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