Review for the Final

I completed the student ratings for this course.

Which topics and theorems do you think are the most important out of those we have studied?

Bijections, set notation, the introduction to analysis stuff, induction. Probably the triangle inequality, considering how frequently it has come up.

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

I think I primarily need to review definitions. Also it would help if I slowed down when doing problems. I need to remember that sets can go to infinity and that my intuition breaks when this is the case. Also, I should review negation rules.

What have you learned in this course? How might these things be useful to you in the future?

I think I have primarily learned how to format and structure solid proofs. This will come in useful in the future. I also learned some cases in which my intuition breaks. Further, I feel that this class improved my ability to think logically, and helped to solidify some of the things I learned over the summer at my math camp.

Thanks for a great semester!

-Batman

Math Talk due 10 December

I read Dr Jarvis' devotional.

I really like how he talks about accepting imperfection. I have found that perfection can be immensely paralyzing, and that settling for "good-enough" is often far, far better than striving for perfection.

At church the other week someone discussed how failure is the Plan of Salvation. The Atonement was there because we would fail, not as a backup plan in case we did. Mistakes are part of the plan. Dr Jarvis seems to be saying this as well.

This guy taught swim lessons and lifeguarded, just like me.😊

Test Prep due 7 December

Which topics and theorems do you think are the most important out of those we have studied?

I think the material surrounding bijections is very important. Also the basics of the delta-epsilon proofs including limits, continuity, sequences, and partial sums.
What kinds of questions do you expect to see on the exam?

I expect to see questions regarding all of the aforementioned things.
What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

The practice tests went very well for me. But there is a lot of material listed on the study guide that was not covered on the practice tests. Thus, I am most worried about the most recent stuff (limits, continuity, sequences, and partial sums), as I have had the least practice with it. I would like to see examples worked out, but I want a variety of problems so that I can get used to identifying which techniques go where. 

Section 36 due 5 December

Theorem 36.7 is cool.

I feel like I have some deep insight about this on the tip of my brain (is that a phrase?), but my conscious mind can't put words to it. So I will think about it and maybe I'll have something cool to say tomorrow. I think it has something to do with the facts that functions are closed under multiplication and addition and that domains can be bounded weirdly. I'm not sure.

Random question in preparation for the midterm: If f: A \rightarrow B and g: A \rightarrow B and we know that f is injective and g is surjective and f \not= g, can we conclude that a bijection must exist? I think so, though we can't conclude what said bijection is based off of the given information. I will try to remember to ask this during office hours.

-Batman

Section 35 due 3 November

There is a problem with my grade. My grade for homework 25 got put in twice (once as assignment 25, once as assignment 26). As such, all of my homework grades since then have been shifted by one. Hence, I have a grade for homework 34 despite not having turned it in yet.

This looks like a delta-epsilon proof.

I like the proof "sketch" descriptions. They are very handy.

"Thus we want 3|x 2| < ", or in other words |x 2| < "/3. This tells us what value for delta we should use. "

How does this tell us what value for delta we should use? Upon further reading, I think I might have figured it out. But it'll be good to review. ...Yeah, I figured it out.

This is really cool. I saw a delta-epsilon proof in high school calculus, but I don't think I fully appreciated it at the time.

Section 34 due 30 November

What on earth are partial sums used for?

More induction!

I have a question about corollary 34.7. If the limit does equal 0, then does a_{n} converge? Both this corollary and theorem 34.6 come very close to saying this, but not quite. Thus, is it not true?
Oh, a note later on in the section answered this. No, the limit equaling 0 does not imply that a_{n} converges.

Section 33 due 28 November

Parts of this look very similar to the material I studied in my independent study calculus 2 class. The rest of the material is new, but it looks like something that would belong in calculus 2. It makes me wonder how comprehensive my class was and just how much I should have learned but didn't...

Woah! What does the star mean? Does that mean that the scratch work is done?

Will we get docked points if the scratch work shows up in the homework?

Proposition 33.16 is weird.

Section 32 due 26 November

This proof is so much more complicated than I thought it would be.

The doubly infinite chain reminds me of how we are taught that there is no beginning nor end to the Gods. Thus each God came from an earth ruled by a God who came from an earth... etc. There is no beginning. It is hard to understand the implications of that.

"[T]he restriction of an injective function is still injective" is such a nice trait.

How does example 32.3 serve as an example of the Schroder-Bernstein theorem? I don't really see how it makes use of the theorem.

Section 31 due 20 November

Is the best way to disproof the lack of a bijection to assume there is a bijection and seek a contradiction? Or should we generally try to find some element that illustrates that the function is not surjective?

I am confused about the proof of the |S| < |P(S)|. Also I'm not following how the barber analogy relates...

Is N the biggest countable set? (And all sets whose cardinality is equivalent to |N|?)

-Batman

Section 30 due 19 November

My intuition tells me that the proof of theorem 30.4 is incomplete. We can continue the pattern to infinity in theory, but because we never actually get to infinity we have no way of knowing down the line if the resulting decimal expansion is truly not achieved in our function... I am going to have to retrain my intuition.

I do not understand the proof that (0,1] has continuum cardinality. What is the purpose of S?

That was a short section...

-Batman

Section 29 due 16 November

How is the diagram used? Is it merely a heuristic for how to count in a way that gets every possible combination exactly once? If so, then it makes sense, though I'm not sure why this is considered a "clever method." It seems pretty obvious, assuming I understand it correctly.

The proofs addressing the rational numbers are interesting.

-Batman

Section 28 due 14 November

WOAH!!! Crazy proofs are happening. See theorem 28.4. That's pretty epic. Though I wonder how we can conclude that the proofs must have the same size, considering the table only portrayed a subset of each set. How do we know that one set doesn't "run out" of elements to pair with the other set?

I was going to ask you if it is an adequate proof of bijection to simply find an inverse function between the same elements as the function. Then I forgot. Then I remembered, but your office hours were super full. I'm hoping this will remind me to ask you when I see you tomorrow.

I would like to settle a debate I was having with a classmate who I will leave unnamed. Take an infinite set and divide it into two subsets such that each subset is mutually exclusive and that between the two of them you cover every element in the set. Can one of these subsets be finite, or do they both have to be infinite sets?

This is super exciting material. I am eager to start the homework.

Section 27 due 12 November

The implications of 27.3 are cool.

Theorem 27.5 is useful. I did not know that, though it is logical.

Woah, surjective reductions are interesting. Are they related to projections in linear algebra?

It keeps messing me up that the range of the function is called the "image." Grrr...

Preimages are cool.

Section 26 due 9 November

I did not know the properties in theorem 26.12. I'm not sure I follow their proofs.

I find it interesting that there are so many instances of an "identity" object in math. For example, $I$ is the identity matrix that does not change any matrix it is multiplied by. In scalar multiplication, 1 times anything equals anything. In addition, 0 plus anything equals anything. Now we have an identity function. I seems somewhat counterintuitive that something that doesn't affect whatever it is applied to is such a powerful tool in math. That's what makes math interesting!

It would have been very nice to have known the technical definition of the inverse relation at my math camp. It would have made my life easier. Now I'm salty.

Section 25 due 7 November

Remark 25.2 is something that I don't think I ever explicitly realized, though it makes perfect sense.

There isn't a whole lot of new material here.

Ellie's new fish is sick (again), so I'm gonna try and comfort her and forgo a longer post here.

-Batman

Section 24 due 5 November

I find the function notation as a set of ordered pairs very interesting.

I LOVE it when all the math comes together. The notation used in previous chapters of this book in combination with the usual function material is a wonderful sight to see. I'd imagine we are going to get into bijections shortly, which will then connect what we are learning in linear algebra to this class, in addition to some number theory I learned at my math camp. I'm excited.

Is the characteristic function mentioned in this chapter similar to the characteristic equation we learned about in linear algebra? I am thinking that the characteristic equation defines a function spanning the domain described by the matrix. If we fall outside of the domain of the function, then the characteristic function returns a 0.

How do equivalence relations fit into all of this?

I'd love to see more (and different) examples of the "well-defined function" thing.

OH MY GOODNESS! I just turned the page and we will be doing injections and surjections in the next chapter!!! Happy day!!! I learned about these at my math camp. They were super fun and I loved working through problems about them with my instructors! I'm a nerd.

-Batman

Test Prep due November 2

Which topics and theorems do you think are the most important out of those we have studied?

Induction!!! Also probably the stuff with binomial coefficients.
What kinds of questions do you expect to see on the exam?

Proofs. I love proofs. I expect to see at least one induction and one strong induction question.
What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

I would love to work through another strong induction problem (on my own, with the option to ask questions if needed). I'd also like to see more examples of the binomial coefficient and theorem (in class).

-Batman

Mathematical Writing Slides 31 October

It really annoys me when the words "obviously," "clearly," and "of course" are used in describing math. It is impossible to ask a follow up question based off of a statement containing these words without confessing that it is not, in fact, obvious to you. I heard this phrase thrown around at the math camp I went to. Some of the students used it as ammo to purposefully talk down to whoever they were having a discussion with.

A lot of this material seemed intuitive. For example, I always try to use good English when I am writing, even if it is math.

-Batman

Section 23 due 29 October

I'm not sure I follow the logic here in the statement immediately after definition 23.3. I would copy and paste it, but the blog doesn't really have the capacity to handle that much mathematical detail. The next example helped to clarify, but I'm still not 100%.

23.C is cool. I'm excited to work with this material.

Section 22 due October 26

I have looked for a way to express the idea behind the floor of x for years! Now I know the official formatting! This is a fantastic day. It shall go down in history. Lol😊

I'm curious to see how we will use transversals. It is a subset of a bigger set, which is useful. Could it play over into linear algebra or multivariable calculus... somehow...? It is a half-baked thought, but intriguing.

-Batman

Section 21 due 24 October

Example 21.15 ended with a triangle in place of the usual QED square. What does the triangle mean?

Determining whether or not something is an equivalence relation seems to be the kind of thing that you just have to practice over and over until it becomes intuitive. It also seems like a powerful tool that we will make use of later, though it seems fairly arbitrary/pointless in a lot of the current examples. I'm excited to see where it will go! I feel like it is used in something I have seen before, but I can't place what it is...

My impression is that we use these blog posts to communicate to you what material we are familiar with (have I seen the content before?) and/or if we don't understand all or part of the content of the section. I'd imagine they also help you check who is actually doing the reading. Am I correct in this assumption? I don't mind doing the posts. I merely ask because I am not entirely certain that the content of my posts is actually fulfilling the purpose behind them. Are these posts the sort of thing you are looking for?

Section 20 due 22 October

This section seemed to mostly just rephrase things we have already been doing for years. I'll be curious to see why it is useful and where it goes.

I like the symmetric, transitive, etc thing. It is very satisfying to read.

That's all I got for this section. Sorry!

Batman

Section 19 due 19 October

I like the comparison of prime numbers to atoms. I've never thought about them like that before.

I'm curious if the proofs of these theorems are more, less, or equally important as the theorems themselves. Normally I think of proofs as a means to demonstrate why we can use an idea. However, this class has been so proof-oriented that it makes me wonder if we are supposed to be gaining more from the theorems or from their proofs. Or maybe we are supposed to be gaining intuition about the types of things we can use proofs for, or why it is important that we can prove things.

I don't understand how example 19.12 was derived as an example of canonical factorization. It doesn't seem to fit the definition...

19.14 is interesting.

Section 18 due 17 October

"Primality" is a weird word. I don't think I've heard it before.

Since GCD(7,9) = 1, does that mean that both a and b are relatively prime? Or are they relatively prime to each other? The latter makes far more sense, as 9 is not prime. Is relative primality unrelated to normal primality? Just wanted to double check. I wonder what good a definition dependent on a relationship to another number will do.

This section was pretty cool. Does this count as number theory?

Section 17 due 15 October

I'm not entirely sure I follow the logic behind algorithm 17.21, but it is clever. However, the algorithm seems fairly tedious. Is it one of those things that you never do by hand because you just plug it into a computer and let it do the work? It doesn't seem hard to program.

As a side note, I can tell you enjoyed making the labs for 495R. It is cool to seem them come together so well.

Section 16 due 12 October

Combinatorics is a cool subject because it takes really simple principles like "n choose k," Pascal's triangle, etc and is able to ascertain quite a lot of new information.

I like the mixing of tools in math to solve more complicated and not at all intuitive problems, such as the proof that binomial coefficients are integers.

Why are induction proofs considered inelegant? They are powerful.

I don't really know what else to say about this chapter...

-Batman

Section 15 due 10 October

The use of variables in the strong induction proof about one way roads and cities is confusing me. Not every road runs either toward or away from X (assuming we are choosing one arbitrary X to use consistently throughout the problem). There are roads that do not go through X...

I have never worked with strong induction before, though I have seen it defined. This will be interesting to experiment with. I'm excited!

Batman

Section 14 due 8 October

I have never seen the "peeling off" technique that was used in both this chapter and in class today. While clever, it seems borderline sloppy - like you're purposely losing information in order to prove something else.

I didn't realize that factorials were a math tool not introduced until college. I think I saw them first in fifth grade. Although, that may have been because my dad showed them to me and not because we actually learned them in class... Though I have definitely seen them between then and now, so it's interesting to me that it is being introduced in this section.


-Batman and her sidekick: the Great Mr Alpha

Section 13 due 5 October

The analogy my professor used to teach us induction is as follows:

You are making a train of dominoes. The goal is to have the whole chain fall over when you push down the first domino. The base case is checking to make sure that the first domino will actually fall over. The induction step is checking to make sure that each domino will knock over the next domino in line. Thus, you know that the whole chain will fall over, because you know the first domino will fall over and you know that it will knock down the second domino, which will knock down each subsequent domino, etc.

I really liked that analogy.


When I read this chapter it was intriguing to me to see a different approach to induction. It is still the same concept, but the way it was explained and some of the formatting differed from what I learned. It's interesting to see how different people approach the same idea.


The pigeonhole-principle is pretty cool. I remember learning about it indirectly in sixth grade. I like how math builds on itself. E.g. the principles that I learned when I was younger prepared me to build upon them to learn more complex principles now.
This principle comes in very handy for solving problems that I wouldn't know how else to approach. We used it a lot at my summer math camp.

Batman

Test Prep due October 3

Which topics and theorems do you think are the most important out of those we have studied?

I think the most important theorems are the ones that provide useful tools that allow us to look at a problem in a new way. That sounds very vague.

Here is an example: The idea that the contrapositive and the original statement have the same truth values. This allows us to approach the problem contrapositively, which can save time in situations where the direct proof would have gotten ugly. I think the theorems that allow us to look at proofs from a different approach are more useful than the smaller theorems one would use within a proof.

What kinds of questions do you expect to see on the exam?

Proofs. 😀 I think that it will be largely focused on proving various statements in a variety of ways, (types of problems similar to those on the homework). However, I hope they are less repetitive than the homework. Also, if we could avoid extremely weird proofs that require some sort of foreknowledge in order to solve or proofs that you have to "just see," that would be phenomenal.

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

I don't understand the solution to problem 4 on the practice exam. It seems as though the power set of {a,b,c} is {{empty set}{a}{b}{c}{a,b}{a,c}{b,c}{a,b,c}}. Thus A, or {a,b,c} is a subset of the power set of A. Or is {a,b,c} an element of the power set of A? If this is the case, why can it not also be a subset?

-Batman

Section 12 due 1 October

I'm not entirely certain that I follow the transition from original statement to the contrapositive assumptions made in example 12.5.

This section just combined previous concepts in new ways. Seems simple enough. I suppose we shall see if it really is that simple in the homework exercises.

This whole course could feasibly pass as a philosophy course. It is more of a course in logical thinking than it is in math thus far. I would love to see this course combined with the material about sets that we are learning in linear algebra.

-Batman

Section 11 due 28 September

"Proposition 11.6. One of the digits of 2 = 1.414213562 . . . occurs infinitely many times.

Proof. There are only finitely many possible digits {0,1,2,3,4,5,6,7,8,9}
but there are infinitely many decimal places. QED"

It seems to me that this proof isn't complete without at least mentioning the pigeonhole principle. They do address it in the explanation after the proof, but since it isn't included in the proof itself, I am not in convinced that this proof is complete. Is it? If so, why? Would this get full credit on the homework/test?

I don't understand the proof of proposition 11.7.


The formatting is so weird. Sorry if the font randomly changes sizes.

-Batman

Section 10 due 26 September

I LOVE set theory! It was a main reason I became a mathematician. I love the idea of defining any set you want and then messing around in the universe defined by that set to see what you can do. I would always ask math teachers in middle and high school how on earth people created new theorems and invented calculus and came up with all of the brilliant math that they did. None of them had an answer for me. When Ken (my professor my senior year) learned that I wanted an answer to that question he began to stay after class to teach me about creating universes through sets and new laws to solve problems and create new math. That is when I fell in love with math and realized I wanted to study it. Now I'm here.

Anyway, as far as something challenging about this section, I am still mixing up the ∩ and ∪ symbols. I just need a good way to remember which is which.  Also, I don't remember learning what a set with a bar over it means. Maybe the negation or inverse or something? We might have learned it and I'm just being forgetful...

How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
I feel that the lecture and the reading did prepare me adequately for the homework. I probably spend about 2-4 hours per class on this class.

What has contributed most to your learning in this class thus far?

I like the current format where we read the chapter to gain an initial impression of the content, then review it in class, then practice it on the homework. I think this strategy of reviewing everything several times helps me learn it more thoroughly and help commit it to long term memory.

What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)

I think there are some homework assignments that get immensely repetitive. Even though there are only 6-10 problems, on some assignments most of the problems require the same technique on nearly all ten problems. I understand that lots of practice is beneficial, but after four problems of nearly identical work, I start to get bored. Maybe cut back on identical homework problems so I am not spending as much time on homework assignments?

Your truly,

Batman

Section 9 due Monday 24 September

Proof by contradiction only works if you are certain that you are working with a scenario that is a boolean. I can imagine a scenario where someone thinks "true" or "false" are the only answers, only to draw a conclusion based on proof by contradiction, when in reality, they are ignoring another possible answer. I do not know how this would play out in practice... maybe in quantum mechanics? I'm not sure if that is even relevant to this. Just a thought...


I would like to hear more of the logic behind the following quote:

"[I]f a proof can be done without contradiction, then that is usually a better option, because you never enter an "imaginary" world where you assume something you are hoping to show is false."

It seems to me that a proof is a proof, right? I don't understand why mathematicians have aversions to certain types of proofs. I have been told that proofs by induction are considered ineloquent and thus should be avoided when possible. Why are some proofs superior to others? "All proofs are equal, but some proofs are more equal than others." :-)


Yours truly,
Batman

Section 8 due 21 September

Proof by cases was something that I have picked up on through contexts in my math classes last year. It was never something that was taught explicitly. It is good to have written patterns confirming what I already figured out. (Indeed, most of this class has been writing out the rules to things that I only ever figured out from context; in the past I could tell you that something works within the context of a proof, but I wouldn't be able to tell you why. This class is changing that!) WLOG was also something that I picked up on through context. So far, the book also seems to address it only through context. I would love to have a more in depth discussion about when it is acceptable to use "without loss of generality" in a proof, as I feel I only have a vague idea of when to use it.

I want to go over the modulus syntax used in this chapter. I think I would have a better understanding if it were explained to me. The modulus is used in a slightly different way than I have ever seen it be used before.

Section 7 due September 19

If a contrapositive restricted to implication statements? Can a contrapositive exist for any other type of statement? If so, how does one go about proving that a contrapositive has the same truth value as the original statement for all statements?

That was my primary thought process reading this section. I thought the section was interesting, but my brain is too tired to form coherent thoughts beyond that. I don't feel well, so I will be going to bed now in the hopes of feeling well enough in the morning to dig deeper into the material during the lecture.

Section 6 due 17 September

This section alternated between clever tricks for solving proofs and inherently obvious observations about mathematics (see definitions 6.1 and 6.2). Parts reminded me of this xkcd post. However, I liked the solutions to finding whether or not the premise is true. That is not how I would have approached that type of problem. See proposition 6.20.

I learned about the fundamentals of proof at the Prove It! Math Academy camp I went to. It is nice to dive back into the same material. Different parts of proofs are emphasized differently in the textbook, classroom, and math camp. It is intriguing to see the similarities and differences between how all mathematicians approach teaching and solving the same concepts. Also, the camp made the word "trivial" into a running joke (sort of by accident), so I have lots of fun memories associated with that word!

Chapter 5 due 14 September

I think it is really cool that a set can have a lower bound that is not actually included in the set. It makes me wonder the purpose of lower bounds, though, as -1, -2, 0, etc are all lower bounds for the set (0,1]. If the lower bound can be any arbitrary selection from anything within a specific domain that is exclusive of the domain of the original set in question, what use does the lower bound have?

I understand the greatest/least element and upper/lower bound thing when I read it on paper, but I think it would sink in better and I would have a more full understanding of it if I were to hear it explained. That was probably the hardest part - connecting the meaning behind the phrases to the bigger picture. I struggle with that when I am reading more so than when it is explained.

Chapter 4 due September 12

I would like to clarify the syntax in the following statement to ensure that I understand.

"As an example, if we let x and y be variables with the domain R, and let P(x,y) be the open sentence
P(x,y) : x>y,
then the sentence ∀x ∈ R,P(x,y) is not a statement, because it still has a variable that has not been specified or quantified (namely y). In order to make it a statement, we need to evaluate y; we could say ∀y ∈ R, ∀x ∈ R, P (x, y), which means that
For all real numbers y and for all real numbers x, it holds that x > y."


Even though the author stated that "x and y [are] variables with the domain R," because this is not explicitly stated in the claim itself ("∀x ∈ R,P(x,y)"), it is not a statement. I am assuming that the aforementioned thing in parenthesis is the only thing I am supposed to be looking at when evaluating the statement, rather than look at the paragraph as a whole. Is that correct?

Also, I was taught not to start sentences with math symbols when writing formal math things. However the book does so: "P implies Q. P only if Q. Q if P." What is the proper standard?

I am wondering what practical application that this section has. I'd imagine that it is more of a thought exercise than an actual useful tool in most cases. However, I have read formal and relevant proofs that did include statements similar to the ones about which we are learning. However, I have not seen them in nearly as much depth as the book is providing.

Chapter 3 due September 10th 2018

Honestly, the hardest part is memorizing all of the symbols and the specific details behind what each symbol means. The concepts themselves aren't particularly difficult.  It is hard for me to keep track of the meanings of ∧ and ∨ because they are so similar in shape.

Understanding logical statements might be a polarizing ability (as in, most people are very/decently good at it or very bad at it). It isn't particularly difficult for me, but I have friends who struggle immensely. It makes me think that some people's brains are hardwired differently than others'. In an English class I took we were examining riddles and logical statements. The teacher and most of the other students had to examine each statement in great detail and still ended up confused, despite spending exorbitant amounts of time on each problem. The teacher ended up coming to me for help when he confused himself to the point where he couldn't help the others. The other students who understood the statements easily were all very good at/interested in math. Maybe the type of people math attracts are also the type of people who have an easier time thinking logically.

Section 1 due on September 7

I had never heard of power sets before. I thought our discussion about them in class was enlightening because it made me realize that the number of elements in a power set is much greater than the number of elements in the original set. However, I do not see an immediate use for power sets. I am eager to see their relevance in the world of advanced mathematics.


Assuming I have a correct understanding, I absolutely adore cardinality because it makes combinatorics possible. I learned about combinatorics at a summer math camp I attended and it was a fascinating subject.

Section 2 due on September 7

I do not understand the point of attempting to graph the unions between sets. Is it meant to be purely a visual representation of the concept of unions, or will graphs of unions play some significant role later on? I'd imagine it is probably the former, though I find that visual representations can provide an alternate way of viewing problems that can prove enlightening.


I find it very exciting and fascinating when I learn math that provides context for simpler math I learned at another point in time. For example,

"The set R2 = R × R is called the Cartesian plane. We view elements in this set as points {(x,y) : x,y ∈ R}."

This is a perfect example of the way the math curriculum simplifies math to make the basics easier to understand. Sometimes I think it is a good idea to simplify the content. After all, why would you learn about sets before you can add? Other times I wish I had been exposed to the "big picture" of math much earlier. When I was a senior in high school I began to witness this bigger picture behind math for the first time. That was ultimately what convinced me to switch from my plans to be a physics major. It makes me sad that so many of my peers will never see the bigger picture because they did not progress far enough in math. It makes me think that we need to drastically rethink the primary education math curriculum to ensure that all students have had enough exposure to math to truly understand whether or not they want to pursue it. Most of the math I learned up until my senior year of high school were merely tools for exploring more complicated subjects; they didn't even address the meat of the subject. (How on earth are you supposed to phrase that sentence correctly? Is "math" inherently singular or plural?)

Introduction due on September 7

Hi!

I suppose you would probably consider me to be a sophomore. However, this is my first year of college. I am majoring in pure math.

I have taken calculus 1, 2, and 3 (multivariable). I have also taken a differential equations class that was supplemented with some linear algebra topics (eigenvectors/values and basic matrix math).

I chose EMC2 because I want to get to know other mathematicians so that I can begin developing a professional network of mathematical colleagues. Also, it is nice to have friends who understand my math jokes!

My math professor my senior year of high school ultimately convinced me to be a math major (I had originally planned to major in physics). He answered my questions in as much detail as he could, and was even willing to stay after class for 45 minutes to teach me more about a theory we had touched on in class but would not be delving further into. He helped me to understand the mathematical community and the sorts of things that I could pursue if I chose to study math. He also gave me the resources to learn more and put me in touch with several experts so that I could get additional opinions. He never once pressured me into switching to math; he was immensely supportive when he learned I wanted to study astrophysics at the beginning of the year.
Tl;dr: He thoroughly answered my questions about 1) the math we were learning in class, 2) math I was learning on my own, and 3) what it would mean for me to study math in the long run. He was my friend who simply wanted me (and worked to enable me) to learn as much as I could about what I thought was interesting.

I am a swimmer. I do not like brown bananas. I only like bananas when they are still slightly green, just about to turn yellow. I like being outside and weightlifting. I love the Enderverse by Orson Scott Card.

I can come to office hours as currently scheduled.