>>10948723 What we're really talking about is whether or not a number has a multiplicative inverse. The multiplicative inverse for a number of a given set is the number which yields the multiplicative identity when multiplied by the first number.
The multiplicative identity for a given set is the number for which other numbers can be multiplied with and return themselves as an answer.
For the set of reals, 1 is the multiplicative identity (1*1=1, 2*1=2, etc.). The multiplicative inverse is 1/x for those numbers that have a multiplicative inverse in this set, and as you already know, 0 does not have a multiplicative inverse in this set i.e. there is no such number which can be multiplied by 0 to yield the multiplicative identity of 1.
For the set of clock hours, the multiplicative identity is also 1 (1*1=1, 2*1=2, etc.). The multiplicative inverse works a little differently with this set compared to the reals in contrast.
Going back to basics, we remember that multiplication is defined in terms of iterated addition. So with the reals 5*5=25 because 5+5=10, 10+5=15, 15+5=20, and 20+5=25.
But with clock hours, 5*5=1 because 5:00+5 hours=10:00, 10:00+5 hours=3:00, 3:00+5 hours=8:00, and 8:00+5 hours=1:00.
Given these starting premises, you'll find not only do numbers other than 0 not have multiplicative inverses in this set, but in fact there are exactly twice as many numbers in this set that don't have a multiplicative inverse (2,3,4,6,8,9,10, and 12) vs. the quantity of numbers that do have one (1,5,7,11). No matter how many hours you clock-multiply 2,3,4,6,8,9,10, or 12 by, you will never get to 1:00 e.g. 2 clock-multiplied can only ever yield 2,4,6,8,10, or 12 and 3 clock-multiplied can only ever yield 3,6, 9, or 12.
tl;dr Having numbers without multiplicative inverses (i.e. you can't make the answer 1 no matter what you multiply them by) is perfectly valid, and 0 happens to be that sort of case for the reals.
