mathematics. Mathematicians involved in research can and do create new theories, mathematical ideas, and new areas of study simply from their use of applied mathematics to solve various problems.
Pure mathematics is the study of mathematics for its own sake, motivated for reasons other than application. It exhibits a trend towards increasing generality and abstraction.
Branches of Mathematics
Arithmetic is the study of combination of numbers. Its basic operations are addition, subtraction, multiplication and division.
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 is concerned with limits and other infinite processes. This subject includes the theory of limits of sequences and series and all forms of statistics, and 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.
Chaos theory is the study of systems that are highly sensitive to initial conditions. That is, small differences in the initial conditions can produce large variations in the long time behaviour. This is known popularly as the Butterfly effect.
The study of discrete structures, such as graphs, latin squares, and block designs. Discrete mathematics can be studied from a pure, theoretical standpoint, or by studying its applications, such as those to theoretical computer science and combinatorial optimization.
Combinatorics is the study of situations in which elements of a set or sets are combined or permuted in various ways. An example of a combinatorics problem would be "in a group of six men and four women, how many possible ways are there to choose three men and two women?" Derangements are another concept in combinatorics. A derangement is a re-ordering of a set so that no element ends up where it was originally; a derangement problem typically asks how many arrangements are possible for a given set, meeting given conditions.
Game theory is the mathematical study of strategic situations, in which the success of an individual making choices depend on the choices of others.
Geometry is the study of shapes and special relationships. It 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.
Logic is the study of reasoning. It examines general forms that arguments can take, and determines which forms are valid, and which forms are fallacies. In mathematics, it is the study of inferences within one formal language.
Set theory is the mathematical study of collections of objects. It is one of the most fundamental areas of mathematics, since 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
Probability can be viewed as the study of processes whose outcome cannot be predicted with certainty, and all that can be done is calculating the like hood of the different possible outcomes. Statistics is the use of numerical data from a small sample of a population to make inferences about the whole population.
Topology is the study of special properties that are preserved under continuous deformation of objects. Put more simply, it studies the properties that don't change unless you poke a hole in the object. One of the most popular examples is that of a coffee cup that can be continuously deformed into a doughnut. Then, the cup and the doughnut are said to be “topologically equivalent”. 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.
In its more basic sense, trigonometry is the study of the relationships between the sides and the angles of triangles. However, trigonometric functions are widely used outside their original realm of describing triangles. For example, sinusoidal functions (sine and cosine) are used to describe oscillatory motion and waves. In fact, there are extremely deep and fruitful connections among those function, the exponential function, complex numbers, Fourier and Laplace transforms, and signal processing.
- ↑ Davis & Hersh, The Mathematical Experience xi (Mariner Books 1981)