Number line

In elementary mathematics, a number line is a picture of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely. The metaphorical association between numbers and points on the line links arithmetical operations on numbers to geometric relations between points, and provides a conceptual scaffold for learning mathematics.

The number line is initially used to teach addition and subtraction of integers, especially involving negative numbers. As students progress, more kinds of numbers can be placed on the line, including fractions, decimal fractions, square roots, and transcendental numbers such as the circle constant π: Every point of the number line corresponds to a unique real 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. An inequality between numbers corresponds to a left-or-right order relation between points. Numerical intervals are associated to geometrical segments of the line. Operations and functions on numbers correspond to geometric transformations of the line. Wrapping the line into a circle relates modular arithmetic to the geometric composition of angles. Marking the line with logarithmically spaced graduations associates multiplication and division with geometric translations, the principle underlying the slide rule. In analytic geometry, coordinate axes are number lines which associate points in a geometric space with tuples of numbers, so geometric shapes can be described using numerical equations and numerical functions can be graphed.

In advanced mathematics, the number line is usually called the real line or real number line, and is a geometric line isomorphic to the set of real numbers, with which it is often conflated; both the real numbers and the real line are commonly denoted R or ⁠${\displaystyle \mathbb {R} }$⁠. The real line is a one-dimensional real coordinate space, so is sometimes denoted R¹ when comparing it to higher-dimensional spaces. The real line is a one-dimensional Euclidean space using the difference between numbers to define the 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 numbers.