To Simplify, or Not To Simplify? A Lesson from Medieval Iraq

Students of elementary algebra are often told to simplify their fractions in the final answer to a test question. For example, \[ \frac{2n+2}{4}\] ought to be simplified to  \[ \frac{n+1}{2}.\]

Should students question why they have to do that, our answer—perhaps like yours—would be a combination of the following arguments:

•  Math is complicated enough; we always want an answer to be as simple as possible;
•  If there is another step in the problem, it will be easier for you to have a simpler starting point for the next step;
•  In case you need to remember the answer, it's easier to memorize a less complicated formula.

All of the above lines of reasoning are generally sound advice; however, there are cases in which these lines of reasoning are actually faulty. The purpose of this short note is to give such an example, and we additionally will point out a surprising connection of this theme with medieval Arabic mathematics.