Continuum

Pronunciation: /kənˈtɪn.ju.əm/ Explain

A continuum is a continuous, unbroken set.^[1] A set is continuous if one can take any two elements of the set and find another element between them. A set is unbroken if one can 'move' from element to element throughout the set without ever leaving the set.

The set of real numbers is a continuum. To see that it is continuous, pick any two numbers, such as 1.53 and 1.54. Is there a number between them? Simply add any decimal digit onto the end of 1.53. 1.531 is between 1.53 and 1.54 (1.53 < 1.531 < 1.54). Table 1 shows that this can be repeated infinitely.

1.53 < 1.531 < 1.54

1.53 < 1.5301 < 1.531

1.53 < 1.53001 < 1.5301

1.53 < 1.530001 < 1.53001

…

Table 1: Continuous real numbers

Click on the blue point and drag it to change the figure. Is there any place on the real number line that does not represent a number?

Manipulative 1 - The Set of Real Numbers is Continuous Created with GeoGebra.

The set of real numbers is unbroken. One go from 1 to 2 without ever leaving the set of real numbers.

References

1. McAdams, David E.. All Math Words Dictionary, continuum. 2nd Classroom edition 20150108-4799968. pg 45. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Péter Komjáth, Vilmos Totik. Problems and Theorems in Classical Set Theory. pp 15-18. Springer. May 2, 2006. Last Accessed 6/25/2018. Buy the book

Cite this article as:

McAdams, David E. Continuum. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/continuum.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
1/5/2010: Added "References". (McAdams, David E.)
11/18/2008: Change manipulative to GeoGebra. Expanded 'More Information' (McAdams, David E.)
4/19/2008: Initial version. (McAdams, David E.)