Editing Number line (section)

{{Short description|Line formed by the real numbers}}
{{About|the mathematical concept|the typesetting practice|Printer's key}}
[[File:Number line with x smaller than y.svg|thumb|upright=1.5|The order of the natural numbers shown on the number line]]

A '''number line''' is a graphical representation of a [[straight line]] that serves as spatial representation of [[number]]s, usually graduated like a [[ruler]] with a particular [[origin (geometry)|origin]] point representing the number [[zero]] and evenly spaced marks in either direction representing [[integers]], imagined to extend infinitely. The association between numbers and [[point (geometry)|points]] on the line links [[elementary arithmetic|arithmetical]] operations on numbers to [[geometry|geometric]] relations between points, and provides a conceptual framework for learning mathematics.

In [[elementary mathematics]], the number line is initially used to teach [[addition]] and [[subtraction]] of integers, especially involving [[negative number]]s. As students progress, more kinds of numbers can be placed on the line, including [[fraction]]s, [[decimal fraction]]s, [[square roots]], and [[transcendental number]]s such as the [[pi|circle constant {{mvar|π}}]]: Every point of the number line corresponds to a unique [[real number]], and every real number to a unique point.<ref>{{cite book | last1=Stewart | first1=James B. | last2 = Redlin | first2 = Lothar | last3=Watson | first3=Saleem | authorlink=James Stewart (mathematician) | title=College Algebra | publisher=[[Brooks Cole]]  | year=2008 | edition = 5th | pages=13&ndash;19 | isbn=978-0-495-56521-5}}</ref>

Using a number line, numerical concepts can be interpreted geometrically and geometric concepts interpreted numerically. An [[inequality (mathematics)|inequality]] between numbers corresponds to a left-or-right order relation between points. Numerical [[interval (mathematics)|intervals]] are associated to geometrical [[line segment|segments]] of the line. Operations and [[Function (mathematics)|functions]] on numbers correspond to [[geometric transformations]] of the line. Wrapping the line into a [[circle]] relates [[modular arithmetic]] to the geometric composition of [[angle]]s. Marking the line with [[logarithm|logarithmically]] spaced graduations associates [[multiplication]] and [[division (mathematics)|division]] with geometric [[translation (geometry)|translations]], the principle underlying the [[slide rule]]. In [[analytic geometry]], [[coordinate axis|coordinate axes]] are number lines which associate points in a geometric space with [[tuple]]s of numbers, so geometric shapes can be described using numerical [[equation]]s and numerical functions can be [[Graph of a function|graphed]].

In advanced mathematics, the number line is usually called the '''real line''' or '''real number line''', and is a geometric line [[isomorphism|isomorphic]] to the [[set (mathematics)|set]] of real numbers, with which it is often conflated; both the real numbers and the real line are commonly denoted {{math|'''R'''}} or {{tmath|\R}}. The real line is a one-[[dimension]]al [[real coordinate space]], so is sometimes denoted {{math|'''R'''<sup>1</sup>}} when comparing it to higher-dimensional spaces. The real line is a one-dimensional [[Euclidean space]] using the difference between numbers to define the [[metric space|distance between points]] on the line. It can also be thought of as a [[vector space]], a [[metric space]], a [[topological space]], a [[measure space]], or a [[linear continuum]]. The real line can be embedded in the [[complex plane]], used as a two-dimensional geometric representation of the [[complex number]]s.