This question revolves around using integers (−1, 0, 1, 2, 3) or simple fractions (½, ⅓, ⅗) vs. real numbers (−1.254, 42.72) in teaching concepts, assigning homework, and preparing tests for math, science, or engineering. For the rest of this question, I will call integer or simple fractions nice and real numbers ugly.
For the sake of simplicity, let’s say that you are teaching a math class, and the first topic is basic addition. The first time that you teach it, I would assume that you would want to teach it using nice numbers. For example, using 2 + 2 = 4 would be preferred over 1.234 + 5.678 = 6.912. Sometimes you can get lost in the weeds of the calculations ("just plug these numbers in here and get the answer out") and completely miss the concepts. While concepts are important, it is important for the students to be able to apply the concepts for more complicated problems. While part of me thinks that learning concepts should be the same for nice numbers and ugly numbers, my personal experience says that there is a difference (perhaps just a small one) between these two.
In order to facilitate better learning and better application of course material to real-world problems, should you also include homework with ugly number inputs and answers? How about tests? During my engineering studies, it seemed like there were lots of problems that had nice inputs and/or answers. Most of the questions did not have really ugly answers. Is this typically done to make it better for the students learning, or is this done to make grading easier? Perhaps calculator use can also influence the type of number being used too. Overall, it would be nice to understand why professors and/or teachers often select nice numbers for assignments.
If it would help to know, the main drive of this question is that I would like to automate some of the homework or maybe even tests for classes. I would like to be able to generate multiple versions of homework or tests so that students can’t simply copy answers from each other. If I am generating homework, it might be tricky to find nice solutions vs ugly solutions. I think I have a method for automatic grading, so that is not a problem. The main thing that I want to maintain is a good learning experience for students.
Note on π and other irrational numbers: For my studies, π was of course in lots of the problems, and this technically makes problems have answers that are irrational. For most problems, it is acceptable to include the symbol π in the answer instead of including the numerical form in calculations. These problems could be still written nicely with implied multiplication like 2π or 3π/5.