Date: Spring 202014
Which is greater: The amount of numbers (like 0.123738) from 0 to 1 or from 0 to 2? Or are they the same?
It is a little strange, but true. There are sets of infinite numbers that are larger/smaller than other sets of infinite numbers. One of the easier to grasp is the set of integers (natural numbers = 1, 2, 3, 4, …. ) There are an infinite number of these “counting” numbers. However, the set of decimal numbers, for example:
(1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, …. ) is also infinite but is larger than the set of integers, since this set contains all the integers and the set of “half integers” 1.5, 2.5 etc. Another example, is the set of rational numbers p/q compared to irrational numbers. Every rational number p/q will eventually end in a repeating decimal, but irrational numbers never ends in a repeating decimal. So the set of irrational numbers is larger than the set of rational numbers. You can find a detailed discussion of this strange behavior on the web site: http://en.wikipedia.org/wiki/Infinity
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Update: November 2011