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Consider the risk-reward tradeoff here.

If you keep quiet and use the wrong interpretation of this "basic" concept, you risk spending the rest of your research time producing nonsense, because you did the equivalent of assuming 2 + 3 = 23.

If you ask your supervisor, you risk looking a bit stupid for five minutes.

The choice seems like a no-brainer to me!

I was on both sides of the fence.

As a PhD student, I asked my supervisor several times about concepts which were maybe obvious for people from that specific field, but not for a newcomer.

Like you, I did my homework and explained my supervisor (a great supervisor, by the way) what I needed, what I searched and why what I found was not helpful. Usually within a few moments I was on the right track.

This was mutually beneficial: he also got to understand what students (MSc level in his field he was a teacher for, outside of the PhD program) may find difficult (and changed some of his notes accordingly), and to realize that advances in other fields may help him as well.

He was welcoming such questions.

As someone who tutored two PhD students (cross disciplinary), I always asked them at some point "do you understand?". I specifically told them that I am asking this question so that they do not feel bad about not knowing (especially when I was sensing a disconnect between the eyes and the brain - for whatever reason).

I was also asking them whether they knew what X was, when I was about to discuss X. I did not want to start before we were on the same page.

To summarize: do ask, show that you did search for an answer, and if your thesis director is a good one he could eventually even encourage this.

You are doing a PhD in a difficult subject. If something is hard for you, then it's hard. Other people, perhaps even many other people, might know how to do it, but that's because they encountered this hard thing before you did. I say this because it's very easy to fall into a trap where you spend all your energy trying to conceal your "ignorance" and never ask the questions you actually need the answers to. That's bad for you, obviously, but it's also bad for the people around you, because the informal discussions that asking such questions spark are one of the main things that drive academic communities forward. So yes, you should ask.

No. It’s better for both of you to understand each other and make sure you’re going in the right direction. Neither of you wants to find out later that what could have been a short conversation has instead been weeks or months doing pointless work based a misunderstanding.

It’s normal to ask for clarification if you’re unsure of anything. Academics do this with each other all the time due to their different backgrounds and specialties. It’s part of a healthy and supportive student-supervisor relationship. They have a responsibility to guide you.

That said: Don’t waste their time. It is your responsibility to show what you’ve done to solve your problems and explain what you need from them clearly. They’re very busy and often working on multiple projects. Still you can and should ask for clarification if you’re unsure you understand their answer, even if it’s on seemingly basic topic. If they don’t have time to explain it to you, they can recommend resources to review the topic. They should never humiliate you for not knowing on the spot. Their role is to guide and train you, not test you every step of the way.

of course not, but that depends on what type of basic questions you are going to ask. Sometimes a basic question will lead to the change of our stereotype knowledge.

For instance, the recent science publication gave us an example. Check "Prolonged milk provisioning in a jumping spider".

Why not... it is the natural right of a student to ask any question whether it belongs to basic theory or modern theory. Actually, students get reluctant to ask basic questions despite not understanding because they feel embarrassed from their coursemates on asking basic questions. You should not at all care what others think and continue to ask as many as basic questions until the time you have a clear conception, because to get a good grip on the latest theory concepts, you must have a clear understanding of the basic theory.

In your question, you've used three rather different terms: "instructor", "supervisor", and "advisor". My understanding is that one generally doesn't have a specific faculty member as an adviser until one has advanced to candidacy (one may have a faculty member advising you, but one wouldn't have an official thesis adviser), and by the time one is a candidate, one should have mastered anything that is truly basic. That being said, while mastering the basics before advancing to candidacy is better than not mastering the basics before advancing to candidacy, not mastering the basics before advancing to candidacy and asking one's adviser for help is better than not mastering the basics before advancing to candidacy and not asking one's adviser for help. Furthermore, not knowing the basics before advancing to candidacy reflects as much on the school as you, as that is the whole point of qualifying exams.

As for an "instructor", it's their job to teach you everything in the course, and if there's material they expect you to come into the course knowing, it's their/the department's job to make that clear in the prerequisites. It's not your job to teach yourself the material; if you were going to do that, you wouldn't need an instructor in the first place.

If your friends are confused on this point as well, then that suggests it's not truly basic, or your program has failed to prepare its students on basic matters. Asking faculty members to explain helps not only you, but your fellow students (assuming you share what you find with them), and the faculty, as it hopefully will clue them in that students don't have a solid foundation in this area.

I'm afraid he'll think that I'm not willing to study the topic myself and that I'm wasting his time.

If you come to a faculty member, and say "These are the books that I've read on this subject, this is what I've learned from them, this is what I still don't understand, this is what my fellow students have said when I asked them for help, what do you have to help?", it would be difficult for them to conclude that you're not willing to put in work yourself.

My PhD supervisor always tells me "You come here to learn, I don't expect you to know everything", so I think there's nothing wrong asking question to your PhD supervisor. Perhaps, you can raise the question in an interesting way, backed with previous information you know about. With that, you can develop a discussion where you get answers to your question.

Don't ask any basic question if you haven't first googled it or looked it up on wikipedia.

If you read a couple of papers, and still don't understand, you can ask.