Pronunciation: /kənˈtɪn yu əm/ ?

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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Manipulative 1: Number line. Created with GeoGebra.
The set of real numbers is unbroken. One go from 1 to 2 without ever leaving the set of real numbers. Click on the blue point in manipulative 1 and drag it to change the figure.


  1. continuum. WordNet. Princeton University. (Accessed: 2011-01-08).
  2. Péter Komjáth, Vilmos Totik. Problems and Theorems in Classical Set Theory, pp 15-18. Springer, May 2, 2006. (Accessed: 2010-01-20).

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Cite this article as:

Continuum. 2010-01-05. All Math Words Encyclopedia. Life is a Story Problem LLC.


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Revision History

2010-01-05: Added "References" (McAdams, David.)
2008-11-18: Change manipulative to GeoGebra. Expanded 'More Information' (McAdams, David.)
2008-04-19: Initial version (McAdams, David.)

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