What are ordered pairs?

An ordered pair is the intuitive idea that objects can be flipped in different positions in such a way that the order in which we take them make different entities.

This "definition" may sound a little abstract, but a few examples should bring the idea comprehensible.

When we think about the basic Cartesian coordinate system of two axes, we immediately think of two "real number lines" intersecting at 90 degrees.

The figure above shows an example of how we intuitively use ordered pairs when we plot graphs of real functions.

In this example, The function is any abstract one-one (1-1) rule Y = f (x). When the variable x on the X-axis assumes or takes the value a, then the function f assigns the value b on the Y-axis to that choice x= a of on the X-axis.

Hence, we are necessarily and intuitively talking about the ordered pair (a, b). This entity (a, b) is an ordered pair because the function f explicitly and uniquely assigns the value b to the unique value a.

The notion of ordered pair is not limited to the usage of the real  numbers.  We can choose the second entry of the ordered pair to be an imaginary number. In that case, the Y-axis is no longer a real-numbers axis, but an imaginary numbers axis. In that case the ordered pair is simply a complex number.

The ordered pairs are very useful when we deal with transformations, specially transformations of plane figures.

For example, the following transformation, made up of two parametric equations:
transforms a circular area of the plane into a  dome in space.

To dramatize the results, a picture of a cat is shown before this transformation, and after the parametric equations are applied to the cat's photo.

In this example we are implicitly using triplets, that is, ordered pairs of three entries, like (x, y, z). The first two entries of the triplet are for the locations of the points of the cat's photo, and the the third entry of the triplet is for the amount of "deformation" applied to each point of the photo.

Transformations and ordered pairs are very interesting subjects, because they are not so abstract after all. 

Interested in more examples of transformations as in the example above? Then download this free E-Book: The Golden E-Book of Graphs of Mathematical Functions: A selection of some beautiful mathematical surfaces from the domain of the real and the transcomplex numbers system.

Interested in an in-depth development of the foundation of the complex numbers from the standpoint of the ordered pairs? Then download this free E-Book: Foundations of Transcomplex Numbers: An extension of the complex number system to four dimensions.

Or simply remember this link:   http://4DLab.info   as a source of free E-Books and critical articles about mathematics and mathematics.