Home > Real line

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. 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. First, since the real numbers are totally ordered, they carry an order topology. 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 equal to the order topology.

As a topological space, the real line is a topological manifold of dimension 1. It is paracompact and second-countable as well as contractible and locally compact. It also has a standard differentiable structure on it, making it a differentiable manifold. (Up to diffeomorphismIn mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. Here is definition Given two differentiable manifolds M and N a bijective map from M to N is called a diffeomorphism if both and its inverse are smooth. Two manifolds M and N a, there is only one differentiable structure that the topological space supports.) 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 fieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil R of real numbers (that is, over itself) of dimension 1. It has a standard inner product, making it an Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers. (The inner product is simply ordinary multiplicationArithmetic In its simplest form, multiplication is a quick way of adding identical numbers. The result of multiplying numbers is called a product''. The numbers being multiplied are called coefficients or factors and individually as the multiplicand and m 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 algebraLinear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is wi, since vector spaces were first studied over R.

R is also a premier example of a ring, even a field. 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. 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.