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

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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.


  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.

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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.)

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