Sentence examples for ordering of real from inspiring English sources

We endow X with the natural ordering of real numbers.

Then ( X, ⪯ ) is a partially ordered set under the natural ordering of real numbers.

Example 8 Let X = ( 0, ∞ ) be endowed with the natural ordering of real numbers ≤.

Example 3.1 Let X = [ − 1, 1 ] with the natural ordering of real numbers.

Example 25 Let X = [ 0, 1 ] endowed with the natural ordering of real numbers.

Then ( X, ≤ ) is a partially ordered set with the natural ordering of real numbers.

For instance, it has a projective well ordering of the reals, and it satisfies the GCH.

A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (a,b) is interpreted as a vector from the origin in the Euclidean plane to the point (a,b) in the plane.

In fact, the example of complex numbers is essentially the same (i.e., it is isomorphic) to the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number as representing the ordered pair in the complex plane then we see that the rules for sum and scalar product correspond exactly to those in the earlier example.

In addition to work on his non-Euclidean geometry, he developed a geometric concept of complex numbers as ordered pairs of real numbers.

A closed interval is an ordered pair of real numbers [x, y], with x ⩽ y.