# Online place to learn proofs?

#### Pupil

I really, really suck at proving things. Do you know of any free online resource one could use to master proving things?

I know enough math up to about one semester of calculus, a bit about sentential logic and a little bit of set theory, but even still, past finding a trig identity or something like that I couldn't prove to save my life. I can follow proofs well, even spot when one isn't complete, but I don't see how one thinks of the steps to prove something. I typically look at a neat proof and come across the third step or something and think "Wow! How the hell did this guy think to do that! I would've never thought of that!" even on a lot of simple proofs involving elementary algebra.

For instance, I was reading this one website that, as an exercise, I was to try to prove that if a/4 is an integer, then a is the difference of two perfect squares. It seems like such a simple thing to prove! I struggled with it, tried 20 different things, ended up with a lot of crumpled paper, a headache, and no progress after quite awhile and gave up. And I bet anyone here could prove that statement after glancing at it once. In other words, I'm proof-challenged. Proof-tarded if you will. Durp.

#### symbolipoint

Homework Helper
Gold Member
Remember when you studied Geometry in high school; or in community college? You learned to do some proofs in that course. You also remember verifying identities in Trigonometry. You should have also done a few derivations in your Algebra 1 and 2. You could spend a few months studying Geometry and Trigonometry again and do several proofs.

#### Pupil

Remember when you studied Geometry in high school; or in community college? You learned to do some proofs in that course. You also remember verifying identities in Trigonometry. You should have also done a few derivations in your Algebra 1 and 2. You could spend a few months studying Geometry and Trigonometry again and do several proofs.
Those were easy proofs. Almost no strange steps necessary to prove the things I learned in geometry or trigonometry. I could still prove pretty much anything in those books either because 1) I remember the proof, or 2) it's pretty straight forward to get from a to b. In geometry, to prove something I usually had to find the perimeter or area of something, drop a perpendicular here or there, stuff like that. Straight forward. Trig identities usually relied on sin^2x + cos^2x = 1 or the sum angle formula, rearranging a few terms, combining, and out popped the identity. Pretty easy stuff. I don't think going back to that is going to help. Any other suggestions?

#### Kaimyn

I find, if all else fails, try wikipedia. It's actually quite reliable. The wikipedia page on mathematical proofs is quite in detail (see: http://en.wikipedia.org/wiki/Mathematical_proof" [Broken])

If that doesn't work for you, have a look at the links down the bottom of the Wikipedia page. I got these two (http://zimmer.csufresno.edu/~larryc/proofs/proofs.html" [Broken]), both of which seem pretty good. If you still can't find anything, try a google search.

As time goes on, you'll attempt more proofs and get better at them. Don't worry if at first you aren't very good at them, your "bag of tricks" will grow after a while and they will become easier.

Also, with the problem you said you had a bit of trouble with, try writing down the first few square numbers and the differences between them (and the differences between the differences).

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#### Cantab Morgan

George Polya's How to Solve It is a classic. I've also heard good things about a book titled How to read and do proofs.

And, what I'm about to suggest may sound strange but... Consider learning to program. There's a discipline of thought that comes from programming that may help your brain find the patterns that seem to leap out of nowhere in certain tricky proofs.

Proofs are essentially a conversation between a proof author and the proof reader. You need to convince your audience, in a rigorous and irrefutable way, that you are correct. I have found that when I disagree with someone, going off and trying to code up a solution can help clarify things enough that one of us realizes the other is right.

#### Kaimyn

George Polya's How to Solve It is a classic. I've also heard good things about a book titled How to read and do proofs.

And, what I'm about to suggest may sound strange but... Consider learning to program. There's a discipline of thought that comes from programming that may help your brain find the patterns that seem to leap out of nowhere in certain tricky proofs.

Proofs are essentially a conversation between a proof author and the proof reader. You need to convince your audience, in a rigorous and irrefutable way, that you are correct. I have found that when I disagree with someone, going off and trying to code up a solution can help clarify things enough that one of us realizes the other is right.
If you do want to learn how to program, I would try something like Python for starters.

#### Pupil

I find, if all else fails, try wikipedia. It's actually quite reliable. The wikipedia page on mathematical proofs is quite in detail (see: http://en.wikipedia.org/wiki/Mathematical_proof" [Broken])

If that doesn't work for you, have a look at the links down the bottom of the Wikipedia page. I got these two (http://zimmer.csufresno.edu/~larryc/proofs/proofs.html" [Broken]), both of which seem pretty good. If you still can't find anything, try a google search.

As time goes on, you'll attempt more proofs and get better at them. Don't worry if at first you aren't very good at them, your "bag of tricks" will grow after a while and they will become easier.

Also, with the problem you said you had a bit of trouble with, try writing down the first few square numbers and the differences between them (and the differences between the differences).
Thanks for the links, I'll look them over. So you're saying to just keep practicing and they'll get easier as I figure more things out?

Thanks for the tip BTW, I think I can nail down the proof, now.

George Polya's How to Solve It is a classic. I've also heard good things about a book titled How to read and do proofs.

And, what I'm about to suggest may sound strange but... Consider learning to program. There's a discipline of thought that comes from programming that may help your brain find the patterns that seem to leap out of nowhere in certain tricky proofs.

Proofs are essentially a conversation between a proof author and the proof reader. You need to convince your audience, in a rigorous and irrefutable way, that you are correct. I have found that when I disagree with someone, going off and trying to code up a solution can help clarify things enough that one of us realizes the other is right.
If you do want to learn how to program, I would try something like Python for starters.
I'll try picking up those books sometime if they aren't textbooks or anything horribly expensive. I had a semester of programming in Pascal, but the hardest thing I did in there was write a program to approximate Pi to any degree the user wants via the pi/4=1 - 1/3 + 1/5 - 1/7... formula (which really sucks in my opinion; it takes around 90,000,000 terms to get Pi right to a measly seven digits). It might help if I get back in to programming. Thanks!

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#### Civilized

I really, I can follow proofs well, even spot when one isn't complete, but I don't see how one thinks of the steps to prove something. I typically look at a neat proof and come across the third step or something and think "Wow! How the hell did this guy think to do that! I would've never thought of that!" even on a lot of simple proofs involving elementary algebra.

Then my advice, which has worked for myself and more than one other real person, is to read as many proofs as you can. The same tricks are used over and over, eventually you will be familiar with all of them. Most people will tell you to practice doing proofs, but my advice is to just practice reading them.

As an analogy, I think that reading the work of other authors is a better way to improve one's own writing then merely practicing. By practicing writing without reading other's work, we would only wallow in our own amateurish naivete. By reading the work of great authors, we get exposed to new possibilities that would have taken ages to stumble upon by practicing.

#### Pupil

Then my advice, which has worked for myself and more than one other real person, is to read as many proofs as you can. The same tricks are used over and over, eventually you will be familiar with all of them. Most people will tell you to practice doing proofs, but my advice is to just practice reading them.

As an analogy, I think that reading the work of other authors is a better way to improve one's own writing then merely practicing. By practicing writing without reading other's work, we would only wallow in our own amateurish naivete. By reading the work of great authors, we get exposed to new possibilities that would have taken ages to stumble upon by practicing.
That's a really good idea, but I've already run into a problem: I don't know where to find an extensive list of proofs to read (also, some of the proofs on Wikipedia are way to hard for me to understand). Any suggestions?

#### snipez90

I agree that looking at random proofs on wikipedia is probably not the best idea. There seems to be a conflation of solving a problem and writing up a proof. While the two are clearly related, pointing out the differences may provide you with another approach to becoming better at proofs.

Assuming that you are dealing with questions that will ask you to prove your assertions, you need to have the basics down cold. I learned the basics from this thread: https://www.physicsforums.com/showthread.php?t=166996

I liked the first one by Hutchings, and the second one was already mentioned in this thread. Those links should provide you with the typical methods of proof and they will likely use examples that are fairly easy to understand.

Of course, the most basic step of any proof is to start by writing down what you want to prove and what your hypotheses are. Working from the hypotheses to the conclusion is really an exercise in problem solving. You should be able to reason out why you think the statement should be true. If you can't, try out concrete examples or small cases of the statement in question to try and come up with a handwavy and/or intuitive explanation for why the statement should be true. Nothing I've mentioned so far requires you to actually start writing a good proof, but ideally, you have written down useful ideas that will help you bridge the hypotheses and the conclusion. Again, this is an exercise in problem solving, and if you are reading someone else's proof, it is your task to understand the motivation for why a certain step was used to reach the conclusion of the proof.

Now obviously, not every proof is as simple a procedure as I have described. Sometimes, solving a problem requiring proof needs a good first step, which may simply be picking the ideal method of proof. Sometimes, going for the proof by contradiction or looking at the contrapositive of your statement helps, and this harks back to understanding the basic proof methods. On the other hand, if you are reading a proof and you don't understand a certain step, you might need to just go back to the definitions and explanatory material of the subject you are pursuing. Since good proofs are meant to be convincing arguments, it is unlikely that you are just bad at proving things in general, but rather you may simply not have seen definitions and theorems used in a creative way.

Thus, my suggestion is to first understand the basics of proof, then really focus on a specific subject. You have taken a semester of calculus, so why not try proving the theorems that you encountered there? This may be a more fruitful approach then trying to read as many proofs as possible.

#### n!kofeyn

This is a response to the original poster.

I agree that Wikipedia is a bad idea. Finding good proofs on there is going to be unreliable, and there often aren't any proofs provided. I also think looking online is going to be hit and miss. I also don't think reading as many proofs is going to improve your ability to create proofs. The only way to learn how to do proofs is doing them and creating them yourself! You want to have as little outside help as possible, unless to maybe check your proofs over. This takes a lot of discipline, but it will be worthwhile.

My suggestion is to find a book and work through it yourself. Published books are reliable sources, plus the authors have often done a good job in sequencing and compiling the subject matter.

https://www.amazon.com/dp/0131481010/?tag=pfamazon01-20 by Steven Lay
This is a superb introduction to proofs and real analysis. It will lead you through the techniques of proof and you will learn meaningful mathematics along the way.

There are many books in real analysis (advanced calculus) and topology that will give you a good start, but I think the above book is the best way to go.

A further suggestion (a non-standard approach) is the book https://www.amazon.com/dp/0883857502/?tag=pfamazon01-20. (I just found out they had reprinted this book, which I think is a testament in that the MAA thinks it is a worthwhile book to republish.) Prof. Wall was very aware of methods of mathematics education. This book will force you to learn how to do proofs in a very mathematical and worthwhile way. I wish I would have worked through this book before entering higher level mathematics. It is a beautiful book, and if you make it through the book, you will have learned a good deal of mathematics where you can be proud that you proved the theorems on your own.

My last comment is that it depends upon what you want to do. If you want to be a computer scientists, my suggestions are not the way to go. If you want to be a mathematician, then I would look into the above material. Furthermore, learning to program will be a waste of time at this point in your career as far as it helping your mathematical proof writing ability. Programming may share some characteristics, but even if you learned Python, you would not have furthered your ability or knowledge to write good mathematical proofs.

Good luck!

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