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For instance, is it correct to say that the number of decimals Numbers between $220$ and $289.999...$ and the number of decimal numbers between $297$ and $297.999...$ are equal? Even though the range of the former is $70$ times bigger?

F. A. Mala
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Jeff Lima
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  • You can find a bijection between them, so yes the cardinalities are the same $\mathfrak c$ – Henry Sep 14 '21 at 16:03
  • The mapping $y=(x-297)\cdot 70+220$ bijectively maps $[297, 298)$ to $[220, 290)$, so yes, the "number" (i.e. cardinality) of (the sets of) numbers in those ranges is the same. –  Sep 14 '21 at 16:07
  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Sep 14 '21 at 16:08
  • No matter how small a real interval is, it is uncountable and has the same cardinality as $\mathbb R$. By the way, why did you not use the numbers $290$ and $298$ , but the weird $0.\bar 9$ - period ? – Peter Sep 14 '21 at 16:40
  • The first interval has size $70$ , the second size $1$, hence the first is $70$ times as big as the second, but as mentioned this does not matter anyway. – Peter Sep 14 '21 at 16:46
  • Since someone asked :The reason for the strange number choices is the example is from the library cataloging Dewey Decimal Classification system, which assigns 220 to 289.999... to books on Christianity and one number for other religions, such as 297 to 297.999... to Islam. It's a somewhat contentious issue in library science. – Jeff Lima Sep 14 '21 at 17:14
  • The punchline is that if talking about the set of reals between $297$ and $298$ versus the set of reals between $220$ and $290$, they are of course the same. If talking about the rationals between $297$ and $298$ versus the set of rationals between $220$ and $290$ they are the same... These coming from the definitions of cardinality which is one of the most common ways of comparing "size" of different sets. You may also instead choose to look at the "measure" of the sets according to your favorite measure in which case they can indeed be said to be different. These are different concepts. – JMoravitz Sep 14 '21 at 17:40
  • As for using these numbers feasibly in practice... if one were to stipulate that all Dewey Decimal numbers have at most eight digits after the decimal... then yes, there are more "legal" numbers between $220$ and $290$ than between $297$ and $298$. We could only say the cardinalities were equal in the case of infinitely many digits possible. – JMoravitz Sep 14 '21 at 17:43
  • In practice, for our public library at least, they try to keep the numbers to about 6 digits if possible. Pretty much any base call number can be made longer by adding a number for location and another for type of person. The longest call number I've seen is 362.20869420941 which is mental health (362.2) of poor people (086942) in the British Isles (0941). If the book covered only a specific aspect of mental health, and only a specific location in England, the call number could be several digits longer. But even in theory, not even all 3 digit or 4 digits are "legal" call numbers. – Jeff Lima Sep 15 '21 at 20:08

1 Answers1

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Consider the intervals $[a,b]$ and $[c,d]$.

The function $f:[a,b]\to[c,d]$ given by $f(x)=\big(\frac{c-d}{a-d}\big)x+\big(\frac{ad-bc}{a-b}\big)$ is a bijection.

This means that any two closed intervals are equinumerous.

Interestingly, this means all line segments contain the same number of points, irrespective of their lengths.

F. A. Mala
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