# How to Self-Study Higher Math Without Solutions

I've been lurking on this site for several months, and as someone studying higher mathematics independently (i.e., outside of a college/institutional setting), this forum has probably been the best reference for gleaning general information about college-level math.

However, recently I've come to realize that few of the major/popular math textbooks tend to have solutions to the exercises contained within them or available for free/online. As someone without access to a professor or peers who can check the accuracy of my answers, working through book solutions becomes very tedious. How do I know if I am getting the right answers and thus understanding the material? I know people are encouraged to post here for help with challenging problems, but it doesn't seem appropriate to post in a forum such as this to simply check the veracity of answers. In addition, certain books (e.g., Pugh's Real Mathematical Analysis) contain hundreds of exercises throughout the text, and it would be neither plausible nor acceptable to constantly ask for help with problem after problem for an entire textbook.

I am not in school nor do I plan to enroll anytime soon, so there's not really an option of waiting until college to talk things over with someone more knowledgeable about mathematics. I am learning math out of pure curiosity and not in preparation for future classes or career goals, and while there are a plethora of resources available to the independent student of mathematics now more than ever before, being able to assess how well the material is being absorbed remains a difficult obstacle to overcome.

• Regarding "it would be neither plausible nor acceptable to constantly ask for help with problem after problem for an entire textbook": while "constantly asking" seems a bit excessive, I'd like to remind you that the (proof-verification) and (solution-verification) tags exist on this site for a reason. You should have no more hesitation in asking those kinds of questions than you would have in asking any other. – Omnomnomnom Jul 29 '14 at 21:00
• As a recommendation, Robert Ash's books all have full solutions. – Mark Fantini Jul 29 '14 at 21:01
• As you learn, you'll build confidence -- eventually you will know whether an argument is valid more or less immediately after you've written it down. That's not to say that you will no longer have questions! But they will be of a different nature. This level of autonomy is more difficult to achieve without a classroom experience, for obvious reasons. But MSE should be able to help you get there, at which point you'll be a very valuable site member. With this in mind, it's beneficial to all if you ask for feedback now. Also, keep in mind that your questions will likely be of use to others. – vociferous_rutabaga Jul 29 '14 at 21:20
• I really respect what you are doing. About solutions its a matter of your own internal sense of rigor. – Rene Schipperus Jul 29 '14 at 21:25
• If there's a book you're interested in, search for a solution's manual prior to purchasing it. You should be able to find a book with a solutions manual for most major topics. You may not always get the text you want, but at least you will be able to verify what you're doing is correct this way. – Vincent Jul 29 '14 at 21:32

I'm kind of in the same 'boat' as you; I've graduated but continue to study in my own time as a leisure activity. I completely feel the same way in that I now self-study but find it difficult to seek critique of my newly acquired knowledge.

These are suggestions and probably will not be a full answer to your question:

1) Participating in this community,

2) Seeking people in the same position as you,

3) Intense 'Googling' and wikipedia,

4) University notes and solutions,

5) 'Crack on' and Critique yourself.

1) As @omnomnomnom says, you are welcome to use this community to verify and give suggestions to your answers by using the relevant tags. You will find that some people have used the materials that you are referring to and will be able to give you hints/solutions/intuitions/motivations etc.

2) You will find that there are many people who are in the same position as you. And, that there will be a similar number of people who seek support and guidance with their study. Have a look if there are similarly minded people in your community (or even f***book) that you can buddy up with.

3) In my experience, it's easier to look for a list of books, in the field of chosen study, and find whether there is solutions for any of the books on that list. Then you can optimize for the best book against the most resourceful solution-base and usually find almost exactly what you want.

Or even work the other way; You could search the internet for solutions in a chosen-field and then find the book the solutions are for. (I did exactly this for Guillemin's & Pollack's Differential Topology)

A tip for finding books is using wiki; search for a subject that you think you might be interested in on wiki, then look at the references and then check these references with review forums etc.

4) As you said there are plenty of sources for mathematical study but try to use university websites. If they are available to the public then it's fair game. They will often be accompanied by tutorials, solutions and amendments.

5) You'll get to a point where you'll be able to see whether your arguments and solutions are correct, overkill, not quite there or just plain wrong. It takes time. Look for counter-examples, always ask what if (and what if not) and make sure a logical statement means what you think it means. Most importantly, you should try to understand (properly) every new 'thing' before you move onto the next.

Lastly, ask your own questions. And, try to answer them. This way you (sort of) have a feeling of whether your answer is rigorous enough. And always write these down, you may come back to them and want to improve (or re-write) them.

Disclaimer: I offer suggestions based on my experience.

• Thank you for your detailed advice, and I thank everyone else for their comments as well. I appreciate the encouragement to join the community and use this site on a more involved level than simply lurking. Over time, receiving feedback here combined with careful study should help me develop the mathematical maturity to proceed without referencing solutions as often. – dubosec Jul 31 '14 at 1:08