## Division Remainders

Ending a division problem with a "remainder" leaves me with the feeling that the problem hasn't been completed. And in a sense it hasn't. You cant divide bread into "\(6\) remainder \(2\)" pieces, or cash a check for "\(3\) remainder \(4\)" dollars. Generally, something has to be done with the remainder for it to become a useful number. Furthermore, remainders are so elusive in nature. For example, \(12/8\), \(6/4\), and \(3/2\) all have the same numerical value, but as a division problem each has a different remainder. A graders have to be on their toes to recognize that "\(1\) remainder \(4\)", "\(1\) remainder \(2\)", and "\(1\) remainder \(1\)" are all correct answers to the problem "what is \(12\) divided by \(8\)". This confusion can be avoided if remainders are always expressed as fractions. Its really not that difficult for students to learn. (I do it by explaining that placing the remainder over the divisor indicates that this part of the division has yet to be completed.)

The word "remainder" reminds me of something leftover. A lot of leftovers get thrown out. Whole-number remainders should fall into that category.

John Schwarz