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When we have a set $A$, it is conventional to denote the cardinality of $A$ as $|A|$. When we have some number $n$, it is conventional to denote the absolute value of said number as $|n|$.

My question is whether there's any 'deeper' reason for this: is there some mathematical connection between the two concepts that explains why the two operators are denoted the same way?

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We tend to use bars to denote some concept of 'size', whether it be cardinality, absolute value, determinant, or a norm. For cardinality, other accepted notations include $\#A$ and $card(A)$.

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  • $\begingroup$ Also Cantor's notation for cardinality, $\overline{\overline A}$. $\endgroup$ – bof Nov 11 '13 at 16:36
  • $\begingroup$ @bof I think that notation might have fallen out of favour with modern authors, but yes that's another alternative :). $\endgroup$ – Dan Rust Nov 11 '13 at 16:49
  • $\begingroup$ Fair enough - I usually use $\#(A)$ to denote set cardinality to avoid ambiguity. $\endgroup$ – Newb Nov 11 '13 at 17:45
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For a finite set S, its cardinality (being a non-negative integer) is sometimes denoted by 𝑛S, or at least it was when I was in primary (junior) school in the 1970's.

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