Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linear continuum.

Just like the set of real numbers, the real line is usually denoted by the symbol R (or alternatively, , the letter “R” in blackboard bold). However, it is sometimes denoted R¹ in order to emphasize its role as the first Euclidean space.

This article focuses on the aspects of R as a geometric space in topology, geometry, and real analysis. The real numbers also play an important role in algebra as a field, but in this context R is rarely referred to as a line. For more information on R in all of its guises, see real number.

As a linear continuum

The real line is a linear continuum under the standard < ordering. Specifically, the real line is linearly ordered by <, and this ordering is dense and has the least-upper-bound property.

Number line

In basic mathematics, a number line is a picture of a straight line that serves as abstraction for real numbers, denoted by . Every point of a number line is assumed to correspond to a real number, and every real number to a point.

Often the integers are shown as specially-marked points evenly spaced on the line. Although this image only shows the integers from −9 to 9, the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers.

In advanced mathematics, the expressions real number line, or real line are typically used to indicate the above-mentioned concept that every point on a straight line corresponds to a single real number, and vice versa.

Drawing the number line

A number line is usually represented as being horizontal, but in a Cartesian coordinate plane the vertical axis (or y-axis) is also a number line. According to one convention, positive numbers always lie on the right side of zero, negative numbers always lie on the left side of zero, and arrowheads on both ends of the line are meant to suggest that the line continues indefinitely in the positive and negative directions. Another convention uses only one arrowhead, which indicates the direction in which numbers grow; the line continues indefinitely in the positive and negative directions according to rules of geometry, which define a line without endpoints as infinite line, a line with one endpoint as ray, and a line with two endpoints as line segment.

Real number

In mathematics, a real number is a value that represents a quantity along a continuous line. The adjective real in this context was introduced in the 17th century by Descartes, who distinguished between real and imaginary roots of polynomials.

The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356…, the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as π (3.14159265…, a transcendental number). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and complex numbers include real numbers.