Which are "real" numbers?

The adjective "real" in the term "real number" causes confusion for many whose mathematics is casual rather than intense.  I like the mathematical definition of a number as real iff it corresponds to a point on the number line -- for this gives the abstract number a geometric counterpart (an attachment to reality) -- but there are others for whom the reality of a number depends on its emotional connections, perhaps used in ways that poet Ginger Andrews uses numbers in the following poem.  

     Divine Mathematics    by Ginger Andrews  

     In her second month of a three-month-long virus,
     which, according to half a dozen fellow victims,
     does not respond to antibiotics, my sister apologizes
     for needing to take her third nap of the day
     on my sofa. Homeless and divorced, she's relieved
     to know that a trip to the doctor most likely wouldn't
     do her any good, especially since she has no insurance
     coverage of any kind, except on her '78 Ford Fairmont,
     with its brand new master cylinder, which thanks to God
     and Les Schwab's low monthly payment plan,
     should be paid for by the end of the year,
     at which time she hopes to get a rotation,
     two new tires, and a badly needed front end alignment,
     all for just under a hundred bucks.

I found Andrews' poem in the online archive for NPR's  Writers Almanac (with poems selected and read by Garrison Keillor); it was read on-the-air on June 20, 2003.  "Divine Mathematics" is from Hurricane Sisters (Story Line Press, 2004).