After calculus first pure math course, having a hard time.

[chimath35]

I just started number theory, and I feel completely lost. I never did proofs in calc. I am used to watching the professor do examples, then with solutions manual at home see how the problems are done and use the manual if I get stuck. This book on NT has no homework problems or solutions. I just don't really know where to start on problems? The professor just basically does proofs every class, and they are very hard to follow. I get lost early in the proof, then don't know what to do. I have done very well through calc, not sure what to do. Suggestions?

 

[NascentOxygen]

Hi chimath35! http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif

Have you tried googling 'number theory exercises'?

There must be others in your class in the same boat. Team up with someone and attack some practice exercises together. Can you get hold of past years' asignments or exam papers in your subject?

Good luck!

 

[chimath35]

Okay, you mean like ask the prof. for past exams or try to find other students who took the course?

 

[eigenperson]

You have a difficult task before you because the skills needed to succeed in this class are almost entirely different from those needed to succeed in a calculus class.

If your professor does not give you homework, you must locate problems in some other way (ask your professor, or look in the library for some of the books listed here), but be aware that "problems" no longer refers to routine exercises in carrying out algorithms, as it did in calculus. You should expect each problem to be a difficult endeavor that requires creativity and cleverness. You will not know how to do the problems at first, and it will take you a very, very long time to solve them (especially while you are just starting), so do not be discouraged.

And above all, do not ever look solutions to the problems before you have solved them! If you don't know how to do the problem, you can either keep thinking about it until you figure it out, or give up, but it is the act of thinking about it that causes you to learn, so don't just look at the solution. If you are not sure if your solution to the problem is correct, ask the professor.

 

[chimath35]

Ya, I just can't seem to figure these out. Even with solutions, it is like the more I look at it the more confused I am. I feel honestly just terrible about this. I don't know what to do. I can't keep getting this mad, not good. Like I have five problems and can solve none. Have gotten straight A's almost and I feel like I would get a 10 percent on an exam for this.

 

[chimath35]

I see formulas that are equal to each other try using them and end up solving none. It is so abstract, and just so many letters and it is like making my head spin. I honestly don't know what to do.

 

[Seydlitz]

Maybe you can post one of the questions in the forum, and see if the mentors can guide you step by step with it?

 

[Petek]

Does your college have an "introduction to proofs" course, or something similar? If so, you might wish to take that course. Proving theorems is very different from calculations found in calculus. You might find this web page useful. The author describes five proof techniques that are used in basic number theory.

 

[Mark44]

There are a couple of books that might help you.
"How to Read and Do Proofs" - Daniel Solow
"The Nuts and Bolts of Proofs" - Antonella Cupillari

If you do a search on Amazon or other bookseller you should find them. Both are paperbacks. As I recall, one was a bit pricy, but the other wasn't that expensive. Either one would be helpful to you, I believe.

 

[chimath35]

These might be simple to you guys but these problems are MUCH harder to me than some hard calculus word problems.
here is my hw:

1) If a l b and that a & b are positive, then a < b unless a = b.

2) If a l b, then a l nb for any integer n. If a l nb for some integer n, can you conclude that a l b?

3) If a l (b + c) and a l (b + d), then a l (d - c) and a l (c - d). Can you conclude that a l b?

4) If a l b, then a l b^{n} for any positive integer n. If a l b^{n} for some positive integer n, can you conclude that a l b?

5) Can there be an even prime greater than 2? Write 365 as a product of primes

 

[Seydlitz]

These might be simple to you guys but these problems are MUCH harder to me than some hard calculus word problems.
here is my hw:

1) If a l b and that a & b are positive, then a < b unless a = b.

2) If a l b, then a l nb for any integer n. If a l nb for some integer n, can you conclude that a l b?

3) If a l (b + c) and a l (b + d), then a l (d - c) and a l (c - d). Can you conclude that a l b?

4) If a l b, then a l b^{n} for any positive integer n. If a l b^{n} for some positive integer n, can you conclude that a l b?

5) Can there be an even prime greater than 2? Write 365 as a product of primes

I think it would definitely help if you can start from the definition first. Like for example I choose question number 5, because I think it can easily be proved by using definition correctly. First of all, what is a prime number? What is the definition of it? Then what is the definition of even number? Can a number be a prime and even at the same time in general?

 

[AlephZero]

If you are completely stuck with a problem, start by playing around with some numbers to see what the problem means, and (with luck) get a clue how to do the proof.

For example the first one: "If a l b and that a & b are positive"

Suppose b = 6 (I'm just picking a number that has a few different divisors, there's nothing special about 6) what values can a have? Why can't a be bigger than 6?

Then, as Seydlitz said, you have to use the definition of what "a | b" means...

 

[chimath35]

Ya, see I have the notes and formulas we are supposed to use, I play with them and end up not being able to solve them. I don't like this math class at all. I was pumped about math after multi v calc. It is like I don't want to spend hours burning out my brain and taking a LOT of time just trying to figure out a few to me boring problems. This kind of math is so boring and time consuming to me not to mention I do have other classes that I need to have time for and don't like burning my brain out with such "max" effort on these problems. I feel like I have to think so ridiculously weird and hard on these.

 

[chimath35]

These problems are like a game to me. I am not a fan of trying and failing, trying and failing to try to solve one in my opinion boring problem. From gradients to this? Are there other types of mathematics that I can study that are not so boring, not so time consuming, and not such a hard game? Like I said I like multi v calc A LOT. I want to do math like that I guess.

 

[chimath35]

Also, I guess I get so bummed from not being able to solve these as well that I honestly get upset and think well I tried and I don't get it; I do have a life and other things to do so I stop. There is really no where for me to go for help at my uni. with this level math either. The help is limited on this.

 

[jgens]

It is like I don't want to spend hours burning out my brain and taking a LOT of time just trying to figure out a few to me boring problems. This kind of math is so boring and time consuming to me not to mention I do have other classes that I need to have time for and don't like burning my brain out with such "max" effort on these problems. I feel like I have to think so ridiculously weird and hard on these.

Many people have difficulty transitioning from the "plug-and-chug" type math in calculus to the more theory oriented math in other courses. The problems are simply going to take longer and you need to think harder about them. That part, unfortunately, will not get better. With some practice, however, the ways of thinking will get more natural and this will expedite things a little. But working theory problems is never going to be as quick as computing a simple gradient.

These problems are like a game to me. I am not a fan of trying and failing, trying and failing to try and solve one in my opinion boring problem. From gradients to this? Are there other types of mathematics that I can study that are not so boring, not so time consuming, and not such a hard game? Like I said I like multi v calc A LOT. I want to do math like that I guess.

Trying and failing is an important part of the process. With time though it will become more clear what will work and what will not work. As to finding this boring, that is unsurprising since elementary number theory largely consists of things you should already know, you just have to prove them this time around. Hopefully you might find other (newer) branches of math more interesting. The time consuming component is not going to get better regardless of what field you choose. That is just something you will have to make peace with.

Also, I guess I get so bummed from not being able to solve these as well that I honestly get upset and think well I tried and I don't get it; I do have a life and other things to do so I stop. There is really no where for me to go for help at my uni. with this level math either. The help is limited on this.

I understand the feeling. When I first started a theory based math course it was discouraging. But give it time. Eventually you will get the hang of things and it will not seem so bad. You can also ask for help here on PF if you ever get stuck!

 

[Mark44]

Many math courses beyond calculus involve problems where you prove a statement rather than, say, find a derivative or evaluate an integral. For example, many students struggle just as hard with linear algebra as you are with number theory. Either of the books I suggested back in post #9 will give you lots of examples of the various types of proofs (e.g., direct proofs, proof by contradiction, contrapositive, and so on).

 

[crownedbishop]

Proof by contradiction sounds like a great way to attack some of the problems you listed. Unless you're a constructivist. In number theory, induction is a very important tool too. For the first one, why can't a divide b if a>b?

 

[Chronos]

My long standing position is number theory is extremely useful in calculus. I had the same problem in my undergrad days. It caused me ... difficulty.

 

[chimath35]

Proof by contradiction sounds like a great way to attack some of the problems you listed. Unless you're a constructivist. In number theory, induction is a very important tool too. For the first one, why can't a divide b if a>b?

beacause it would be less than one, not a positive integer

 

[chimath35]

So do you think if I take discrete math concurrently, it will help?

 

[chimath35]

Also if you are bored, here is a number theory problem:

An integer is divisible by 9 if and only if the sum of its digits is divisible by 9

Proof by induction?

I am stuck ac=b 9c=b ex. 9c=81 so c is an int. thus divisible let b= d+dn so 9c=d+dn then 9c=d+dn+dn+9

 

[jgens]

An integer is divisible by 9 if and only if the sum of its digits is divisible by 9

Write n = ∑a_k10^k and notice that n is divisible by 9 if and only if n ≡ 0 (mod 9). So this means n is divisible by 9 if and only if ∑a_k ≡ 0 (mod 9) and the result follows.

 

[Vahsek]

Also if you are bored, here is a number theory problem:

An integer is divisible by 9 if and only if the sum of its digits is divisible by 9

Proof by induction?

I am stuck ac=b 9c=b ex. 9c=81 so c is an int. thus divisible let b= d+dn so 9c=d+dn then 9c=d+dn+dn+9

Write n = ∑a_k10^k and notice that n is divisible by 9 if and only if n ≡ 0 (mod 9). So this means n is divisible by 9 if and only if ∑a_k ≡ 0 (mod 9) and the result follows.

Hi jgens, I am pretty sure that your sketch of the proof is very elegant and efficient, but I feel "mod" is an unnecessary artifice for somebody trying to understand (and like) this type of pure math course.

I have something which is a little bit simpler to handle for a novice:


Suppose k is a positive integer. Then k = a_n10ⁿ + a_n-110^n-1 + ... + a₃10³ + a₂10² + a₁10¹ + a₀, for some non-negative integers n, a_n, a_n-1, ..., a₃, a₂, a₁, a₀.
(Note that the a_i's are the digits of k.)

Dividing the positive integer k by 9 yields the following equation:
k/9 = (a_n10ⁿ + a_n-110^n-1 + ... + a₃10³ + a₂10² + a₁10¹ + a₀)/9.

The above equation can be re-written as follows:
k/9 = [a_n(10ⁿ -1 +1) + a_n-1(10^n-1 -1 +1) + ... + a₃(10³ -1 +1) + a₂(10² -1 +1) + a₁(10¹ -1 +1) + a₀]/9.

It follows that
k/9 = [a_n(10ⁿ -1) + a_n-1(10^n-1 -1) + ... + a₃(10³ -1) + a₂(10² -1) + a₁(10¹ -1)]/9 + [ a_n+ a_n-1 + ... + a₃ + a₂ + a₁ + a₀]/9.

Clearly, the first term on the right hand-side of the above equation is an integer since a_n(10ⁿ -1) + a_n-1(10^n-1 -1) + ... + a₃(10³ -1) + a₂(10² -1) + a₁(10¹ -1) is divisible by 9.
(Note how each (10ⁱ -1) contains only 9's as digits, so each (10ⁱ -1) is divisible by 9.)

Now suppose that k is not divisible by 9. Then k/9 is not an integer. It follows that [ a_n+ a_n-1 + ... + a₃ + a₂ + a₁]/9 is not an integer, so the sum of the digits of k is not divisible by 9. Thus, if the sum of the digits of k is divisible by 9, k is divisible by 9.

Next suppose the sum of the digits of k is not divisible by 9. Then [ a_n+ a_n-1 + ... + a₃ + a₂ + a₁]/9 is not an integer. It follows that k/9 is not an integer, and hence k is not divisible by 9. Thus, if k is divisible by 9, the sum of its digits is divisible by 9.

Therefore, we conclude that integer k is divisible by 9 if and only if the sum of its digits is divisible by 9. This completes the proof.

 

[chimath35]

Ya I think I may just change the class from credit to audit; considering I have not taken an intro to proofs class, and the fact that my gpa is almost a 4. My uni. does not offer proof intro this sem. I think this way I will still build up my mathematical thinking abstractly and with proofs without the consequence of possibly tanking my gpa. My professor also just states theorems and then proves them. I am not sure if that is typical or not? The way that I learn the best I think is seeing the theorem and then doing examples of similar problems. Not just stating a theorem and then taking almost the entire lecture time to be complete and prove the theorem; I don't think him being complete and proving the theorems helps with homework or test problems.

 

[Vahsek]

The way that I learn the best I think is seeing the theorem and then doing examples of similar problems. Not just stating a theorem and then taking almost the entire lecture time to be complete and prove the theorem; I don't think him being complete and proving the theorems helps with homework or test problems.

I would advise you to be careful and not to adopt such an attitude. Think about it for a second: you posted some of your homework problems, and evidently, all of them involved proofs. Given that you don't have much experience with proof-writing, I believe that it would be wiser to carefully follow the proofs (and especially, the proof strategies that he uses) of your professor in the lectures, and then try to apply those similar proof strategies on your homework problems.

Concerning your decision to audit the course instead of taking it for a credit, I think that was a smart move on your part. However, you should still try to make the most out of this course (especially about proof-writing) because most if not all of your future courses might be heavily proof-based.

Another important thing: luck seems to be smiling at you, as you'll have this summer before your next semester at college. It would be in your interest to pick up a book on proof-writing (or if you will have already mastered proof-writing by summer, pick up another pure math text like introductory real analysis or abstract algebra), and work through it. Trust me, that's the best way to really "get" proofs if you are not a "natural" (it worked out pretty well for me).