- talking in class
- chewing gum
- resting your feet on the seats
- using electronic devices

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Evangelos K. (12/6/2012 at 4:35:16 AM) Amy, that number is not on Mertens' formula. It is on the elementary proof of the prime number theorem. What one does is to begin with an initial estimate which is of the form |R(z)/z|<a for x>x_o (this is an assumption that one makes) and improve it to |R(z)/z|<a(1-a^2/300).What Selberg did was to describe this general mechanism. If this mechanism gets started with some initial a<8 then it produces estimates that show that |R(z)/z| becomes arbitrarily small.

Amy C. (12/5/2012 at 3:46:48 PM) Professor, we are a little bit confused about where the number "a" came from in the proof done based on Mertens' formula. Does "a" represent the 4*delta? Also, is "a" sequence or is it a number on a number line?

Nadine K. (11/25/2012 at 5:46:52 PM) will do then. thanks

Evangelos K. (11/25/2012 at 5:44:54 PM) Nadine, you can bring your laptop along tomorrow so that we can take a look but otherwise it does not make too much sense.

Nadine K. (11/25/2012 at 2:24:44 PM) Professor, I redownloaded the miktex complete version so the tikz could work, but it still isn't working out. Do you have any recommendations?

Evangelos K. (11/20/2012 at 10:29:40 AM) Maryam, thanks for catching this. I have taken care of it.

Maryam M. (11/20/2012 at 9:45:17 AM) Professor, I think there is no link from the drop-down list to submit the current homework assignment. Can you please take a look at it, thank you.

Evangelos K. (11/14/2012 at 7:13:42 PM) Nicholas, the program that you are using (texworks) does this for you automatically. The pdf file should be found in the same directory as your source file once you run LaTeX.

Nicholas L. (11/14/2012 at 6:02:00 PM) Professor, How can I convert a LaTeX document to a PDF file?

Evangelos K. (11/9/2012 at 4:28:20 AM) Nadine, I would have to take a look at the code. However if you center the picture after some text, then there should be no problem determining how it is going to show.

Nadine K. (11/9/2012 at 12:38:11 AM) what do you do to a picture in latex that refuses to move from a given position? maria and i have the hilbert curve picture which regardless of where it is in the coded version, still ends up being in the same spot after typesetting. how do we fix this?

Evangelos K. (11/8/2012 at 6:07:50 PM) Mohit, please leave graphics and pictures for later. Let's focus on the text now.

Mohit D. (11/8/2012 at 4:34:14 PM) Professor, how is one able to insert pictures in LaxTex from the internet?I tried using \begin{figure} \centering \includegraphics{../../Downloads/quad.jpg} \label{fig:quad}\end{figure}but the software does not read the extension.

Evangelos K. (11/6/2012 at 8:19:21 AM) Nadine, as I told you in class we will have to run some experiments together and see how to include the graphics in your paper.

Nadine K. (11/4/2012 at 9:27:42 PM) Professor, I am trying to input codes to create some fractals but a lot of the parts are suppose to be entered into the preamble. How do I do this though if I already have codes for the preamble for the actual writing of the paper?

Evangelos K. (11/2/2012 at 3:42:16 PM) Paul, I will have to check if this particular proof is due to Gauss.

As for your second question, we proved that it was bounded. We argued by contradiction. If it were not bounded, then it would have a subsequence that tends to infinity. But then P(that subsequence) would also tend to infinity.

Abel, that was a contradiction argument. We wanted to show that z_n was bounded. This is why we supposed that it was not bounded and under that assumption it has a subsequence that tends to infinity. But then P(that subsequence) would also tend to infinity , which contradicts the fact that it tends to a.

Abel T. (11/1/2012 at 7:49:08 PM) Professor,In my notes, we showed that P(z_n_k) goes to infinity. But I thought that P(z) was bounded? Which means that its subset shouldnt go to infinity?

Paul S. (11/1/2012 at 6:35:07 PM) Professor, how do we know that the sequence z_n is bounded again?

Paul S. (11/1/2012 at 5:14:59 PM) Professor, was the proof we did one of Gauss's proofs because I found out he did four different types of proofs for the FTA

Evangelos K. (11/1/2012 at 2:08:16 PM) Abel, you will have twenty minutes for this task. You are only limited by the time. Your presentation should contain an outline of the proof with some history and biographical notes with a level of details that is only restricted by the time that you have at your disposal for this task.

Abel T. (11/1/2012 at 11:58:56 AM) Professor,Are we simply supposed to mention a little about the mathematicians whenever appropriate, or are we to dedicate a significant amount of work on these mathematicians?

Evangelos K. (11/1/2012 at 5:50:13 AM) Paul, when we write the expression z\epsilon C, this means z belongs to C. In other words \epsilon stands for "belongs to". The truth is that it is typeset as \in and not as \epsilon.

The sequence z_n is a sequence that realizes the infimum of |p(z)|. In other words it is a sequence of complex numbers such that |p(z_n)| approaches the infimum of the set {|p(z)| as z is in C}. Remember the description that we gave in class: Imagine that this infimum is 1. Then we would choose z_1 so that |p(z_1)| is between 1 and 2. We would then choose z_2 so that |p(z_2)| is between 1 and 1+1/2. In general we would choose z_n so that |p(z_n)| is between 1 and 1+1/n.

As for your last question, suppose that the polynomial p(z) has the minimum of its values |p(z)| at w. Then the polynomial p_1(z)=p(z+w) has its minimum at 0. This is because in order to get w in the expression p(z+w) which is known to minimize the expression |p(.)|, we need to set z=0.

Please let me know if you have any further questions.

Paul S. (11/1/2012 at 12:43:54 AM) Professor, can you also explain again how to do the linear change in variable to assume that w=0 and p(0)=1

Paul S. (10/31/2012 at 11:07:38 PM) Professor, does|p(z_n)|represent the sequence of complex numbers within the polynomial |p(z)| or is it a subsequence of |p(z)|? Also, what did z \epsilon C mean again in a=inf{|p(z)|:z \epsilon C}

Evangelos K. (10/31/2012 at 2:49:49 PM) Euler, Gauss, Bolzano, Weierstrass would certainly be some names to consider. But you can do your own research and surprise me too, if you like. Everything should be in the context of the proof of the theorem of course.

Nicholas L. (10/31/2012 at 1:53:05 PM) Professor, for this Friday's quiz on the fundamental theorem of algebra, which mathematician did you want us to give background information on? I thought it was Euler, but there is not very much information about his work on the fundamental theorem of algebra.

Avni B. (10/22/2012 at 8:12:45 AM) Thanks, Abel and Professor!

Evangelos K. (10/22/2012 at 4:20:53 AM) Sandy, Nadine is right. By submitting it on Thursday you ensured extra credit but I did ask you to resubmit it for technical purposes.

Nadine K. (10/21/2012 at 11:01:18 PM) i think he said to resubmit that just for technical reasons. it won't hurt if u do

Sandy E. (10/21/2012 at 8:57:38 PM) Dr. Kobotis, do we have to re-submit the paper about quadratic reciprocity from last week? I submitted it on Thursday...

Evangelos K. (10/21/2012 at 7:25:18 PM) Avni, Abel's remark is correct. The dollar signs should open in the beginning of a mathematical expression and should close in the end. Also, to get nice big parentheses surrounding p/q you can try this:

$\left(\dfrac{p}{q}\right)=\left(\dfrac{q}{p}\right)(-1)^{{\frac{p-1}{2}}{\frac{q-1}{2}}}$

Abel T. (10/21/2012 at 6:42:16 PM) Avni, if you still need the help, i think the reason your having trouble is because your missing dollar signs for "(-1)^{{\frac{p-1}{2}}{\frac{q-1}{2}}}"

Avni B. (10/20/2012 at 2:19:22 PM) Hi Professor, I'm having an issue with the typesetting of this line: ($\frac{p}{q}$)=($\frac{q}{p}$)(-1)^{{\frac{p-1}{2}}{\frac{q-1}{2}}}. Can you help me figure out why? Thanks!

Evangelos K. (10/18/2012 at 7:49:51 PM) Maria, exactly. There is nothing for tomorrow. Tomorrow we will cover quadratic residues and you will have to do the homework for Monday. I mentioned that it would be a good idea to take a look at tomorrow's summary in advance.

Maria K. (10/18/2012 at 4:50:21 PM) Just to be sure: Professor, you mentioned that the tasks for 10/19 were also for the weekend and that we hadn't gone over the entire summary, yes? Are the tasks technically due on Monday?

Evangelos K. (10/18/2012 at 12:09:57 PM) Feedback for all recent paper submissions has been posted.

Evangelos K. (10/17/2012 at 6:07:31 AM) Gargi is exactly right!! Don't forget [1]. [1] is always among the congruence classes coprime to n.

Gargi V. (10/16/2012 at 11:32:32 PM) Maria, Nadine, and Avni: 1 is coprime to any number, so you might want to add [1] to your list!

Maria K. (10/16/2012 at 10:48:08 PM) Also dealing with a similar issue to Nadine & Avni!

Nadine K. (10/16/2012 at 9:01:27 PM) avni i came across the same problem and was very confused as well!

Avni B. (10/16/2012 at 5:52:03 PM) Hi, I think I'm a little confused. [5], [7], [11] are the only things coprime to [12], correct? But when I plug in 12(1-1/2)(1-1/3), that equals 4. So shouldn't there be one more class that is coprime to [12]?

Evangelos K. (10/16/2012 at 2:09:08 PM) Nicholas, the formula for the number of congruence classes that are coprime to n is given in terms of the prime factorization of n. Look at the summary for the formula. The formula does not work only if the prime factors appear with exponent 2 (as in the case of 36). They work in all cases. For example 12=2^2*3. Just apply the formula based on that factorization.

Nicholas L. (10/16/2012 at 10:53:18 AM) Maybe I misunderstood the lecture. I noticed that (for the example 36), you had to find two perfect squares that multiply to get 36. Those perfect squares were 4 and 9. There are no two perfect squares that multiply to get the number 12. Can I use three perfect squares?

Evangelos K. (10/15/2012 at 8:23:55 PM) Nicholas, of course it is not possible to use square root symbols in the case of congruence classes. What congruence class are you trying to invert and in which set?

Nicholas L. (10/15/2012 at 2:10:22 PM) Professor, I am a bit puzzled by the second assignment (congruence classes in $\mathbb{Z}$/ 12 $\mathbb{Z}$). In our class example, we had 2^2 and 3^2 = 36 with $\frac{1}{2}$ and $\frac{1}{3}$ being the multiplicative inverses. For the homework problem, it seems that \sqrt{3} must the denominator in one of the multiplicative inverses. Is this possible?

Evangelos K. (10/14/2012 at 7:47:23 AM) Feedback on yesterday's submissions has now been posted.

Evangelos K. (10/13/2012 at 11:23:54 AM) I posted comments and scores for all the recent submissions. The projects are proceeding very well and I hope that we will done with all the major proofs this week. I will check for more submissions and post new comments tomorrow morning.

Evangelos K. (10/13/2012 at 11:22:48 AM) Amy, it is $\mbox{Li}(x)$ not \nbox.

Amy C. (10/13/2012 at 10:01:57 AM) Professor, for writing Li(x), I typed in $\nbox{Li}(x)$, and it is not working.. what is the correct way to type it in?

Evangelos K. (10/10/2012 at 2:58:15 PM) I took care of the technical error. I have successfully received papers from the following groups: MN, AN, AM and KP. If you are not among these groups and you submitted your paper today, please submit it again!

Nicholas L. (10/10/2012 at 8:32:30 AM) Professor Kobotis, the same problem is coming up again when I try to submit the project paper. I tried using a different computer, and I still cannot get it to work. Are other students having the same trouble? Maybe I am doing something wrong.

Nadine K. (10/10/2012 at 12:34:35 AM) okay yes it was much larger and easier to see in this format. thank you

Evangelos K. (10/10/2012 at 12:24:42 AM) Nadine, I am sure you found \binom{m}{n}. However, let me make a remark. It shows better if you issue the command \displaystyle right before it, i.e. \displaystyle\binom{m}{n}.

Nadine K. (10/9/2012 at 8:02:27 PM) nevermind! got it!

Nadine K. (10/9/2012 at 7:53:17 PM) Professor, how do you write in the latex the format for combinations and permutations? for example m choose n

Nicholas L. (10/2/2012 at 7:29:03 PM) It is working fine now. Thank you.

Evangelos K. (10/2/2012 at 5:17:21 PM) Nicholas, I see what you mean. It should be fine now. Please let me know if you have further problems.

Nicholas L. (10/2/2012 at 2:42:32 PM) Professor Kobotis,I am having trouble again submitting the next version of the project paper. An error message comes up: "This Page Cannot Be Displayed." I believe I had the same problem before, but you fixed it. Can you please take a look at that and see if there is anything wrong?

Evangelos K. (9/25/2012 at 7:50:57 PM) Paul, you will have to mimic the work that we did in class for the Fibonacci sequence. That work has not been completed; so it is conceivable that at this point you can only take the steps that we took in class together. After tomorrow's lecture, you should be able to complete the process of finding the formula.

Paul S. (9/25/2012 at 2:16:25 PM) Professor Kobotis, do you want us to just give you the direct formula for the sequence, or do you want us to find a couple of the terms of the sequence and then express the direct forumla?

Evangelos K. (9/24/2012 at 4:23:50 PM) Abel, yes you guessed right! It is $a_{n+2}$.

Nicholas, we found two sequences that satisfy the relation a_{n+2}=a_{n+2}+a_n. Those where ((1+sqrt(5))/2)^n and ((1-sqrt(5))/2)^n. It is also true that any sequence of the form A((1+sqrt(5))/2)^n+B((1-sqrt(5))/2)^n for any two numbers A and B will satisfy the same recursive relation. The whole point is to choose A and B appropriately so that we get the same two first terms as the Fibonacci sequence. This is why we first set n=1 and set the whole thing equal to 1 (the first term of the Fibonacci sequence) and then we set n=2 and make the whole thing equal to 2 (the second term of the Fibonacci sequence).

Nicholas L. (9/24/2012 at 1:45:53 PM) Dr. Kobotis,I understood the process of finding the Golden Ratio, but I am not quite sure what you did to get the formula for the Fibonacci sequence.

Abel T. (9/24/2012 at 1:24:32 PM) Never mind I figured it out

Abel T. (9/24/2012 at 12:33:33 PM) Dr. Kobotis, how do you make subscript include more than n? When I put $a_n+2$, it only subscripts the n and not the +2.

Evangelos K. (9/22/2012 at 7:27:38 PM) Sandy, this is OK. The reason I might prefer a little the order in which we did things is because it is abundantly clear why each step is taken. My small objection with the approach that you suggest is that it is not as clear as in the other approach why you need to add and subtract b^2/4a. Other than that, this solution is also perfectly correct.

Sandy E. (9/22/2012 at 7:18:24 PM) Dr. Kobotis I have a question about the quiz. Do I have to show the proof exactly the same way as shown in the class? My first step involves adding and subtracting b^2/4a. Is this fine?

Evangelos K. (9/21/2012 at 7:08:51 PM) Paul, you can do this with an underscore. For example $x_n$.

Paul S. (9/21/2012 at 4:59:33 PM) Professor Kobotis, how do you code a variable to have a sub unit underneath it?

Evangelos K. (9/20/2012 at 8:03:41 PM) Sandy, all of your papers have been successfully submitted. I have not compiled them yet - this will happen over the weekend - but if they gave you no error, then the same will be true when I compile them too.

Sandy E. (9/20/2012 at 6:17:19 PM) Dr Kobotis, I just want to make sure that you've been receiving my papers. Can you check if you've received "four" papers from me thus far. Are they compiled correctly?Also, regarding the first version of the project, it doesn't have to be long, correct? It is just an explanation of the statement of theorem.

Evangelos K. (9/20/2012 at 1:14:08 PM) Amy, yes it is possible to do that. It is possible to insert any type of picture in a LaTeX document and we will possibly do this for the final project paper but not for the papers that are due each week. So please don't try that for those.

Amy C. (9/20/2012 at 11:37:24 AM) Professor Kobotis, is there any way to insert pictures from internet on the latex? When I use the "insert picture" and compile, it won't show the pictures, but it only shows the address of the picture

Evangelos K. (9/19/2012 at 7:57:23 PM) Nicholas, it was a technical error and it was just taken care of. I have received your other papers and you should now be ready to upload the Al Khwarizmi paper.

Nicholas L. (9/19/2012 at 7:48:49 PM) Professor Kobotis,I tried to upload my paper on Al-Khwarizmi, but there was an error message. The last three papers and the project paper seemed to work, but I am having problems with this one. Can you please let me know if you received it? Have you received the last 3 papers from me? I just want to make sure, since there is no way for me to tell if you have them.

Evangelos K. (9/19/2012 at 7:20:35 PM) Sandy, the presentation should include the proof of an equation with abstract coefficients: ax^2+bx+c=0 and yes I do want the detailed proof of how one gets the roots this way.

Sandy E. (9/19/2012 at 7:03:16 PM) Dr. Kobotis, do you want us to show the proof for the completing square method (Quadratic equation), or you just want us to practice an example and solve it? In other words, what should the presentation include? Thanks.

Evangelos K. (9/18/2012 at 8:46:09 PM) Nadine, unlike the project paper, every new version overrides the previous ones for the usual homework papers. This means that your initial upload has already been erased.

Nadine K. (9/18/2012 at 8:42:53 PM) Professor, the first version that I uploaded of the Archimedes paper did not include my references so please ignore that copy!

Evangelos K. (9/18/2012 at 6:31:06 PM) Maria, yes that would be a good idea.

Maria K. (9/18/2012 at 5:42:48 PM) Professor, If the main reference we used for writing some of the essays you requested was "The Math Book" would you still want us to cite it at the end of the copy/paste to "My Page"?

Paul S. (9/18/2012 at 4:12:35 PM) Alrighty thanks!

Evangelos K. (9/18/2012 at 2:27:13 PM) Paul, yes unless it is contained in an equation with more things, for example:

$\sin\pi=-1$

or

$e^{i\pi}+1=0$

Paul S. (9/18/2012 at 9:50:32 AM) Ok so it's like this? $\pi$

Evangelos K. (9/18/2012 at 6:32:12 AM) Paul, it is \pi (always in dollar signs)

Paul S. (9/17/2012 at 10:22:59 PM) Professor Kobotis, I was wondering how you could code the pi character onto LaTeX

Evangelos K. (9/17/2012 at 8:17:11 PM) Sandy, this is precisely what I am saying.

Sandy E. (9/17/2012 at 7:24:24 PM) Dr. Kobotis, so you are saying that the volume displaced would always be the same no matter what type of fluid we are using?

Evangelos K. (9/17/2012 at 7:04:25 PM) Dear Sandy, volume has nothing to do with density. You could perform Archimedes' experiment (i.e. measure the volume of an irregular object) by using any liquid. All you need to see is how much the level of the liquid is rising in order to infer the volume.

Sandy E. (9/17/2012 at 6:42:29 PM) Dr Kobotis, I have a question regarding Archimedes Principle. Why does the principle states that volume of displaced "fluid" equals the volume of irregular shaped object. I think that the word "fluid" is inaccurate since it might imply fluids like alcohols, oils, etc.. which have different densities than water. I thought that we should always use the water since it has a density of 1 which doesn't affect the volume/mass measurement.

Evangelos K. (9/17/2012 at 1:27:53 PM) Amy, this is what you get if you stop at (n-1)^2. If instead you go to n^2 then you get what we got in class today. Alternatively, take today's formula and change n into n-1. Then you are going to get exactly the formula that you just mentioned.

Amy C. (9/17/2012 at 12:37:04 PM) Professor, on the proof for getting the volume of the sphere, the sum of sequence n^2 (1^2+ 2^2+...(n-1))^2) is shown as n*(n-1)*(2n-1)/6 which is different from what we learned today?

Evangelos K. (9/14/2012 at 5:41:52 AM) Paul, if the power has more than one digits, you have to put it in curly brackets. For example: $8.06\times 10^{63}$.

Paul S. (9/13/2012 at 4:42:14 PM) Oh alright. Another thing, if I wanted to express a number raised to the tenth, such as 8.06 x 10^ 63, how would I express that in LaTeX?

Evangelos K. (9/13/2012 at 4:37:56 PM) Paul, as usual just for one of them.

Paul S. (9/13/2012 at 2:25:54 PM) Professor, when said to find a topic that related to the readings, did you mean that we have to find a topic (via. internet or library) that is applicable to all five reading? Or, a presentation on the description of one of the readings?

Evangelos K. (9/13/2012 at 6:59:53 AM) Paul the main part of the proof is the computation of the volume of the hemisphere of radius 1. This is where we apply the method of exhaustion and this is where our focus should be.

Avni, Mochizuki has been claiming proof of the famous abc conjecture since August but I have not seen any official reaction. Sometimes, it takes time to check these proofs, especially since he has built a whole new theory. The proof of this conjecture would not imply the existence of any particular pattern for prime numbers. However it would enhance our understanding of additive number theory and it would imply a number of significant other results including Fermat's last theorem that we are going to mention in our course. Thanks for sharing the link!

Paul S. (9/13/2012 at 12:52:35 AM) Professor Kobotis, Im a little unsure as to what we need to study for the volume of the sphere. Do you want us to study how to find the volume of a hemisphere through the usage of the cylinders, or simply the volume of the sphere of radius r?

Avni B. (9/12/2012 at 9:32:38 PM) I just ran across this and thought I should share:http://www.livescience.com/23075-mathematician-claims-proof-of-connection-between-prime-numbers.html

Evangelos K. (9/12/2012 at 3:20:42 PM) Nicholas, I want you to know the volume of the sphere.

Daniel, I will respond to your email.

Daniel M. (9/12/2012 at 2:58:10 PM) Hey Professor Kobotis, I was making sure that you received my email?

Nicholas L. (9/12/2012 at 1:00:30 PM) Professor Kobotis, do you want us to know the proof for the volume of a sphere or the area of a circle?

Nicholas L. (9/12/2012 at 1:00:18 PM) Professor Kobotis, do you want us to know the proof for the volume of a sphere or the area of a circle?

Evangelos K. (9/11/2012 at 7:30:34 PM) Paul, not yet.

Paul S. (9/11/2012 at 4:18:56 PM) Hey professor, I was just wondering if you made any updates as to how we should turn our work in

Evangelos K. (9/8/2012 at 03:56:15 PM) Your topics have been assigned. Here is the complete list:

DR: Combinations and Permutations

MM: Symmetric Polynomials

BU: Transcendence of e

AN: Irrationality of pi

AG: Set Theory and Cardinalities

AS: The Prime Number Theorem

CE: Discovering Prime Numbers

MN: Fractal sets

KP: Solving the cubic and quartic equation

AM: The quadratic reciprocity law

MT: Fermat's little theorem

I have named each group by using the initials of your first names. We are going to be using this coding in the preparation of our papers. One thing that I want everybody to understand is that you are not going to be on your own when you are preparing your paper. I am going to discuss the topics with you, explain the math, suggest sources and do everything possible to make the process as smooth and instructive as possible. This coming week I am going to let you know about the workflow.

Evangelos K. (9/7/2012 at 3:37:12 PM) Paul, not for the essay. Not for any of the essays. The only time that I will ask you to use BibTeX is for your final paper.

Paul S. (9/7/2012 at 3:21:42 PM) Professor Kobotis, I'm assuming that we won't have to use bibTex to site our sources for this essay.

Evangelos K. (9/7/2012 at 1:58:52 PM) Nicholas, you are absolutely right and thanks for catching this. I meant to say "handwritten". It has now been corrected.

Nicholas L. (9/7/2012 at 1:52:08 PM) Professor Kobotis,The homework due for 9/10 says to do the following in LaTeX.By using a ruler and a compass construct:an equilateral triangle of given sidethe tangent to a circle at a given point,a circle that goes through three non collinear points.How can this be done in LaTeX? Did you mean to label this assignment as "handwritten"?

Sandy E. (9/5/2012 at 9:51:42 PM) Problem solved! Thank you so much, Dr. Kobotis.

Evangelos K. (9/5/2012 at 5:37:24 PM) Hi Sandy, yes I got it. You are getting this problem because the LaTeX program is producing a DVI file as opposed to pdf file. From the typesetting options, make sure that a pdf file is produced. Then this file will be easily printable.

Sandy E. (9/5/2012 at 4:41:34 PM) Dr. Kobotis: Did you get my email? Thanks

Evangelos K. (9/5/2012 at 3:25:37 AM) Maria, I got your corrected version and I did send you some comments.

Maria K. (9/5/2012 at 1:26:42 AM) I believe that I have figured it out. Professor Kobotis: Would I be able have you double check to be sure after class? Thanks!

Maria K. (9/4/2012 at 9:23:18 PM) The attachment I sent to you contained my old Latex pdf. I resent the correct one! Please let me know. I am very sorry for the inconvenience!

Evangelos K. (9/4/2012 at 7:50:41 PM) Maria, I did not get your email. Please resend it. My email is evangelos@math.uic.edu

Maria K. (9/4/2012 at 7:40:39 PM) Professor Kobotis: Did you receive my email? Please contact me if there are any errors. If not I will Typeset it exactly the way it is and disregard the errors it is notifying me with!

Evangelos K. (9/4/2012 at 7:24:05 PM) Nadine, I don't need anything too sophisticated for the references. In fact, LaTeX has its own very sophisticated reference system (called BibTeX) but we will use that only for the term paper.

Nadine K. (9/4/2012 at 7:12:36 PM) Oh and also for the question I just posted, easybib uses word document so would we have to then put it in latex?

Evangelos K. (9/4/2012 at 7:07:31 PM) Nadine, yes that would be fine. We are going to say a few more things tomorrow about using references.

Nadine K. (9/4/2012 at 6:58:26 PM) To name our sources, can we simply use easybib.com? Or is there a special format for the citation page?

Evangelos K. (9/4/2012 at 5:36:41 PM) Raghava, "the paper on Euclid's Elements" (not the Euclid write up...) will be due this Friday.

Raghava M. (9/4/2012 at 3:31:03 PM) So the Euclid write up is due on Friday?

Evangelos K. (9/4/2012 at 2:43:47 PM) Since the statement for the paper on Euclid's elements was clearly ambiguous (gotta love the oxymoron), I specified what I precisely want and gave an extension for Friday.

Evangelos K. (9/4/2012 at 2:16:13 AM) Abel, it will be done in LaTeX using \documentclass[11pt]{article}. The rest is taken care of by this command.

Abel T. (9/3/2012 at 10:45:53 PM) Is the Elements paper supposed to be double spaced?

Evangelos K. (9/3/2012 at 9:14:10 PM) Bartosz, as with any paper, you will have to name your sources.

Maria, email me your paper and I will try to give you some hints.

Abel T. (9/3/2012 at 7:53:19 PM) Maria K., did you remember to use dollar signs? and the forward slashes? or the squiggly parenthesis? all of these have to be exact. I would also just use the LaTeX Template, it does all the set up for you.

Maria K. (9/3/2012 at 4:24:25 PM) Hello, everyone! I've been coming across errors once I press "Typeset" on TeXShop. It will not show me the information as numbers/equations. What can I do? It keeps highlighting some of the codes we were given on the board in class as errors.

Bartosz G. (9/3/2012 at 2:15:02 PM) Do we have to do a citation page for our paper on Euclid?

Evangelos K. (9/2/2012 at 6:00:56 AM) Paul, yes TeXworks installs automatically once you have installed Miktex and you can use it in order to typeset your LaTeX documents.

Paul S. (9/1/2012 at 10:48:48 PM) I have a PC. Was TeXworks the program that came after installing the LaTex engine from www.miketex.org?

Abel T. (8/30/2012 at 7:30:31 PM) Also, on Mac, are there any significant differences between TeXshop and TeXworks? The user interface is slightly different but other than that I haven't noticed any major differences.

Abel T. (8/30/2012 at 7:28:03 PM) Maaria, I might know what you mean. You need to click on "Typeset" in order for your information to display as numbers/equations. It will display them on a new page which is your pdf file that you print out and turn in.

Evangelos K. (8/30/2012 at 7:15:09 PM) No it's not but I am not sure what you mean.

Maaria S. (8/30/2012 at 5:40:21 PM) okay is it suppose to change to numbers and equations when i press enter? because its not?

Evangelos K. (8/30/2012 at 5:21:36 PM) Maaria, yes that's right! This is the name of the program that I was forgetting in class yesterday: TeXWorks Thanks!

Maaria S. (8/30/2012 at 5:13:26 PM) Am i suppose to use the TeXworks for writing the article?

Evangelos K. (8/30/2012 at 2:20:41 PM) Abel, I went through that too a few years ago... And I still love to do things with this program. Take a look at the following website:

http://www.texample.net/

Abel T. (8/30/2012 at 12:24:17 PM) Not gonna lie, this is pretty addicting...

Evangelos K. (8/30/2012 at 8:05:18 AM) Nicholas, I would like to have a hard copy actually. I am also going to discuss the code with you tomorrow, since it is our first assignment. I hope that soon the use of LaTeX becomes a routine component of our class.

Nicholas L. (8/30/2012 at 6:29:41 AM) Would you like the final LaTeX document sent to your email, or do you want a hard copy?

Evangelos K. (8/29/2012 at 9:41:52 PM) Avni, to write a in italics, so that it is a number, just put it in dollar signs. You would write something like Let $a$ be a positive number etc...

Evangelos K. (8/29/2012 at 9:40:39 PM) Paul, that's precisely right. Upon taking the squares of 3k, 3k+1, 3k+2, you will realize that the squares of non-multiples of 3 are non-multiples of 3 and the squares of multiples of 3 are multiples of 3.

Avni B. (8/29/2012 at 9:20:31 PM) And to Paul S - that's what I did and (I think) it worked out for me.

Avni B. (8/29/2012 at 9:19:38 PM) Is there a problem in my formatting? I am receiving error messages: here is one of the lines causing issues. If \textsl{a} is a positive number with \textsl{a^2=3}, \textsl{a} is not rational.i was attempting to italicize. Thanks!

Paul S. (8/29/2012 at 8:57:53 PM) Tell me if I am correct. The non multiples of 3 are 3k +1 and 3K + 2? Then would you square them along with 3K?

Evangelos K. (8/29/2012 at 8:47:43 PM) Nicholas, now you will have to play with multiples of 3, not with multiples of 2. In other words, if n^2=3m^2, then n has to be a multiple of 3 (read my comment to Keane below), which means that n=3u. Therefore,(3u)^2=3m^2 etc.

Nicholas L. (8/29/2012 at 8:40:31 PM) To prove the square root of 3 irrational using the same method we used for 2, I end up with n^2 = 3m^2 which doesn't prove that n is an odd number or an even number. Am I taking this problem the wrong way?

Evangelos K. (8/29/2012 at 5:22:10 PM) Keane, today's proof was based on the fact that squares of even numbers are even and squares of odd numbers are odd. You have to prove something similar for 3. In particular you need to prove that squares of multiples of 3 are multiples of 3 and squares of non-multiples of 3 are non-multiples of 3. This will enable you to use an argument almost identical to the one that we presented in the morning. Instead of simplifying all powers of 2, you will begin by assuming that you have simplified with all powers of 3. I hope this helps. Please let me know if you have any further questions.

Keane H. (8/29/2012 at 5:13:40 PM) What was the tip you gave us to prove square root of 3 is irrational?

Evangelos K. (8/29/2012 at 5:43:19 AM) Welcome to our section's online message board. Please, post here any comments and questions that may be to our class material and schedule. I will try to respond to them in a timely manner but you may also get an answer from one of your classmates.

Amy C. (12/5/2012 at 3:46:48 PM) Professor, we are a little bit confused about where the number "a" came from in the proof done based on Mertens' formula. Does "a" represent the 4*delta? Also, is "a" sequence or is it a number on a number line?

Nadine K. (11/25/2012 at 5:46:52 PM) will do then. thanks

Evangelos K. (11/25/2012 at 5:44:54 PM) Nadine, you can bring your laptop along tomorrow so that we can take a look but otherwise it does not make too much sense.

Nadine K. (11/25/2012 at 2:24:44 PM) Professor, I redownloaded the miktex complete version so the tikz could work, but it still isn't working out. Do you have any recommendations?

Evangelos K. (11/20/2012 at 10:29:40 AM) Maryam, thanks for catching this. I have taken care of it.

Maryam M. (11/20/2012 at 9:45:17 AM) Professor, I think there is no link from the drop-down list to submit the current homework assignment. Can you please take a look at it, thank you.

Evangelos K. (11/14/2012 at 7:13:42 PM) Nicholas, the program that you are using (texworks) does this for you automatically. The pdf file should be found in the same directory as your source file once you run LaTeX.

Nicholas L. (11/14/2012 at 6:02:00 PM) Professor, How can I convert a LaTeX document to a PDF file?

Evangelos K. (11/9/2012 at 4:28:20 AM) Nadine, I would have to take a look at the code. However if you center the picture after some text, then there should be no problem determining how it is going to show.

Nadine K. (11/9/2012 at 12:38:11 AM) what do you do to a picture in latex that refuses to move from a given position? maria and i have the hilbert curve picture which regardless of where it is in the coded version, still ends up being in the same spot after typesetting. how do we fix this?

Evangelos K. (11/8/2012 at 6:07:50 PM) Mohit, please leave graphics and pictures for later. Let's focus on the text now.

Mohit D. (11/8/2012 at 4:34:14 PM) Professor, how is one able to insert pictures in LaxTex from the internet?I tried using \begin{figure} \centering \includegraphics{../../Downloads/quad.jpg} \label{fig:quad}\end{figure}but the software does not read the extension.

Evangelos K. (11/6/2012 at 8:19:21 AM) Nadine, as I told you in class we will have to run some experiments together and see how to include the graphics in your paper.

Nadine K. (11/4/2012 at 9:27:42 PM) Professor, I am trying to input codes to create some fractals but a lot of the parts are suppose to be entered into the preamble. How do I do this though if I already have codes for the preamble for the actual writing of the paper?

Evangelos K. (11/2/2012 at 3:42:16 PM) Paul, I will have to check if this particular proof is due to Gauss.

As for your second question, we proved that it was bounded. We argued by contradiction. If it were not bounded, then it would have a subsequence that tends to infinity. But then P(that subsequence) would also tend to infinity.

Abel, that was a contradiction argument. We wanted to show that z_n was bounded. This is why we supposed that it was not bounded and under that assumption it has a subsequence that tends to infinity. But then P(that subsequence) would also tend to infinity , which contradicts the fact that it tends to a.

Abel T. (11/1/2012 at 7:49:08 PM) Professor,In my notes, we showed that P(z_n_k) goes to infinity. But I thought that P(z) was bounded? Which means that its subset shouldnt go to infinity?

Paul S. (11/1/2012 at 6:35:07 PM) Professor, how do we know that the sequence z_n is bounded again?

Paul S. (11/1/2012 at 5:14:59 PM) Professor, was the proof we did one of Gauss's proofs because I found out he did four different types of proofs for the FTA

Evangelos K. (11/1/2012 at 2:08:16 PM) Abel, you will have twenty minutes for this task. You are only limited by the time. Your presentation should contain an outline of the proof with some history and biographical notes with a level of details that is only restricted by the time that you have at your disposal for this task.

Abel T. (11/1/2012 at 11:58:56 AM) Professor,Are we simply supposed to mention a little about the mathematicians whenever appropriate, or are we to dedicate a significant amount of work on these mathematicians?

Evangelos K. (11/1/2012 at 5:50:13 AM) Paul, when we write the expression z\epsilon C, this means z belongs to C. In other words \epsilon stands for "belongs to". The truth is that it is typeset as \in and not as \epsilon.

The sequence z_n is a sequence that realizes the infimum of |p(z)|. In other words it is a sequence of complex numbers such that |p(z_n)| approaches the infimum of the set {|p(z)| as z is in C}. Remember the description that we gave in class: Imagine that this infimum is 1. Then we would choose z_1 so that |p(z_1)| is between 1 and 2. We would then choose z_2 so that |p(z_2)| is between 1 and 1+1/2. In general we would choose z_n so that |p(z_n)| is between 1 and 1+1/n.

As for your last question, suppose that the polynomial p(z) has the minimum of its values |p(z)| at w. Then the polynomial p_1(z)=p(z+w) has its minimum at 0. This is because in order to get w in the expression p(z+w) which is known to minimize the expression |p(.)|, we need to set z=0.

Please let me know if you have any further questions.

Paul S. (11/1/2012 at 12:43:54 AM) Professor, can you also explain again how to do the linear change in variable to assume that w=0 and p(0)=1

Paul S. (10/31/2012 at 11:07:38 PM) Professor, does|p(z_n)|represent the sequence of complex numbers within the polynomial |p(z)| or is it a subsequence of |p(z)|? Also, what did z \epsilon C mean again in a=inf{|p(z)|:z \epsilon C}

Evangelos K. (10/31/2012 at 2:49:49 PM) Euler, Gauss, Bolzano, Weierstrass would certainly be some names to consider. But you can do your own research and surprise me too, if you like. Everything should be in the context of the proof of the theorem of course.

Nicholas L. (10/31/2012 at 1:53:05 PM) Professor, for this Friday's quiz on the fundamental theorem of algebra, which mathematician did you want us to give background information on? I thought it was Euler, but there is not very much information about his work on the fundamental theorem of algebra.

Avni B. (10/22/2012 at 8:12:45 AM) Thanks, Abel and Professor!

Evangelos K. (10/22/2012 at 4:20:53 AM) Sandy, Nadine is right. By submitting it on Thursday you ensured extra credit but I did ask you to resubmit it for technical purposes.

Nadine K. (10/21/2012 at 11:01:18 PM) i think he said to resubmit that just for technical reasons. it won't hurt if u do

Sandy E. (10/21/2012 at 8:57:38 PM) Dr. Kobotis, do we have to re-submit the paper about quadratic reciprocity from last week? I submitted it on Thursday...

Evangelos K. (10/21/2012 at 7:25:18 PM) Avni, Abel's remark is correct. The dollar signs should open in the beginning of a mathematical expression and should close in the end. Also, to get nice big parentheses surrounding p/q you can try this:

$\left(\dfrac{p}{q}\right)=\left(\dfrac{q}{p}\right)(-1)^{{\frac{p-1}{2}}{\frac{q-1}{2}}}$

Abel T. (10/21/2012 at 6:42:16 PM) Avni, if you still need the help, i think the reason your having trouble is because your missing dollar signs for "(-1)^{{\frac{p-1}{2}}{\frac{q-1}{2}}}"

Avni B. (10/20/2012 at 2:19:22 PM) Hi Professor, I'm having an issue with the typesetting of this line: ($\frac{p}{q}$)=($\frac{q}{p}$)(-1)^{{\frac{p-1}{2}}{\frac{q-1}{2}}}. Can you help me figure out why? Thanks!

Evangelos K. (10/18/2012 at 7:49:51 PM) Maria, exactly. There is nothing for tomorrow. Tomorrow we will cover quadratic residues and you will have to do the homework for Monday. I mentioned that it would be a good idea to take a look at tomorrow's summary in advance.

Maria K. (10/18/2012 at 4:50:21 PM) Just to be sure: Professor, you mentioned that the tasks for 10/19 were also for the weekend and that we hadn't gone over the entire summary, yes? Are the tasks technically due on Monday?

Evangelos K. (10/18/2012 at 12:09:57 PM) Feedback for all recent paper submissions has been posted.

Evangelos K. (10/17/2012 at 6:07:31 AM) Gargi is exactly right!! Don't forget [1]. [1] is always among the congruence classes coprime to n.

Gargi V. (10/16/2012 at 11:32:32 PM) Maria, Nadine, and Avni: 1 is coprime to any number, so you might want to add [1] to your list!

Maria K. (10/16/2012 at 10:48:08 PM) Also dealing with a similar issue to Nadine & Avni!

Nadine K. (10/16/2012 at 9:01:27 PM) avni i came across the same problem and was very confused as well!

Avni B. (10/16/2012 at 5:52:03 PM) Hi, I think I'm a little confused. [5], [7], [11] are the only things coprime to [12], correct? But when I plug in 12(1-1/2)(1-1/3), that equals 4. So shouldn't there be one more class that is coprime to [12]?

Evangelos K. (10/16/2012 at 2:09:08 PM) Nicholas, the formula for the number of congruence classes that are coprime to n is given in terms of the prime factorization of n. Look at the summary for the formula. The formula does not work only if the prime factors appear with exponent 2 (as in the case of 36). They work in all cases. For example 12=2^2*3. Just apply the formula based on that factorization.

Nicholas L. (10/16/2012 at 10:53:18 AM) Maybe I misunderstood the lecture. I noticed that (for the example 36), you had to find two perfect squares that multiply to get 36. Those perfect squares were 4 and 9. There are no two perfect squares that multiply to get the number 12. Can I use three perfect squares?

Evangelos K. (10/15/2012 at 8:23:55 PM) Nicholas, of course it is not possible to use square root symbols in the case of congruence classes. What congruence class are you trying to invert and in which set?

Nicholas L. (10/15/2012 at 2:10:22 PM) Professor, I am a bit puzzled by the second assignment (congruence classes in $\mathbb{Z}$/ 12 $\mathbb{Z}$). In our class example, we had 2^2 and 3^2 = 36 with $\frac{1}{2}$ and $\frac{1}{3}$ being the multiplicative inverses. For the homework problem, it seems that \sqrt{3} must the denominator in one of the multiplicative inverses. Is this possible?

Evangelos K. (10/14/2012 at 7:47:23 AM) Feedback on yesterday's submissions has now been posted.

Evangelos K. (10/13/2012 at 11:23:54 AM) I posted comments and scores for all the recent submissions. The projects are proceeding very well and I hope that we will done with all the major proofs this week. I will check for more submissions and post new comments tomorrow morning.

Evangelos K. (10/13/2012 at 11:22:48 AM) Amy, it is $\mbox{Li}(x)$ not \nbox.

Amy C. (10/13/2012 at 10:01:57 AM) Professor, for writing Li(x), I typed in $\nbox{Li}(x)$, and it is not working.. what is the correct way to type it in?

Evangelos K. (10/10/2012 at 2:58:15 PM) I took care of the technical error. I have successfully received papers from the following groups: MN, AN, AM and KP. If you are not among these groups and you submitted your paper today, please submit it again!

Nicholas L. (10/10/2012 at 8:32:30 AM) Professor Kobotis, the same problem is coming up again when I try to submit the project paper. I tried using a different computer, and I still cannot get it to work. Are other students having the same trouble? Maybe I am doing something wrong.

Nadine K. (10/10/2012 at 12:34:35 AM) okay yes it was much larger and easier to see in this format. thank you

Evangelos K. (10/10/2012 at 12:24:42 AM) Nadine, I am sure you found \binom{m}{n}. However, let me make a remark. It shows better if you issue the command \displaystyle right before it, i.e. \displaystyle\binom{m}{n}.

Nadine K. (10/9/2012 at 8:02:27 PM) nevermind! got it!

Nadine K. (10/9/2012 at 7:53:17 PM) Professor, how do you write in the latex the format for combinations and permutations? for example m choose n

Nicholas L. (10/2/2012 at 7:29:03 PM) It is working fine now. Thank you.

Evangelos K. (10/2/2012 at 5:17:21 PM) Nicholas, I see what you mean. It should be fine now. Please let me know if you have further problems.

Nicholas L. (10/2/2012 at 2:42:32 PM) Professor Kobotis,I am having trouble again submitting the next version of the project paper. An error message comes up: "This Page Cannot Be Displayed." I believe I had the same problem before, but you fixed it. Can you please take a look at that and see if there is anything wrong?

Evangelos K. (9/25/2012 at 7:50:57 PM) Paul, you will have to mimic the work that we did in class for the Fibonacci sequence. That work has not been completed; so it is conceivable that at this point you can only take the steps that we took in class together. After tomorrow's lecture, you should be able to complete the process of finding the formula.

Paul S. (9/25/2012 at 2:16:25 PM) Professor Kobotis, do you want us to just give you the direct formula for the sequence, or do you want us to find a couple of the terms of the sequence and then express the direct forumla?

Evangelos K. (9/24/2012 at 4:23:50 PM) Abel, yes you guessed right! It is $a_{n+2}$.

Nicholas, we found two sequences that satisfy the relation a_{n+2}=a_{n+2}+a_n. Those where ((1+sqrt(5))/2)^n and ((1-sqrt(5))/2)^n. It is also true that any sequence of the form A((1+sqrt(5))/2)^n+B((1-sqrt(5))/2)^n for any two numbers A and B will satisfy the same recursive relation. The whole point is to choose A and B appropriately so that we get the same two first terms as the Fibonacci sequence. This is why we first set n=1 and set the whole thing equal to 1 (the first term of the Fibonacci sequence) and then we set n=2 and make the whole thing equal to 2 (the second term of the Fibonacci sequence).

Nicholas L. (9/24/2012 at 1:45:53 PM) Dr. Kobotis,I understood the process of finding the Golden Ratio, but I am not quite sure what you did to get the formula for the Fibonacci sequence.

Abel T. (9/24/2012 at 1:24:32 PM) Never mind I figured it out

Abel T. (9/24/2012 at 12:33:33 PM) Dr. Kobotis, how do you make subscript include more than n? When I put $a_n+2$, it only subscripts the n and not the +2.

Evangelos K. (9/22/2012 at 7:27:38 PM) Sandy, this is OK. The reason I might prefer a little the order in which we did things is because it is abundantly clear why each step is taken. My small objection with the approach that you suggest is that it is not as clear as in the other approach why you need to add and subtract b^2/4a. Other than that, this solution is also perfectly correct.

Sandy E. (9/22/2012 at 7:18:24 PM) Dr. Kobotis I have a question about the quiz. Do I have to show the proof exactly the same way as shown in the class? My first step involves adding and subtracting b^2/4a. Is this fine?

Evangelos K. (9/21/2012 at 7:08:51 PM) Paul, you can do this with an underscore. For example $x_n$.

Paul S. (9/21/2012 at 4:59:33 PM) Professor Kobotis, how do you code a variable to have a sub unit underneath it?

Evangelos K. (9/20/2012 at 8:03:41 PM) Sandy, all of your papers have been successfully submitted. I have not compiled them yet - this will happen over the weekend - but if they gave you no error, then the same will be true when I compile them too.

Sandy E. (9/20/2012 at 6:17:19 PM) Dr Kobotis, I just want to make sure that you've been receiving my papers. Can you check if you've received "four" papers from me thus far. Are they compiled correctly?Also, regarding the first version of the project, it doesn't have to be long, correct? It is just an explanation of the statement of theorem.

Evangelos K. (9/20/2012 at 1:14:08 PM) Amy, yes it is possible to do that. It is possible to insert any type of picture in a LaTeX document and we will possibly do this for the final project paper but not for the papers that are due each week. So please don't try that for those.

Amy C. (9/20/2012 at 11:37:24 AM) Professor Kobotis, is there any way to insert pictures from internet on the latex? When I use the "insert picture" and compile, it won't show the pictures, but it only shows the address of the picture

Evangelos K. (9/19/2012 at 7:57:23 PM) Nicholas, it was a technical error and it was just taken care of. I have received your other papers and you should now be ready to upload the Al Khwarizmi paper.

Nicholas L. (9/19/2012 at 7:48:49 PM) Professor Kobotis,I tried to upload my paper on Al-Khwarizmi, but there was an error message. The last three papers and the project paper seemed to work, but I am having problems with this one. Can you please let me know if you received it? Have you received the last 3 papers from me? I just want to make sure, since there is no way for me to tell if you have them.

Evangelos K. (9/19/2012 at 7:20:35 PM) Sandy, the presentation should include the proof of an equation with abstract coefficients: ax^2+bx+c=0 and yes I do want the detailed proof of how one gets the roots this way.

Sandy E. (9/19/2012 at 7:03:16 PM) Dr. Kobotis, do you want us to show the proof for the completing square method (Quadratic equation), or you just want us to practice an example and solve it? In other words, what should the presentation include? Thanks.

Evangelos K. (9/18/2012 at 8:46:09 PM) Nadine, unlike the project paper, every new version overrides the previous ones for the usual homework papers. This means that your initial upload has already been erased.

Nadine K. (9/18/2012 at 8:42:53 PM) Professor, the first version that I uploaded of the Archimedes paper did not include my references so please ignore that copy!

Evangelos K. (9/18/2012 at 6:31:06 PM) Maria, yes that would be a good idea.

Maria K. (9/18/2012 at 5:42:48 PM) Professor, If the main reference we used for writing some of the essays you requested was "The Math Book" would you still want us to cite it at the end of the copy/paste to "My Page"?

Paul S. (9/18/2012 at 4:12:35 PM) Alrighty thanks!

Evangelos K. (9/18/2012 at 2:27:13 PM) Paul, yes unless it is contained in an equation with more things, for example:

or

Paul S. (9/18/2012 at 9:50:32 AM) Ok so it's like this? $\pi$

Evangelos K. (9/18/2012 at 6:32:12 AM) Paul, it is \pi (always in dollar signs)

Paul S. (9/17/2012 at 10:22:59 PM) Professor Kobotis, I was wondering how you could code the pi character onto LaTeX

Evangelos K. (9/17/2012 at 8:17:11 PM) Sandy, this is precisely what I am saying.

Sandy E. (9/17/2012 at 7:24:24 PM) Dr. Kobotis, so you are saying that the volume displaced would always be the same no matter what type of fluid we are using?

Evangelos K. (9/17/2012 at 7:04:25 PM) Dear Sandy, volume has nothing to do with density. You could perform Archimedes' experiment (i.e. measure the volume of an irregular object) by using any liquid. All you need to see is how much the level of the liquid is rising in order to infer the volume.

Sandy E. (9/17/2012 at 6:42:29 PM) Dr Kobotis, I have a question regarding Archimedes Principle. Why does the principle states that volume of displaced "fluid" equals the volume of irregular shaped object. I think that the word "fluid" is inaccurate since it might imply fluids like alcohols, oils, etc.. which have different densities than water. I thought that we should always use the water since it has a density of 1 which doesn't affect the volume/mass measurement.

Evangelos K. (9/17/2012 at 1:27:53 PM) Amy, this is what you get if you stop at (n-1)^2. If instead you go to n^2 then you get what we got in class today. Alternatively, take today's formula and change n into n-1. Then you are going to get exactly the formula that you just mentioned.

Amy C. (9/17/2012 at 12:37:04 PM) Professor, on the proof for getting the volume of the sphere, the sum of sequence n^2 (1^2+ 2^2+...(n-1))^2) is shown as n*(n-1)*(2n-1)/6 which is different from what we learned today?

Evangelos K. (9/14/2012 at 5:41:52 AM) Paul, if the power has more than one digits, you have to put it in curly brackets. For example: $8.06\times 10^{63}$.

Paul S. (9/13/2012 at 4:42:14 PM) Oh alright. Another thing, if I wanted to express a number raised to the tenth, such as 8.06 x 10^ 63, how would I express that in LaTeX?

Evangelos K. (9/13/2012 at 4:37:56 PM) Paul, as usual just for one of them.

Paul S. (9/13/2012 at 2:25:54 PM) Professor, when said to find a topic that related to the readings, did you mean that we have to find a topic (via. internet or library) that is applicable to all five reading? Or, a presentation on the description of one of the readings?

Evangelos K. (9/13/2012 at 6:59:53 AM) Paul the main part of the proof is the computation of the volume of the hemisphere of radius 1. This is where we apply the method of exhaustion and this is where our focus should be.

Avni, Mochizuki has been claiming proof of the famous abc conjecture since August but I have not seen any official reaction. Sometimes, it takes time to check these proofs, especially since he has built a whole new theory. The proof of this conjecture would not imply the existence of any particular pattern for prime numbers. However it would enhance our understanding of additive number theory and it would imply a number of significant other results including Fermat's last theorem that we are going to mention in our course. Thanks for sharing the link!

Paul S. (9/13/2012 at 12:52:35 AM) Professor Kobotis, Im a little unsure as to what we need to study for the volume of the sphere. Do you want us to study how to find the volume of a hemisphere through the usage of the cylinders, or simply the volume of the sphere of radius r?

Avni B. (9/12/2012 at 9:32:38 PM) I just ran across this and thought I should share:http://www.livescience.com/23075-mathematician-claims-proof-of-connection-between-prime-numbers.html

Evangelos K. (9/12/2012 at 3:20:42 PM) Nicholas, I want you to know the volume of the sphere.

Daniel, I will respond to your email.

Daniel M. (9/12/2012 at 2:58:10 PM) Hey Professor Kobotis, I was making sure that you received my email?

Nicholas L. (9/12/2012 at 1:00:30 PM) Professor Kobotis, do you want us to know the proof for the volume of a sphere or the area of a circle?

Nicholas L. (9/12/2012 at 1:00:18 PM) Professor Kobotis, do you want us to know the proof for the volume of a sphere or the area of a circle?

Evangelos K. (9/11/2012 at 7:30:34 PM) Paul, not yet.

Paul S. (9/11/2012 at 4:18:56 PM) Hey professor, I was just wondering if you made any updates as to how we should turn our work in

Evangelos K. (9/8/2012 at 03:56:15 PM) Your topics have been assigned. Here is the complete list:

DR: Combinations and Permutations

MM: Symmetric Polynomials

BU: Transcendence of e

AN: Irrationality of pi

AG: Set Theory and Cardinalities

AS: The Prime Number Theorem

CE: Discovering Prime Numbers

MN: Fractal sets

KP: Solving the cubic and quartic equation

AM: The quadratic reciprocity law

MT: Fermat's little theorem

I have named each group by using the initials of your first names. We are going to be using this coding in the preparation of our papers. One thing that I want everybody to understand is that you are not going to be on your own when you are preparing your paper. I am going to discuss the topics with you, explain the math, suggest sources and do everything possible to make the process as smooth and instructive as possible. This coming week I am going to let you know about the workflow.

Evangelos K. (9/7/2012 at 3:37:12 PM) Paul, not for the essay. Not for any of the essays. The only time that I will ask you to use BibTeX is for your final paper.

Paul S. (9/7/2012 at 3:21:42 PM) Professor Kobotis, I'm assuming that we won't have to use bibTex to site our sources for this essay.

Evangelos K. (9/7/2012 at 1:58:52 PM) Nicholas, you are absolutely right and thanks for catching this. I meant to say "handwritten". It has now been corrected.

Nicholas L. (9/7/2012 at 1:52:08 PM) Professor Kobotis,The homework due for 9/10 says to do the following in LaTeX.By using a ruler and a compass construct:an equilateral triangle of given sidethe tangent to a circle at a given point,a circle that goes through three non collinear points.How can this be done in LaTeX? Did you mean to label this assignment as "handwritten"?

Sandy E. (9/5/2012 at 9:51:42 PM) Problem solved! Thank you so much, Dr. Kobotis.

Evangelos K. (9/5/2012 at 5:37:24 PM) Hi Sandy, yes I got it. You are getting this problem because the LaTeX program is producing a DVI file as opposed to pdf file. From the typesetting options, make sure that a pdf file is produced. Then this file will be easily printable.

Sandy E. (9/5/2012 at 4:41:34 PM) Dr. Kobotis: Did you get my email? Thanks

Evangelos K. (9/5/2012 at 3:25:37 AM) Maria, I got your corrected version and I did send you some comments.

Maria K. (9/5/2012 at 1:26:42 AM) I believe that I have figured it out. Professor Kobotis: Would I be able have you double check to be sure after class? Thanks!

Maria K. (9/4/2012 at 9:23:18 PM) The attachment I sent to you contained my old Latex pdf. I resent the correct one! Please let me know. I am very sorry for the inconvenience!

Evangelos K. (9/4/2012 at 7:50:41 PM) Maria, I did not get your email. Please resend it. My email is evangelos@math.uic.edu

Maria K. (9/4/2012 at 7:40:39 PM) Professor Kobotis: Did you receive my email? Please contact me if there are any errors. If not I will Typeset it exactly the way it is and disregard the errors it is notifying me with!

Evangelos K. (9/4/2012 at 7:24:05 PM) Nadine, I don't need anything too sophisticated for the references. In fact, LaTeX has its own very sophisticated reference system (called BibTeX) but we will use that only for the term paper.

Nadine K. (9/4/2012 at 7:12:36 PM) Oh and also for the question I just posted, easybib uses word document so would we have to then put it in latex?

Evangelos K. (9/4/2012 at 7:07:31 PM) Nadine, yes that would be fine. We are going to say a few more things tomorrow about using references.

Nadine K. (9/4/2012 at 6:58:26 PM) To name our sources, can we simply use easybib.com? Or is there a special format for the citation page?

Evangelos K. (9/4/2012 at 5:36:41 PM) Raghava, "the paper on Euclid's Elements" (not the Euclid write up...) will be due this Friday.

Raghava M. (9/4/2012 at 3:31:03 PM) So the Euclid write up is due on Friday?

Evangelos K. (9/4/2012 at 2:43:47 PM) Since the statement for the paper on Euclid's elements was clearly ambiguous (gotta love the oxymoron), I specified what I precisely want and gave an extension for Friday.

Evangelos K. (9/4/2012 at 2:16:13 AM) Abel, it will be done in LaTeX using \documentclass[11pt]{article}. The rest is taken care of by this command.

Abel T. (9/3/2012 at 10:45:53 PM) Is the Elements paper supposed to be double spaced?

Evangelos K. (9/3/2012 at 9:14:10 PM) Bartosz, as with any paper, you will have to name your sources.

Maria, email me your paper and I will try to give you some hints.

Abel T. (9/3/2012 at 7:53:19 PM) Maria K., did you remember to use dollar signs? and the forward slashes? or the squiggly parenthesis? all of these have to be exact. I would also just use the LaTeX Template, it does all the set up for you.

Maria K. (9/3/2012 at 4:24:25 PM) Hello, everyone! I've been coming across errors once I press "Typeset" on TeXShop. It will not show me the information as numbers/equations. What can I do? It keeps highlighting some of the codes we were given on the board in class as errors.

Bartosz G. (9/3/2012 at 2:15:02 PM) Do we have to do a citation page for our paper on Euclid?

Evangelos K. (9/2/2012 at 6:00:56 AM) Paul, yes TeXworks installs automatically once you have installed Miktex and you can use it in order to typeset your LaTeX documents.

Paul S. (9/1/2012 at 10:48:48 PM) I have a PC. Was TeXworks the program that came after installing the LaTex engine from www.miketex.org?

Abel T. (8/30/2012 at 7:30:31 PM) Also, on Mac, are there any significant differences between TeXshop and TeXworks? The user interface is slightly different but other than that I haven't noticed any major differences.

Abel T. (8/30/2012 at 7:28:03 PM) Maaria, I might know what you mean. You need to click on "Typeset" in order for your information to display as numbers/equations. It will display them on a new page which is your pdf file that you print out and turn in.

Evangelos K. (8/30/2012 at 7:15:09 PM) No it's not but I am not sure what you mean.

Maaria S. (8/30/2012 at 5:40:21 PM) okay is it suppose to change to numbers and equations when i press enter? because its not?

Evangelos K. (8/30/2012 at 5:21:36 PM) Maaria, yes that's right! This is the name of the program that I was forgetting in class yesterday: TeXWorks Thanks!

Maaria S. (8/30/2012 at 5:13:26 PM) Am i suppose to use the TeXworks for writing the article?

Evangelos K. (8/30/2012 at 2:20:41 PM) Abel, I went through that too a few years ago... And I still love to do things with this program. Take a look at the following website:

http://www.texample.net/

Abel T. (8/30/2012 at 12:24:17 PM) Not gonna lie, this is pretty addicting...

Evangelos K. (8/30/2012 at 8:05:18 AM) Nicholas, I would like to have a hard copy actually. I am also going to discuss the code with you tomorrow, since it is our first assignment. I hope that soon the use of LaTeX becomes a routine component of our class.

Nicholas L. (8/30/2012 at 6:29:41 AM) Would you like the final LaTeX document sent to your email, or do you want a hard copy?

Evangelos K. (8/29/2012 at 9:41:52 PM) Avni, to write a in italics, so that it is a number, just put it in dollar signs. You would write something like Let $a$ be a positive number etc...

Evangelos K. (8/29/2012 at 9:40:39 PM) Paul, that's precisely right. Upon taking the squares of 3k, 3k+1, 3k+2, you will realize that the squares of non-multiples of 3 are non-multiples of 3 and the squares of multiples of 3 are multiples of 3.

Avni B. (8/29/2012 at 9:20:31 PM) And to Paul S - that's what I did and (I think) it worked out for me.

Avni B. (8/29/2012 at 9:19:38 PM) Is there a problem in my formatting? I am receiving error messages: here is one of the lines causing issues. If \textsl{a} is a positive number with \textsl{a^2=3}, \textsl{a} is not rational.i was attempting to italicize. Thanks!

Paul S. (8/29/2012 at 8:57:53 PM) Tell me if I am correct. The non multiples of 3 are 3k +1 and 3K + 2? Then would you square them along with 3K?

Evangelos K. (8/29/2012 at 8:47:43 PM) Nicholas, now you will have to play with multiples of 3, not with multiples of 2. In other words, if n^2=3m^2, then n has to be a multiple of 3 (read my comment to Keane below), which means that n=3u. Therefore,(3u)^2=3m^2 etc.

Nicholas L. (8/29/2012 at 8:40:31 PM) To prove the square root of 3 irrational using the same method we used for 2, I end up with n^2 = 3m^2 which doesn't prove that n is an odd number or an even number. Am I taking this problem the wrong way?

Evangelos K. (8/29/2012 at 5:22:10 PM) Keane, today's proof was based on the fact that squares of even numbers are even and squares of odd numbers are odd. You have to prove something similar for 3. In particular you need to prove that squares of multiples of 3 are multiples of 3 and squares of non-multiples of 3 are non-multiples of 3. This will enable you to use an argument almost identical to the one that we presented in the morning. Instead of simplifying all powers of 2, you will begin by assuming that you have simplified with all powers of 3. I hope this helps. Please let me know if you have any further questions.

Keane H. (8/29/2012 at 5:13:40 PM) What was the tip you gave us to prove square root of 3 is irrational?

Evangelos K. (8/29/2012 at 5:43:19 AM) Welcome to our section's online message board. Please, post here any comments and questions that may be to our class material and schedule. I will try to respond to them in a timely manner but you may also get an answer from one of your classmates.

Here, you can submit the score(s) of your in-class quizzes. Please do the following:

- Choose your quiz number from the drop-down box.
- Write your quiz score in the box next to it.
- Write your UIN.
- Click the SUBMIT button.

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This is your page. You can do the following things:

Here, you can submit the LaTeX version of your paper. Please do the following:

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