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 |
Manipulative 1: Number line. Created with GeoGebra. |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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