# Effective Research Notes

Note-taking for research is vital to your success as a mathematician. As I look back at some of my handwritten notes, I realized how poor they were. I had thought to myself, “What happened?” I was confused on how I should format it, if I should develop a proof of the proof, or what I should write down. I did not know if I should write down a proof or a intuition of the theorem, proposition, etc. This topic has really gotten to me as a young mathematician and is really putting me down. I want to strike a change, however the thought is not coming to me.

I would like to hear some of your advice and feedback on the research note-taking process. Any examples? Tips? Intuition and proofs? Personal experience? Etc.? The main problems I am having deal with the formality, neatness, and intuition as a factor. In addition, the greatest thing I want to be able to receive out of this is the ability to take notes with the right mindset.

In referring to intuition, I have yet found my way of incorperating intuition and proofs as a way of understanding the abstract concepts of mathematics. I have found through my experiences that the “how” of something is more intuitive to you understanding than the “why.” As a final remark, I am a young mathematician that has a lot more to learn. Understanding my intuition for mathmatics here could develop me into a well-structured mathematician.

As this is a matter of opinion, I can only offer my opinion. If you write 5 pages for 6 pages of reading (as you mention in the comments), you should certainly change something. Personally, I always attempt to maximize the ratio content/length. Generally speaking it also depends on what I intend to do; if it is supposed to be a conceptual summary, I would focus on intuition, some key examples, and without proofs. Also I would attempt to write down the intuitive meaning of certain terms as precisely as possible, and omit the formal definition (which you have to know anyhow). As for proofs, I generally omit them. (Not because I think that they are unimportant, but because you can look them more easily.) Perhaps write a few things about some standard proof techniques, e.g. partition of unity in differential geometry, or some key ideas. In any case don’t attempt to rewrite the textbook, it is likely a waste of time. Also always express things in your own words. If you want a concrete example, tell me about the 6 pages which you read, and I make a one page note.

I would summarize the content of chapter 2 as follows. Note that I did not include any definitions, but emphasized intuitive meaning, structures, and how the notions are in relation to one another. Also I emphasized what you can do with $\mathbf{R}$ (perform all kinds of operations, compare elements, etc.) and not “what $\mathbf{R}$ is” (i.e. how $\mathbf{R}$ is constructed). For thinking, this is often more convenient. Consider the related question: what are the natural numbers $\mathbf{N}$? The construction is: $0:=\emptyset$, $1:=\{\emptyset\}$, $2:=\{ \emptyset,\{\emptyset\}\}$,… but it is not useful to think about $\mathbf{N}$ in this way, because it wastes brain capacity and the only things you really use are the Peano axioms, the principle of induction, and the well-ordering principle.

The real numbers form a complete ordered field, and by this property it is determined up to unique order isomorphism (2-9). Thus the real numbers are endowed with the following structures: i) an algebraic structure (2-1 to 2-3), the field structure which governs the arithmetic ($+$ and $\cdot$) of $\mathbf{R}$. ii) an order structure which admits comparing the elements of $\mathbf{R}$. This order structure is compatible with the algebraic structure (this is the order axiom, 2-3). The order structure induces a metric space structure (via the absolute value), giving a notion of distance(2-5 to 2-6). By the order structure we have a notion of boundedness for subsets of $\mathbf{R}$, and for bounded above (resp. bounded below sets) one as has the notion of a supremum/least upper bound (resp. infimum/greatest lower bound). There is a certain duality between the supremum and the infimum, replacing $\leqslant$ by $\geqslant$. [If you want to understand this in detail: see here.]

The completeness axiom states that every nonempty bounded above subset of $\mathbf{R}$ has a supremum. (Intuitively this means that $\mathbf{R}$ “has no holes”, like $\mathbf{Q}$ (this is related to the fact that $\sqrt{2}\notin\mathbf{Q}$, learn the proof of this: every student has to know it). There is a unique element of $\mathbf{R}$ that is both positive and squares to $2$, it is denoted $\sqrt{2}$ (this is proven by the Archimidean principle, read the proof but there is no need to learn it by heart or something). [Theorem 2.3.3 is quite important for technical purposes, but I would return to it when I need it.]

The real numbers contains the rational numbers, which is not a complete field. In fact the real numbers is the completion of $\mathbf{Q}$ (i.e. “$\mathbf{R}$ is $\mathbf{Q}$ made complete”). Since $\sqrt{2}$ is irrational, $\mathbf{Q}$ is properly included in $\mathbf{R}$ and in fact there are more irrational numbers than rational numbers: $\mathbf{Q}$ is countable, but $\mathbf{R}$ is not (proof by Cantor’s diagonal argument).

In response to the comments: When reading something new, I first try to figure out what is the core of the text. (“Try” because what consider to be the core will necessarily depend on the level of your understanding, and thus the “core” is time-dependent.) Meaning, I understand the most important definitions first, then the main theorems (which form the core) without looking at proofs or anything. Then I consider more specific results. For instance for chapter 3: Definitions: Sequence, Cauchy-Sequence, Convergence of a sequence, Subsequence. Theorems: Thm. 3.1.1., Cor. 3.1.3, Thm. 3.1.4, Thm. 3.1.5, Thm. 3.3.3, Thm. 3.4.1, Thm. 3.6.1. At a first reading I would omit sections 3.2 and 3.5 altogether, e.g. becaue it is rather specific material (but useful, come back later!). Always try to make the link to what you have learned already. (E.g. how is the fact that every Cauchy-sequence in $\mathbf{R}$ converges related to the completeness axiom?) Pictures can help, but shouldn’t be taken literally. For finding out what is the core, I cannot really tell you how to do it, it seems to be a matter of experience. I never said that you should not write something, you must write. But don’t just copy all the theorems, try to understand them in several ways, their relation to one another, consider examples. At some point come back, and ask yourself what you have learned: and write a short note as I did. It might also help you if you know why you are reading the text: do you want to know something specific? Do you want to get a general overview? Do you want to calculate something?