Anumber line is a graphical representation of astraight line that serves as spatial representation ofnumbers, usually graduated like aruler with a particularorigin point representing the numberzero and evenly spaced marks in either direction representingintegers, imagined to extend infinitely. The association between numbers andpoints on the line linksarithmetical operations on numbers togeometric relations between points, and provides a conceptual framework for learning mathematics.
Inelementary mathematics, the number line is initially used to teachaddition andsubtraction of integers, especially involvingnegative numbers. As students progress, more kinds of numbers can be placed on the line, includingfractions,decimal fractions,square roots, andtranscendental numbers such as thecircle constantπ: Every point of the number line corresponds to a uniquereal number, and every real number to a unique point.[1]
Using a number line, numerical concepts can be interpreted geometrically and geometric concepts interpreted numerically. Aninequality between numbers corresponds to a left-or-right order relation between points. Numericalintervals are associated to geometricalsegments of the line. Operations andfunctions on numbers correspond togeometric transformations of the line. Wrapping the line into acircle relatesmodular arithmetic to the geometric composition ofangles. Marking the line withlogarithmically spaced graduations associatesmultiplication anddivision with geometrictranslations, the principle underlying theslide rule. Inanalytic geometry,coordinate axes are number lines which associate points in a geometric space withtuples of numbers, so geometric shapes can be described using numericalequations and numerical functions can begraphed.
In advanced mathematics, the number line is usually called thereal line orreal number line, and is a geometric lineisomorphic to theset of real numbers, with which it is often conflated; both the real numbers and the real line are commonly denotedR or. The real line is a one-dimensionalreal coordinate space, so is sometimes denotedR1 when comparing it to higher-dimensional spaces. The real line is a one-dimensionalEuclidean space using the difference between numbers to define thedistance between points on the line. It can also be thought of as avector space, ametric space, atopological space, ameasure space, or alinear continuum. The real line can be embedded in thecomplex plane, used as a two-dimensional geometric representation of thecomplex numbers.
The first mention of the number line used for operation purposes is found inJohn Wallis'sTreatise of Algebra (1685).[2] 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 inJohn Napier'sA Description of the Admirable Table of Logarithmes (1616), which shows values 1 through 12 lined up from left to right.[3]
Contrary to popular belief,René Descartes's originalLa 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.[4]
A number line is usually represented as beinghorizontal, but in aCartesian coordinate plane the vertical axis (y-axis) is also a number line.[5] The arrow on the line indicates the positive direction in which numbers increase.[5] Some textbooks attach an arrow to both sides, suggesting that the arrow indicates continuation. This is unnecessary, since according to the rules of geometry a line without endpoints continues indefinitely in the positive and negative directions. A line with one endpoint as aray, and a line with two endpoints as aline segment.
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 ofsubtraction.
Thus, for example, the length of aline segment between 0 and some other number represents the magnitude of the latter number.
Two numbers can beadded 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 bemultiplied 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).
The section of the number line between two numbers is called aninterval. 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 aray. If the ray includes the particular point, it is a closed ray; otherwise it is an open ray.
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 thelogarithmic scale, which is a representation of thepositive 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 has10×10 = 100, then10×100 = 1000 = 103, then10×1000 = 10,000 = 104, etc. Similarly, one inch to the left of 1, one has1/10 = 10–1, then1/100 = 10–2, etc.
This approach is useful, when one wants to represent, on the same figure, values with very differentorder of magnitude. For example, one requires a logarithmic scale for representing simultaneously the size of the different bodies that exist in theUniverse, typically, aphoton, anelectron, anatom, amolecule, ahuman, theEarth, theSolar System, agalaxy, and the visible Universe.
Logarithmic scales are used inslide rules for multiplying or dividing numbers by adding or subtracting lengths on logarithmic scales.
A line drawn through the origin at right angles to the real number line can be used to represent theimaginary numbers. This line, calledimaginary line, extends the number line to acomplex number plane, with points representingcomplex numbers.
Alternatively, one real number line can be drawn horizontally to denote possible values of one real number, commonly calledx, and another real number line can be drawn vertically to denote possible values of another real number, commonly calledy. Together these lines form what is known as aCartesian 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 calledz. 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.
The real line is alinear continuum under the standard< ordering. Specifically, the real line islinearly ordered by<, and this ordering isdense and has theleast-upper-bound property.
In addition to the above properties, the real line has nomaximum orminimum element. It also has acountabledensesubset, namely the set ofrational numbers. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element isorder-isomorphic to the real line.
The real line also satisfies thecountable chain condition: every collection of mutuallydisjoint,nonempty openintervals inR is countable. Inorder theory, the famousSuslin problem asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic toR. This statement has been shown to beindependent of the standard axiomatic system ofset theory known asZFC.
The real line forms ametric space, with thedistance function given by absolute difference:
Themetric tensor is clearly the 1-dimensionalEuclidean metric. Since then-dimensional Euclidean metric can be represented in matrix form as then-by-n identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1.
Ifp ∈R andε > 0, then theε-ball inR centered atp is simply the openinterval(p −ε,p +ε).
This real line has several important properties as a metric space:
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The real line carries a standardtopology, which can be introduced in two different, equivalent ways. First, since the real numbers aretotally ordered, they carry anorder topology. Second, the real numbers inherit ametric topology from the metric defined above. The order topology and metric topology onR are the same. As a topological space, the real line ishomeomorphic to the open interval(0, 1).
The real line is trivially atopological manifold ofdimension1. Up to homeomorphism, it is one of only two different connected 1-manifolds withoutboundary, the other being thecircle. It also has a standard differentiable structure on it, making it adifferentiable manifold. (Up todiffeomorphism, there is only one differentiable structure that the topological space supports.)
The real line is alocally compact space and aparacompact space, as well assecond-countable andnormal. It is alsopath-connected, and is thereforeconnected as well, though it can be disconnected by removing any one point. The real line is alsocontractible, and as such all of itshomotopy groups andreduced homology groups are zero.
As a locally compact space, the real line can be compactified in several different ways. Theone-point compactification ofR is a circle (namely, thereal projective line), and the extra point can be thought of as an unsigned infinity. Alternatively, the real line has twoends, and the resulting end compactification is theextended real line[−∞, +∞]. There is also theStone–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 thelower limit topology or theZariski topology. For the real numbers, the latter is the same as thefinite complement topology.
The real line is avector space over thefieldR of real numbers (that is, over itself) ofdimension1. It has the usual multiplication as aninner product, making it aEuclidean vector space. Thenorm defined by this inner product is simply theabsolute value.
The real line carries a canonicalmeasure, namely theLebesgue measure. This measure can be defined as thecompletion of aBorel measure defined onR, 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 aHaar measure on alocally compact group.
WhenA is a unitalreal algebra, the products of real numbers with 1 is a real line within the algebra. For example, in thecomplex planez =x + iy, the subspace {z :y = 0} is a real line. Similarly, the algebra ofquaternions
has a real line in the subspace {q :x =y =z = 0 }.
When the real algebra is adirect sum then aconjugation onA is introduced by the mapping of subspaceV. In this way the real line consists of thefixed points of the conjugation.
For a dimensionn, thesquare matrices form aring that has a real line in the form of real products with theidentity matrix in the ring.
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