A continuum is a continuous, unbroken set. 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.|
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