Yet another math column in the NY Times by Cornell mathematician Steven Strogatz. This week it's titlted Division and it's Discontents. He makes the point that the natural numbers (1,2,3...) are good enough if all we want to do is count (addition and multiplication). He covered this in the From Fish to Infinity and Rock Groups columns. Once we start subtracting, however, we need to add negative numbers to our counting numbers to form the integers (...-2,-1,0,1,2...). The was dealt with in The Enemy of My Enemy column.
Division, now dealt with in this column, creates an entire new set of numbers - the rational numbers (they can be expressed as a ratio like 1/2 or 22/7). Turns out that this infinite set of rational numbers is actually the exception - most numbers are actually irrational - different levels of infinity (I'm sure Strogatz will discuss the concept of infinity in a later column).
This is probably where many people's understanding of math starts to fall apart. Concepts start to become less concrete and outside of our day-to-day experience.