Question: What Is A Point

Question: What is a point

 

Answer:

In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, meaning that a point cannot be defined in terms of previously defined objects.

Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". In two-dimensional Euclidean space, a point is represented by an ordered pair (x, y) of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by an ordered triplet (x, y, z) with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms, (a1, a2, … , an) where n is the dimension of the space in which the point is located.

Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points; As an example, a line is an infinite set of points of the form {\displaystyle \scriptstyle {L=\lbrace (a_{1},a_{2},...a_{n})|a_{1}c_{1}+a_{2}c_{2}+...a_{n}c_{n}=d\rbrace }} \scriptstyle {L = \lbrace (a_1,a_2,...a_n)|a_1c_1 + a_2c_2 + ... a_nc_n = d \rbrace}, where c1 through cn and d are constants and n is the dimension of the space. Similar constructions exist that define the plane, line segment and other related concepts. A line segment consisting of only a single point is called a degenerate line segment.

In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, that any two points can be connected by a straight line. This is easily confirmed under modern extensions of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts known at the time. However, Euclids postulation of points was neither complete nor definitive, and he occasionally assumed facts about points that did not follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these assumptions.