# Space

Pronunciation: /speɪs/ Explain Figure 1: Graph of y = 2x - 3 in Euclidean 2-space.

A mathematical space is a set of properties governing the interaction of objects contained in the space.

A common example of a space is Euclidean 2-space. An x-y graph exists in Euclidean 2-space. The word Euclidean means that objects have the properties associated with Euclidean geometry, most importantly the fifth postulate which controls the intersection of lines. The 2 in 2-space refers to the fact that this space has two dimensions: length and width.

The term n-space is used to define a space with n dimensions. For example, a 2-space is a space with 2 dimensions. A 3-space is a space with 3 dimensions.

Some spaces that are often used in mathematics are:

• Euclidean n-space: A space following the rules of Euclidean geometry with n dimensions.
• Minkowski space: Euclidean 3-space with time as a fourth dimension. This space was used by Albert Einstein for his theory of Special Relativity.
• Hyperbolic geometry: The difference between Euclidean geometry and Hyperbolic geometry is that, given a line and a point not on the line, in Euclidean geometry there is a unique line passing through the point that are parallel to the given line. In Hyperbolic geometry, there are infinite lines passing through the point that are parallel to the given line.

One attribute that a space can have is a metric space. In a metric space, location can be measured from the origin, and the distance between one location and another can be measured.

1. McAdams, David E.. All Math Words Dictionary, space. 2nd Classroom edition 20150108-4799968. pg 168. Life is a Story Problem LLC. January 8, 2015. Buy the book

• Euclid of Alexandria. Elements. Clark University. 9/6/2018. https://mathcs.clarku.edu/~djoyce/elements/elements.html.

McAdams, David E. Space. 5/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/s/space.html.

### Revision History

5/13/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)