I, personally, gained a lot of value from having almost all (70%ish) "ugly" problems on my homework, as these can expose any flaws in my understanding that I would otherwise not discover with nice numbers

While I do not think that your question is off-topic here, it is almost certainly better suited for Mathematics Educators. In fact, a duplicate already exists.

For the sake of simplicity, let’s say that […] the first topic is basic addition. – I don’t think this is a valid simplification. Addition is one of the most basic and thus first math concepts that students encounter. It takes children years of primary school to get a good grasp of it. You cannot compare teaching addition to e.g. teaching integration by residuals. Therefore it may actually help to be more specific of what subjects we are talking about.

When I was in my first two semesters, our maths professor did not care about numbers at all. We were supposed to provide a symbolic solution. Calculators were not allowed in the exams. I appreciated that (because at school I spent most of my time during exams triple-checking that I had typed all numbers correctly into my calculator).

Do you allow calculators in your class?

.. math, science, or engineering .. Which of those are you teaching, and at what level?

I'm having trouble finding an actual question here. Is this just asking for discussion? Can make your question obvious? What you are getting here are opinions.

Last spring, due to the coronavirus, I gave an online multiple choice test in probability with answer options of the form a) 0.45323456439109, b) 0.45378987668990, c) 0.4502345987342, d) 0.40674887840. The difference between a correct answer that realizes certain events are not independent, and an incorrect answer that pretends events are independent, can be very small. Incorrect approaches can be reinforced if students always round to answers such as 0.45. Written exams allow grading based on approach, multiple choice do not.

Unless your students are all going to become pure mathematicians, everything which uses the math you are teaching will involve "ugly numbers".

What platform do you intend to use for your tests? E.g., On Blackboard one can make personalized-to-each-student questions with decimal answers, auto-grading, and dialed-to-preference precision in how it grades. Seems great for science/engineering. On the other hand, it cannot do this for symbolic answers, say, with variables or pi.

@Polygnome As for calculators, I think it is impossible to restrict usage for homework. Students will use things much worse than calculators to complete homework. For tests, if the numbers are ugly, it would be not be fair or conducive to good learning to not allow calculators (unless you want the students to learn that their professor is evil). If learning is better with ugly numbers, it would be worth using calculators (simple, non-programmable ones) during tests. Whether or not learning is better with nice numbers is the focus of the question.

@ZachS if learning is better with ugly numbers, that isn't a justification for using them on the test. You shouldn't be learning anything on a test, you should be just demonstrating you know something. Things that are good for learning go in the learning material, IE, homework.

@Wrzlprmft The question may be similar, but the kind of answers received are very different and both are useful. On math educators the answers seem to have more of a math background, whereas here there are more engineers and both perspectives can be important.

Answers with nice numbers are at least easier to grade, but if you overdo it students will start to only ever expect nice numbers in their answer. This may or may not be an unwanted side-effect.

@whn You shouldn't be learning anything on a test — I strongly disagree. Every question is an opportunity to practice. ("Know" is not an on/off switch.)