Logarithmic scale

A log-log plot of y = x (blue), y = x² (green), and y = x³ (red).
Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1.

On the number line, the distance between two points is the unit length if and only if the difference of the represented numbers equals 1. Other choices are possible.

One of the most common choices is the logarithmic scale, which is a representation of the positive numbers on a line, such that the distance of two points is the unit length, if the ratio of the represented numbers has a fixed value, typically 10. In such a logarithmic scale, the origin represents 1; one inch to the right, one has 10, one inch to the right of 10 one has 10×10 = 100, then 10×100 = 1000 = 10³, then 10×1000 = 10,000 = 10⁴, etc. Similarly, one inch to the left of 1, one has 1/10 = 10^–1, then 1/100 = 10^–2, etc.

This approach is useful, when one wants to represent, on the same figure, values with very different order of magnitude. For example, one requires a logarithmic scale for representing simultaneously the size of the different bodies that exist in the Universe, typically, a photon, an electron, an atom, a molecule, a human, the Earth, the Solar System, a galaxy, and the visible Universe.

Logarithmic scales are used in slide rules for multiplying or dividing numbers by adding or subtracting lengths on logarithmic scales.

The two logarithmic scales of a slide rule

Combining number lines

A line drawn through the origin at right angles to the real number line can be used to represent the imaginary numbers. This line, called imaginary line, extends the number line to a complex number plane, with points representing complex numbers.

Alternatively, one real number line can be drawn horizontally to denote possible values of one real number, commonly called x, and another real number line can be drawn vertically to denote possible values of another real number, commonly called y. Together these lines form what is known as a Cartesian coordinate system, and any point in the plane represents the value of a pair of real numbers. Further, the Cartesian coordinate system can itself be extended by visualizing a third number line "coming out of the screen (or page)", measuring a third variable called z. Positive numbers are closer to the viewer's eyes than the screen is, while negative numbers are "behind the screen"; larger numbers are farther from the screen. Then any point in the three-dimensional space that we live in represents the values of a trio of real numbers.

As a linear continuum

Each set on the real number line has a supremum.

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.

In addition to the above properties, the real line has no maximum or minimum element. It also has a countable dense subset, namely the set of rational numbers. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is order-isomorphic to the real line.

The real line also satisfies the countable chain condition: every collection of mutually disjoint, nonempty open intervals in R is countable. In order theory, the famous Suslin problem asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to R. This statement has been shown to be independent of the standard axiomatic system of set theory known as ZFC.

As a metric space

The metric on the real line is absolute difference.

An ε-ball around a number a

The real line forms a metric space, with the distance function given by absolute difference:

${\displaystyle d(x,y)=|x-y|.}$

The metric tensor is clearly the 1-dimensional Euclidean metric. Since the n-dimensional Euclidean metric can be represented in matrix form as the n-by-n identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1.

If p ∈ R and ε > 0, then the ε-ball in R centered at p is simply the open interval (p − ε, p + ε).

This real line has several important properties as a metric space:

• The real line is a complete metric space, in the sense that any Cauchy sequence of points converges.
• The real line is path-connected and is one of the simplest examples of a geodesic metric space.
• The Hausdorff dimension of the real line is equal to one.

As a topological space

The real line can be compactified by adding a point at infinity.

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 inherit a metric topology from the metric defined above. The order topology and metric topology on R are the same. As a topological space, the real line is homeomorphic to the open interval (0, 1).

The real line is trivially a topological manifold of dimension 1. Up to homeomorphism, it is one of only two different connected 1-manifolds without boundary, the other being the circle. It also has a standard differentiable structure on it, making it a differentiable manifold. (Up to diffeomorphism, there is only one differentiable structure that the topological space supports.)

The real line is a locally compact space and a paracompact space, as well as second-countable and normal. It is also path-connected, and is therefore connected as well, though it can be disconnected by removing any one point. The real line is also contractible, and as such all of its homotopy groups and reduced homology groups are zero.

As a locally compact space, the real line can be compactified in several different ways. The one-point compactification of R is a circle (namely, the real projective line), and the extra point can be thought of as an unsigned infinity. Alternatively, the real line has two ends, and the resulting end compactification is the extended real line [−∞, +∞]. There is also the Stone–Čech compactification of the real line, which involves adding an infinite number of additional points.

In some contexts, it is helpful to place other topologies on the set of real numbers, such as the lower limit topology or the Zariski topology. For the real numbers, the latter is the same as the finite complement topology.

As a vector space

The bijection between points on the real line and vectors

The real line is a vector space over the field R of real numbers (that is, over itself) of dimension 1. It has the usual multiplication as an inner product, making it a Euclidean vector space. The norm defined by this inner product is simply the absolute value.

As a measure space

The real line carries a canonical measure, namely the Lebesgue measure. This measure can be defined as the completion of a Borel measure defined on R, where the measure of any interval is the length of the interval.

Lebesgue measure on the real line is one of the simplest examples of a Haar measure on a locally compact group.

In real algebras

When A is a unital real algebra, the products of real numbers with 1 is a real line within the algebra. For example, in the complex plane z = x + iy, the subspace {z : y = 0} is a real line. Similarly, the algebra of quaternions

q = w + x i + y j + z k

has a real line in the subspace {q : x = y = z = 0 }.

When the real algebra is a direct sum ${\displaystyle A=R\oplus V,}$ then a conjugation on A is introduced by the mapping ${\displaystyle v\to -v}$ of subspace V. In this way the real line consists of the fixed points of the conjugation.

For a dimension n, the square matrices form a ring that has a real line in the form of real products with the identity matrix in the ring.