In mathematics, the real line is simply the set R of real numbers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the real numbers may be described informally in several different ways However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The real line has been studied at least since the days of the ancient Greeks, but it was not rigorously defined until 1872. Before and since that date, it has been a prolific example that has played a significant role in many branches of mathematics.
The real line carries a standard topology which can be introduced in two different, equivalent ways. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. First, since the real numbers are totally ordered, they carry an order topology. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics, an order topology is a certain Topology that can be defined on any Totally ordered set. With respect to this topology, the real line is a linear continuum. In Mathematics, an ordered set, S, is said to be a linear continuum if it satisfies the following properties a S has the Least upper Second, the real numbers can be turned into a metric space by using the metric given by the absolute value
This metric induces a topology on R equivalent to the order topology. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.
The real line is trivially a topological manifold of dimension 1. In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity It is paracompact and second-countable as well as contractible and locally compact. In Mathematics, a paracompact space is a Topological space in which every Open cover admits an open locally finite refinement. In Topology, a second-countable space is a Topological space satisfying the " second Axiom of countability " In Mathematics, a Topological space X is contractible if the Identity map on X is Null-homotopic, i In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks It also has a standard differentiable structure on it, making it a differentiable manifold. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. (Up to diffeomorphism, there is only one differentiable structure that the topological space supports. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable ) Indeed, R was historically the first example to be studied of each of these mathematical structures, so that it serves as the inspiration for these branches of modern mathematics. (Indeed, many of the terms above can't even be defined until R is already in place. )
As a vector space, the real line is a vector space over the field R of real numbers (that is, over itself) of dimension 1. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity It has a standard inner product, making it an Euclidean space. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. (The inner product is simply ordinary multiplication of real numbers. ) As a vector space, it is not very interesting, and thus it was in fact 2-dimensional Euclidean space that was first studied as a vector space. However, we can still say that R inspired the field of linear algebra, since vector spaces were first studied over R. Linear algebra is the branch of Mathematics concerned with
R is also a premier example of a ring, even a field. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division It is in fact a real complete field, and was the first such field to be studied, so that it inspired that branch of abstract algebra as well. In the mathematical area of Order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules However, in such purely algebraic contexts, R is rarely called a "line".
For more information on R in all of its guises, see real number. In Mathematics, the real numbers may be described informally in several different ways
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic