real line

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In elementary mathematics, a number line is a picture of a graduated straight
line Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclide ...
that serves as visual representation of the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s. Every point of a number line is assumed to correspond to a
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
, and every real number to a point. The
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
s are often shown as specially-marked points evenly spaced on the line. Although the image only shows the integers from –3 to 3, the line includes all
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, continuing forever in each direction, and also numbers that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving
negative number In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is inequality (mathematics), less than 0 (number), zero. Negative numbers are often used to represent the magnitude of a loss ...
s. In advanced mathematics, the number line can be called as a real line or real number line, formally defined as the set of all real numbers, viewed as a geometric
space Space is the boundless Three-dimensional space, three-dimensional extent in which Physical body, objects and events have relative position (geometry), position and direction (geometry), direction. In classical physics, physical space is often ...
, namely the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
of dimension one. It can be thought of as a
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
(or affine space), a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
, 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 (or alternatively, $\mathbb$, the letter “ R” in blackboard bold). However, it is sometimes denoted in order to emphasize its role as the first Euclidean space.

# History

The first mention of the number line used for operation purposes is found in John Wallis's ''Treatise of algebra''. In his treatise, Wallis describes addition and subtraction on a number line in terms of moving forward and backward, under the metaphor of a person walking. An earlier depiction without mention to operations, though, is found in John Napier's ''A description of the admirable table of logarithmes'', which shows values 1 through 12 lined up from left to right. Contrary to popular belief, Rene Descartes's original La Géométrie does not feature a number line, defined as we use it today, though it does use a coordinate system. In particular, Descartes's work does not contain specific numbers mapped onto lines, only abstract quantities.Núñez, Rafael (2017). ''How Much Mathematics Is "Hardwired", If Any at All'' Minnesota Symposia on Child Psychology: Culture and Developmental Systems, Volume 38. http://www.cogsci.ucsd.edu/~nunez/COGS152_Readings/Nunez_ch3_MN.pdf pp. 98

# Drawing the number line

A number line is usually represented as being horizontal, but in a Cartesian coordinate plane the vertical axis (y-axis) is also a number line.Introduction to the x,y-plane
"Purplemath" Retrieved 2015-11-13
According to one convention,
positive number In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique thi ...
s always lie on the right side of zero,
negative number In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is inequality (mathematics), less than 0 (number), zero. Negative numbers are often used to represent the magnitude of a loss ...
s 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 the rules of geometry which define a line without endpoints as an ''infinite line'', a line with one endpoint as a ''ray'', and a line with two endpoints as a ''line segment''.

# Comparing numbers

If a particular number is farther to the right on the number line than is another number, then the first number is greater than the second (equivalently, the second is less than the first). The distance between them is the magnitude of their difference—that is, it measures the first number minus the second one, or equivalently the absolute value of the second number minus the first one. Taking this difference is the process of subtraction. Thus, for example, the length of a
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...
between 0 and some other number represents the magnitude of the latter number. Two numbers can be added by "picking up" the length from 0 to one of the numbers, and putting it down again with the end that was 0 placed on top of the other number. Two numbers can be multiplied as in this example: To multiply 5 × 3, note that this is the same as 5 + 5 + 5, so pick up the length from 0 to 5 and place it to the right of 5, and then pick up that length again and place it to the right of the previous result. This gives a result that is 3 combined lengths of 5 each; since the process ends at 15, we find that 5 × 3 = 15. Division can be performed as in the following example: To divide 6 by 2—that is, to find out how many times 2 goes into 6—note that the length from 0 to 2 lies at the beginning of the length from 0 to 6; pick up the former length and put it down again to the right of its original position, with the end formerly at 0 now placed at 2, and then move the length to the right of its latest position again. This puts the right end of the length 2 at the right end of the length from 0 to 6. Since three lengths of 2 filled the length 6, 2 goes into 6 three times (that is, 6 ÷ 2 = 3). File:Number line with x smaller than y.svg, The ordering on the number line: Greater elements are in direction of the arrow. File:Number line with addition of -2 and 3.svg, The difference 3-2=3+(-2) on the real number line. File:Number line with addition of 1 and 2.svg, The addition 1+2 on the real number line File:Absolute difference.svg, The absolute difference. File:Number line multiplication 2 with 1,5.svg, The multiplication 2 times 1.5 File:Number line division 3 with 2.svg, The division 3÷2 on the real number line

# Portions of the number line

The section of the number line between two numbers is called an interval. If the section includes both numbers it is said to be a closed interval, while if it excludes both numbers it is called an open interval. If it includes one of the numbers but not the other one, it is called a half-open interval. All the points extending forever in one direction from a particular point are together known as a ray. If the ray includes the particular point, it is a closed ray; otherwise it is an open ray.

# Extensions of the concept

## Logarithmic scale

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 , then , then , etc. Similarly, one inch to the left of 1, one has , then , etc. This approach is useful, when one wants to represent, on the same figure, values with very different
order of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic d ...
. For example, one requires a logarithmic scale for representing simultaneously the size of the different bodies that exist in the
Universe The universe is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmology, cosmological description of the development of ...
, typically, a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
, an
electron The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...
, an
atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, l ...
, a
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
, a
human Humans (''Homo sapiens'') are the most abundant and widespread species of primate, characterized by bipedality, bipedalism and exceptional cognitive skills due to a large and complex Human brain, brain. This has enabled the development of ad ...
, the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large list of largest lakes and seas in the Solar System, volumes of water can be found throughout the Solar System, only water distributi ...
, 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.

## 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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
s. 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 A Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of number, numerical coordinates, which are the positive and negative numbers, signed distance ...
, 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

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 In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
dense
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
, 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 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 . This statement has been shown to be
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independen ...
of the standard axiomatic system of set theory known as ZFC.

## As a metric space

The real line forms a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
, with the distance function given by absolute difference: : $d\left(x, y\right) = , x - y, .$ The metric tensor is clearly the 1-dimensional Euclidean metric. Since the -dimensional Euclidean metric can be represented in matrix form as the -by- identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1. If and , then the - ball in centered at is simply the open interval . 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 In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point (geometry), po ...
of the real line is equal to one.

## As a topological space

The real line carries a standard
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, 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 are the same. As a topological space, the real line is homeomorphic to the open interval . The real line is trivially a topological manifold of dimension . 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 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 real line is a
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
over the field of real numbers (that is, over itself) of dimension . 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 In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...
.

## 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 , 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

The real line is a one-dimensional subspace of a real algebra ''A'' where R ⊂ ''A''. For example, in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, ...
''z'' = ''x'' + i''y'', the subspace is a real line. Similarly, the algebra of quaternions :''q'' = ''w'' + ''x'' i + ''y'' j + ''z'' k has a real line in the subspace . When the real algebra is a direct sum $A = R \oplus V,$ then a conjugation on ''A'' is introduced by the mapping $v \mapsto -v$ of subspace ''V''. In this way the real line consists of the fixed points of the conjugation.