On Nov 25, 2012, at 8:23 PM, Jonathan Crabtree <firstname.lastname@example.org> wrote:
> What is the best answer to the simple question: > > "Where does the rational number line start?" > > Yes it goes on forever in both directions and we can all agree on that. Yet does that mean a line that goes forever in two directions does not have any starting point? > > For on-topic discussion purposes, I'm talking about a single rational number line and not about the real continuous line or complex number plane. > > So where does the rational number line start? >
It goes on forever in both directions. :)
At least with a number line, you can talk about it going on forever from zero, in opposite directions, like two oppositely directed rays. In fact, the term "directed number" is (was) used in algebra texts to indicate signed numbers.
Isn't formal mathematics the ultimate destination to this discussion? Don't we have to formally define "start" and "end" and several other things as well?
I assume that "start" would be the least rational number and "end" the greatest. Using that definition, a "line" has neither a start nor an end. A "ray" however, would have a start but not an end. You could say that if we pick a particular rational number (like in the case of zero above) that we divide the line into two rays, have a common start but no ends.
> Or do we tell a child it doesn't start anywhere because it doesn't end anywhere?
I think this would be false. A line doesn't have a "start" by definition. The fact that it doesn't have an "end" (which is also by definition) plays no deciding part in it not having a start. A ray has a start and no end.
How about "Can you have an end but no start?" I am thinking of something like the set of rational numbers greater than zero and less than OR EQUAL to 1, indicated as (0,1]. In this case we have something like a "start" but it isn't really the same thing. Instead of "start" we would call zero the "lower bound" because, unlike our definition of "start" above, a lower bound doesn't have to be in the set, it just has to be lower or equal to any member of the set. I guess we would be more specific and call zero the "greatest lower bound". In any event, it isn't a "start" because it isn't contained in the set.
PS: When was the parenthesis / bracket set notation introduced?