Here, you can ask me any questions ranging in the Gr. 9 to second year university difficultly level. Put your questions in the comments below and I’ll be sure to edit this page to answer your questions.

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Here, you can ask me any questions ranging in the Gr. 9 to second year university difficultly level. Put your questions in the comments below and I’ll be sure to edit this page to answer your questions.

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Hi there, I’m in math finance with cs/pmath minors (4A) and apparently we took so many same courses. No questions, just wanted to say hi. Good luck with pmath 450 midterm!

Hi alexisonfire,

Thanks for visiting! While I am in math finance, I’m only in 2B, so I’m still not completely emerged in the PMATH’s and CS’s just yet. If you’re looking at the course list, the blue ones are the ones taken by a friend of mine who is doing a PMATH/CS major.

So how have you liked the program so far? Any interesting co-op jobs that you’ve received because of the program (if you’re in co-op)?

Hi; I am working on the realization of a Website of a University research Unit in Italy; we are updating our website and we would like to ask you if it is possible to use one of the images from your gallery. Can you give us authorization for this; I can be more precise by email or by messaging

Sure thing! Just by sure to provide the appropriate acknowledgments on your website.

Hi! I downloaded your “Assignment” template (current version) and it looks great. However, I’m a very rookie lyx user and can’t seem to find the way to edit the header line: I can’t enter the Season, etc. Any help with this? Thanks in advance,

Francisco

Hi Francisco,

Thanks for writing! To edit the header line, go to ‘Documents > Settings’ and in the sub-menu on the left, scroll down to ‘Latex Preamble’. In the text box, you should see a part that says ‘\lhead{[Season] 20[XX]}’. Change what’s in the curly brackets to change the text that appears on the left side of the header. The same formatting can be done for \chead{} and \rhead{} as well as \lfoot{}, \cfoot{}, and \rfoot{} for text on the footer.

Incidentally, you add the line \renewcommand{\headrulewidth}{0pt} to the preamble to remove the horizontal line in the header if it is too distracting.

Let me know if you need any more LyX or LaTeX related help and I’ll be glad to answer your questions.

Thanks a lot, worked like a charm 🙂

Once again, congrats on these beautiful templates.

Cheers,

F

Hello,

First of all, I would like to congratulate you on the amazing work you’ve done on this website: it is so inspiring!

Secondly, I’m Ludi, an IB student in her first year. And I too have chosen (or anyway am seriously considering) Mathematics as my Extended Essay subject. I too am in a school that looks at you like an alien for wanting to do something like Maths.

So, I think what I’m really asking is some outsider’s point of view. I am in a bit of a rocky situation in terms of choosing my topic, never mind the research question!

I am looking to talk about applied maths rather than pure maths.

I had initially thought of the typical gaming questions: probability and frequency in the lottery etcetc. But now I found an interesting course – that I am planning to attend – that focuses on ‘Prime Numbers and Cryptography’. I find the whole idea of maths concealing secret codes quite enchanting.

I guess other options are proofs or exploration of theorems. But writing 3000-4000 words there sounds rather dooming.

I think it’s quite obvious that I’m roaming in troubled waters of confusion, so if you have any extra advice, opinions, suggestions… anything really would be welcome!

Again, thank you for all the interesting material on this website and good luck with whatever course of study you are on now.

Ludi

Hi Ludi,

Thanks for writing! It’s nice to hear from someone else who is considering doing a topic in maths during high school. I can personally say that if it was an interesting ride for me studying a maths topic that was completely outside of the Canadian high school curriculum. It was, however, very much well worth it and led me closer to seeing what mathematics, in general, is.

So first, allow me to offer some advice about writing the maths EE (Extended Essay). To begin, I can tell you that unless you are entirely okay with mathematical formalities and building up structure in your essay from a small set of theorems, I would stay far away from a proof/theorem type of essay. You will only see that kind of presentation in university level maths and only in the honours classes and above. Now I’m not saying that proofs, theorems, lemmas, axioms (the works) are not important (they definitely are!), but for a student in high school such as yourself, you should spend most of the essay outlining your discoveries in the topic of your choice and not let the topic dictate what you write.

What is key here is that your essay should include a sort of adventurous spirit, where you detail interesting failures that led to interesting discoveries or connections that you may not have made before. For example, consider the problem

“How many rational pairs (a/b, c/d) lie on the unit circle (the curve defined by x^2 + y^2 = 1) where a, b, c, and d are integers?”

Perhaps you might first start by finding patterns in the pairs that lead them to being on the circle. Maybe you might decide that the answer is that you really can’t count them at all and there are infinitely many, but you are convinced that this is a “pattern matching problem”. And you may be right. There are many ways of solving this by just repeating all of the rules that you learned in school over and over again until you get what you want. But that wouldn’t be very fun, would it?

With some more out of the box thinking, however, this problem, surprisingly, without giving too much away, can actually be intricately tied to the prime numbers and in particular how many prime Pythagorean triples can be enumerated (it’s a nice exercise for you to think about!). Whole numbers and curves. Not what you normally would expect, huh? There are many interesting problems out there that connect the many fields of mathematics together. The Extended Essay is just a nice way of nudging you towards them.

As for the essay itself, these are the various types of essays that I generally think of when writing a maths EE:

– An essay about my problem solving methods and applying them to a VERY hard problem

– Researching an obscure field of mathematics (suitable for the determined high school student) along with personal opinions and conclusions

– Solving a problem very differently from how it is conventional solved

For reference, I went with method #2.

Finally, it’s nice to hear that you’re thinking of learning more about prime numbers and cryptography! It’s a very beautiful and thought provoking field as the methods for cryptography don’t end just at prime numbers but you can also think of “equations” that act like prime numbers. How do prime numbers “act” and behave? What exactly is randomness? These would be topics that could go well over the 4000 word limit if you wanted them to.

As a final bonus, if you want see a brief glimpse at mathematics at the university level but with language that’s clear enough for a high school student to understand, I recommend Hiroshi Yuki’s novel series, “Math Girls”, now translated in English. I, myself, have both books and the problems (with solutions) are challenging enough to keep you thinking but not too advanced to overload you with notation.

Anyways, if you ever need someone as a reference to help out with how to construct an maths essay, set a direction in maths that relates to your goals, or chat to clarify a maths topic, I’m always willing and able (as a maths major) to help!

Cheers,

William

Thank you for all the advice, it is proving very useful to get a clearer idea of what I am aiming for in this EE!

I am definitely going for the second method, as you have because I feel it is the most interesting and inspiring way to approach the EE. My friend has lent me a book on Cryptography – ‘Cryptography: the science of secret writing’ by Laurence Dwight Smith – which is great, but really more for background knowledge on cryptography rather than any mathematical explanation. I am sure the course will require or anyway provide some reading material, but in the meantime do you have any particular suggestion as to a maths book that could help me in this topic? Does the ‘Math Girls’ cover the prime number stuff required? (The same friend said something about phi equations – where could I get more beginners information on this?).

Also, between cryptography, probability in gaming (lottery and roulette), and something to do with the Golden Ratio, which one would you consider best for an EE?

Once I have made my mind up on the topic and research question I will let you know…

In the meantime, thanks for all the help!

Ludi

With regards to math books, the novel series “Math Girls” is more like a series of short high-school life novels that are interwoven with some very interesting math problems. It won’t have anything related to cryptography, but it is a good source for inspiration if you’re stumped on trying to find a topic.

If you really would like something to do with cryptography and numbers in general, try “An Introduction to Mathematical Thinking: Algebra and Number Systems” by Gilbert and Vanstone. You can probably find a copy at any large reference library. The content in there is pretty interesting and related to how mathematicians think about discrete numbers like integers. In terms of difficulty, a sharp high school student is more than able to read it and for reference, it was one of my first university textbooks.

If you’re up for real challenge to see what “rigourous” cryptography is about (away from Caesar ciphers and translation ciphers), check out “An Introduction to Mathematical Cryptography” by Hoffstein, Pipher, and Silverman. You can incidentally find an online copy here:

http://www.ic.unicamp.br/~rdahab/hddn/IMC.pdf

About the “phi functions”, I think your friend might be referring to the Euler-Totient function (http://en.wikipedia.org/wiki/Euler's_totient_function) which is used in RSA cryptography (what powers encryption across in the Interwebs). There are some interesting mathematical properties about this function but I can’t go into them because there may be notation that may not have learned just yet.

To answer your question about your EE topic, all three are good subject but it really depends on which of the three you that you think you could commit to for a year or so. Cryptography, in my opinion is the easiest to dive into at the high school level. Probability in gaming can be a bit of chore because the tools at your disposal in high school are fairly limited (although if you learned more about distributions and Bayesian probabilities there’s more to explore and discover). If you like reasoning about making combinations and permutations, it can be quite fun. The golden ratio essays are usually for students who love geometry and shapes. It’s both difficult and fun to see why many flowers like sunflowers and pine cones contain natural golden spirals and a fibonacci number of petals (hint: this is an optimization problem). You might even get into “fractals” (try googling it) going this route! In short, you should think hard about where your preferences lie and also how much time you have on hand to study your topic of choice.

If you need any more clarification about the above just ask.

Hello

I am grade 12 student. I am taking advanced functions class. Do you have any advanced books to recommend?

Hello Simon,

Thanks for visiting! To answer your question, there are a few resources that I do know of, depending on your background. If you feel that you are fluent in the material and content in G12 Advanced Functions, check out Santos’ online book “Precalculus, An Honours Course” which can be found here:

http://faculty.utpa.edu/gkioulekase/OGS/Santos/santos-precalculus.pdf

If you’re feeling a little more bold about exploring more areas of mathematics, and have had a little taste of real mathematical rigour, check out the resources at this website (although a few links may be broken):

http://faculty.utpa.edu/gkioulekase/OGS/Santos/santos-precalculus.pdf

If you feel like you’re comfortable with all of the books above, check out the main page in the second link for even more material:

http://hbpms.blogspot.ca/

Finally, if there is a specific area in mathematics that you may be interested in, I would be more than willing to help point you to some introductory material.

Hi William,

I am interested to know what co-op positions have you had since your first year and if they were hard to land. I am applying for B. Math/FARM this September and any input and advice will be much appreciated

Hi Joe,

Thanks for visiting! Here is a quick summary of past co-op jobs that I’ve had to date:

– Instructional Technology Support Assistant at Humber College (a teaching job)

– Pension Analyst at Morneau Shepell (pension and Excel VBA work)

– Enterprise Risk Management Analyst at TD (database work)

– [Current] Risk Modeling Analyst at TD (R&D, database, modeler work)

To comment on the difficulty of landing each job, I would say that because I had the right skill set and lacked skills in other areas at the time, each of these jobs were the most suited for me in their respective terms. In short, they were easy to land because they were relevant to my own skill set and I could talk quite a bit about that area of expertise.

As for advice, there are quite a lot of things that I could say about the FARM program, the math faculty, and co-op in general. However, I don’t want to dilute your understanding by offering information that you wouldn’t be interested in. Could you be a little more specific about what aspects that you would like me to talk about?

Hello!

I am also doing my EE in maths, is it possible for me to see a sample of your paper?

Im planning to do it on mathematcs in nature based on the 1998 movie PI.

“Restate my assumptions: One, Mathematics is the language of nature. Two, Everything around us can be represented and understood through numbers. Three: If you graph the numbers of any system, patterns emerge. Therefore, there are patterns everywhere in nature. Evidence: The cycling of disease epidemics;the wax and wane of caribou populations; sun spot cycles; the rise and fall of the Nile. So, what about the stock market? The universe of numbers that represents the global economy. Millions of hands at work, billions of minds. A vast network, screaming with life. An organism. A natural organism. My hypothesis: Within the stock market, there is a pattern as well… Right in front of me… hiding behind the numbers. Always has been.”

If you could send it to me on my email; frenchsasha@gmail.com

i would be greatfull 🙂

Hi Alexander,

You can find a copy of my paper in the Resource tab of this blog.