In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, 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 R (or alternatively, the letter “R” in blackboard bold). However, it is sometimes denoted R1 in order to emphasize its role as the first Euclidean space.
This article focuses on the aspects of R as a geometric space in topology, geometry, and real analysis. The real numbers also play an important role in algebra as a field, but in this context R is rarely referred to as a line. For more information on R in all of its guises, see real number.
Read more about Real Line: As A Linear Continuum, As A Metric Space, As A Topological Space, As A Vector Space, As A Measure Space
Famous quotes containing the words real and/or line:
“If we consider what happens in conversation, in reveries, in remorse, in times of passion, in surprises, in the instructions of dreams, wherein often we see ourselves in masquerade,—the droll disguises only magnifying and enhancing a real element, and forcing it on our distinct notice,—we shall catch many hints that will broaden and lighten into knowledge of the secret of nature.”
—Ralph Waldo Emerson (1803–1882)
“A line in long array, where they wind betwixt green islands;
They take a serpentine course—their arms flash in the sun—hark to the musical clank;”
—Walt Whitman (1819–1892)