Mentor: Now we should look at the division operation. A good way to visualize what will be going on is to think of a piece of rope. If we have a piece of rope that is 12 units long, how many 4 unit pieces can we cut from it?

Student: If we cut 4 unit pieces of rope from a 12 unit long rope, we will end up with three pieces.

Mentor: Correct. To express this mathematically we would write 12 / 4, or 12 divided by 4. What do you think happens when we divide a number by 1 or 0? Be careful, though, 0 is tricky.

Student: If a piece of rope were r units long we cut off 1 unit pieces at a time, then we could get r pieces. So any number divided by 1 is that number. If you divided rope of length r into pieces that were each 0 units long. . . wait, can a piece of rope without length exist?

Mentor: No, that would be like having an object without mass, it cannot exist. For this reason we say that whenever a number is divided by 0, the result is undefined.

Student: That seems to make sense. Getting back to the first example, though, what happens if there is leftover rope? Say, for instance, we wanted to take a 12 unit long rope and cut off 5 unit long pieces. We could cut off 2 pieces that are 5 units long, but we would be left with one piece that is 2 units long. What do we do with a situation like that?

Mentor: If you end up with an amount leftover that is smaller than the unit we are dividing by, then we call the leftover amount the remainder. In the case you described we would say that 12 divided by 5 equals 2 with remainder 2.