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I'm a second year math student. And I've the following problem.

When I prepare myself for an exam, I can distinguish two phases. First I'm mainly interested in whatever is necessary to pass the exam. This means that I do not always read the theory too intensely, and I'm more focused on finding the "tricks" that I need to solve the problems.

Second phase: After some time, I feel like I've found most of the tricks, and at this point, I usually spend more time in understanding the theory. Just because, I know I will pass the exam, and I think it is fun to understand the theory better.

However, sometimes when I really try to understand the theory, I feel like understanding the foundation of something, doesn't make me feel I understand it better, it makes me only feel like I never really understood it. And this kind of frustrates me, and I'm not sure if it is even smart to try to understand a concept in that way as an undergraduate student.

For example, I'm studying complex analysis. I skipped the formal definition of complex numbers a little bit, and I just began making calculation and showing if mappings are conformal/holomorphic etc. Now I'm back trying to understand the construction of the complex numbers as pairs of real numbers etc. My brain is getting more and more confused the more I think about it.

My brains goes like: "If you define $\mathbb{C}$ as pairs of real numbers with complex addition and complex multiplication in the usual way. Okay that makes sense, we clearly don't have $\mathbb{R}$ as a subset of $\mathbb{C}$, but we can construct a bijection $f:\mathbb{R} \to \{(x,y):y=0\}:x\mapsto (x,0)$ that preserves every single property you want real numbers to have. So in that sense there is no difference between $\mathbb{R}$ and $\{(x,y):y=0\}$, and if we would define the real numbers as $\{(x,y):y=0\}$, then we have $\mathbb{R} \subset \mathbb{C}$. "Euphoric" Two seconds later, but wait... now I basicly define $\mathbb{R}$ as $\mathbb{R} \times 0$.... this doesn't make sense.... "Frustration" (just to give an example, I wouldn't post the whole list of things about theory of complex analysis that confuses me)

Sometimes I wonder if it there is any use of trying to understand mathematical concepts in this way as an undergraduate student. Should I just accept that it doesn't totally make sense and go on with solving problems?

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The question is what objects you "take for granted". If you take the complex numbers for granted, then $\Bbb R$ is just the fixed field by the conjugation map; if you take the rational numbers for granted, then $\Bbb R$ is their completion defined in several ways, and already you run into a similar problem that the rational numbers have "two copies" that you can work with. It's the same problem, and however you solved it in that case, you can solve it again here. (And if you construct the rationals from the integers, and those from the natural numbers, and so on, you run into that again.) –  Asaf Karagila Mar 12 at 20:20
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A student must be smarter than Sherlock Holmes to decipher what most authors mean by words like 'clearly' , 'obvious' , 'elementary' , 'trivial' ,etc. Especially for beginning students. Too bad the author is usually not available for Q&A. Join/form a study group. :) –  neofoxmulder Mar 12 at 20:40
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Well, you can't "define" $\mathbb{R}$ as $\mathbb{R}\times \{0\}$ since you're using the defined term in the definition... But $\mathbb{R}$ is isomorphic to $\mathbb{R}\times\{0\}$, which means they can be used for all the same things. It's nothing to be confused about; what would you expect to happen when you have one structure that basically duplicates another? –  Malice Vidrine Mar 12 at 21:00
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Perhaps I'm chasing your one example too far; apologies if so. As someone mentions, it could be true. And further, it doesn't matter if it is true. You can take the thing that is a subset of $\mathbb{C}$ and interchange it with $\mathbb{R}$ to your heart's content. This should lead you to notice that there's a notion of "sameness of structure", distinct from literal identity, that comes into play here, and that this notion tells you more about $\mathbb{R}$ than its actual construction does. –  Malice Vidrine Mar 12 at 21:33
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@Kasper : $\:$ The same thing happens for $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$. $\;\;\;\;$ –  Ricky Demer Mar 12 at 22:36

14 Answers 14

up vote 37 down vote accepted

Mathematics is really about relations between things. Therefore, while constructions are good and useful, you should never take them very seriously.

Construction agnosticism

Consider the real and complex numbers. Given $\mathbb R$, you can construct a ring which is isomorphic to $\mathbb C$ by taking pairs of real numbers and defining addition and multiplication in the usual way. Note here that I say that you can define a ring isomorphic to $\mathbb C$. We can ask the following question:

  • Is the ring we have defined actually the ring of complex numbers $\mathbb C$ itself?

But you shouldn't ask yourself that question. (Just because you can ask a question, doesn't mean you should.) It won't do you any harm; it's just not useful to.

What really matters? That there is a ring which has all the properties that we want from the complex numbers — such a ring exists. Then we can say: let $\mathbb C$ be such a ring, and then study the maps $f: \mathbb C \to \mathbb C$.

When we say "the" complex number field, the word "the" is a red herring; we just want to talk about some ring which works that way. Similarly, we don't really care about "the" real numbers. If $\mathbb C$ is "the" complex number field, then it contains a unique subfield $F$ such that $[\mathbb C : F] = 2$; we can then call these "the" real numbers. Just as we don't care if "the" complex numbers consist of ordered pairs of objects from some ring $S$, we don't care if "the" real numbers are the ring $S$ or some set of ordered pairs $(s,0)$ for $s \in S$ and $0$ the additive identity of $S$. It just doesn't matter — we just care about the proper subfield $F \subseteq \mathbb C$ which has all of the same properties as the real numbers, so we may as well adopt the convention that this subfield is the field of real numbers.

This applies to the integers as well. The von Neumann construction of the natural numbers makes $3 = \{ \varnothing, \{\varnothing\}, \{\varnothing,\{\varnothing\}\}$. Does this mean that $3$ "really is" a set which e.g. has the empty set as a member? Not really, because these ideas are totally irrelevant to what we care about the number $3$. We could consider any other "construction" of the natural numbers, in which case $3$ might not be a set at all (for instance, if we consider a set theory in which the natural numbers are atoms), in which case it is not only irrelevant to consider the maps $f:3\to3$, but these would not even be defined. All we care about is that $3$ is part of a collection of objects $\mathbb N$ which forms a monoid with some specific properties. The "true identity" of $3$ is beside the point.

Mathematical "interfaces" (in place of "foundations")

If this ambiguity bothers you, you can think of it axiomatically as follows: treat each of the sets we care about — such as $\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ — as underspecified objects, where we specify all of the properties about them that we could care about, and only those properties.

  • von Neumann's construction of ordinals describes a certain countable well-ordered monoid: we don't define $\mathbb N$ to be that monoid, but merely say that it is isomorphic to it, leaving further details to be filled in later.

  • From equivalence classes of ordered pairs of elements of $\mathbb N$, you can define an ordered ring $Z$, which contains a monoid $M \cong \mathbb N$ and which is the closure of $M$ under differences. Now, $\mathbb N$ can never be a subset of this set of equivalence classes; but it can be a subset of some other set. We never pretended to characterize precisely what object $\mathbb N$ is, so who is to say that $\mathbb N$ is not itself contained in a ring which is isomorphic to $Z$? Nobody, that's who; without loss of generality we may define $\mathbb Z$ to be a ring isomorphic to $Z$, and declare as a refinement of the earlier specification that in fact $\mathbb N$ is contained in $\mathbb Z$.

  • We may similarly declare that $\mathbb Z \subseteq \mathbb Q$, where $\mathbb Q$ is isomorphic to the ring of equivalence classes of ordered pairs over $\mathbb Z$ in the usual way. We also declare that $\mathbb Q \subseteq \mathbb R$, where $\mathbb R$ is isomorphic to the field of Dedekind cuts over $\mathbb Q$, or (equivalently) to the field of equivalence classes of Cauchy sequences, or any of the typical constructions of the real numbers. The set $\mathbb R$ isn't defined to be any particular one of these constructions, because (a) any of these constructions is as good as the others, and (b) we don't really care about any of the details lying underneath any of the constructions, so long as the properties we care about hold for each.

You should think of these refinements as axioms which we add during the process of doing mathematics. A definition is in the first place is only an axiom: one which defines a constant, such as defining by asserting ∀x:¬(x∈∅). These mathematical underspecifications — mathematical interfaces — are also axioms: having proven that a certain sort of monoid exists satisfying the Peano axioms, we assert that $\mathbb N$ is such a monoid, saying nothing more until it suits us to; and similarly declaring that $\mathbb C$ is a sort of number field of a kind which we've proven exists, and which happens to contain the field $\mathbb R$ which we mentioned previously without quite defining it completely.

Fundamentally, this approach to mathematics is not really all that different from what we usually do: it merely substitutes complete descriptions (what Bertrand Russell would call simply "a description") for objects, with partial descriptions. But pragmatically, in the real world as in mathematics, partial descriptions tend to be all that we care about (and in the real world, they are all that we ever have access to). Embracing this allows you to focus on what really matters.

If you are a mathematical "realist", in which the real numbers has an identity separate from our descriptions of it and has some fixed location in the mathematical firmament, this sort of wishy-washiness as to the "exact identity" of these objects may bother you. After all, if you imagine the possible identities of the objects $\mathbb N$, $\mathbb Z$, $\mathbb R$, etc. as you subsume them into more and more complicated objects, it would seem that the set of objects with which a set such as $\mathbb N$ could be identified recedes to infinity as our mathematical framework grows more elaborate. To this I can only say, "so much the worse for realism". If you want the freedom to construct objects and only concern yourself with the relationships that matter, in the end it is better to abandon this preoccupation with the precise identity of a mathematical object, and engage in mathematics as the creative, descriptive, and above all incomplete and ongoing endeavor that it is.

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Or the realist could say each construction of the reals observes the form of the real number. –  Jacob Wakem Mar 13 at 3:10
    
@JacobWakem: I suppose that by "form" you mean something suitably Platonic (otherwise, I could easily substitute 'form' for 'properties', and we're back on the slippery slope to the notion of mathematical "interfaces" outlined above). Does this mean that all of our usual constructions of the reals are akin to our constructions of chairs, in that we can by construction but imitate a dimly recalled/perceived shadow of the perfect form, which evokes but falls short of true "chairiness"? A true but unknowable definition of the reals is not likely to satisfy the OP very much... –  Niel de Beaudrap Mar 13 at 7:15
    
Good answer. $\!\;$ –  goblin Mar 13 at 10:53
    
I learned really much from your post, thank you ! Allthough you don't really answer my question, I wanted to show my appreciation. –  Kasper Mar 14 at 15:59
    
I thought to address the question of foundations which you touch on in your question. If you repeatedly get hung up on such details, this is the most important thing to know. You should accept someone else's answer if you find that my answer is tangential, but thanks for the compliment! –  Niel de Beaudrap Mar 14 at 21:58

It really depends on what kind of things you are talking about.

You certainly shouldn't accept theorems as true just because someone tells you they hold - especially not at an undergraduate level. Any good math curriculum should start with the very basics (say, natural numbers), and work its way up from there, proving every theorem that is encountered as the curriculum progresses.

What you seem to be struggling with, however, doesn't seeem to be a particular theorem, but rather notation. There, my advice is to just not take notation too seriously.

Take $\mathbb{R}$ for example. People will often say "... the set of real numbers $\mathbb{R}$...". That makes it sound as if there was exactly one set of real numbers, and that set, and nothing else, went by the name of $\mathbb{R}$. The problem with this is - it isn't true. There are different ways to construct the real numbers (as dedekind cuts, as sequences of rationals, as infinite decimals, ...), and strictly speaking all of these constructions yield a different set. After all, the set of all possible sequences of rationals certainy isn't the same as the set of all infinite decimals, which in turn isn't the same as all dedekind cuts of the rationals. But you can define an ordering and the basic arithmetic operations on all of these sets, and then find bijections between these sets which are compatible with these operations and the ordering. Thus, for all intents and purposed, it doesn't matter which of these sets of real numbers you pick, and hence $\mathbb{R}$ really stands for any of those, instead of for one particular one.

What you have discovered that there's another way to arrive at $\mathbb{R}$ - you can start with $\mathbb{C}$, and view $\mathbb{R}$ as a subset. Now, it's true that, strictly peaking, $\mathbb{R}$ cannot be that subset if you also want $\mathbb{C}$ to be the set $\mathbb{R}^2$. But if you view $\mathbb{R}$ as just something that stands for any representation of the real numbers, that problem goes away. $\mathbb{C} = \mathbb{R}^2$ then just tells you that one way to view $\mathbb{C}$ is as the set of pairs of real numbers, and $\mathbb{R} \subset \mathbb{C}$ tells you that one way to view $\mathbb{R}$ is as the set of complex numbers with imaginary part zero.

So again - don't take notation too seriously. It's just a tool used to communicate ideas, just as any language is. And just as any other language, it's sometimes ambiguous, and sometimes a bit confusing. The trick is to try to see beneath that.


Since you seem to be dealing with complex analysis currently, here's a list of things (off the top of my head, surely incomplete, and in no particular order) that I think are worthwhile to think about and try to understand

  • What does $f$ being holomophic mean geometrically (compare to the situation in $\mathbb{R}^2$)
  • Why $z^n$ is integrable, except for $n=-1$. Watch how that single fact shapes the whole subject (c.f. laurent series)
  • Exponential function and logarithms. Especially the fact that $x \to e^x$ is not injective over $\mathbb{C}$.
  • What does exponentation mean over $\mathbb{C}$. $z^n$ for $n \in \mathbb{Z}$ is clear enough, but what about $z^q$ for $q \in \mathbb{Q}$?
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I agree that any good math curriculum should start with the very basics (say, natural numbers), and work its way up from there. Stating theorems explicitly is an absolute must. However, I STRONGLY disagree with the assertion that every theorem that is encountered needs to be proven. The whole point of a theorem is that you can use it without knowing why its true. In fact, I'd go as far as to say that in certain courses, it is appropriate to prove none of the theorems, because they're problem solving courses and 90% of the students will just zone out of you show them a proof. –  goblin Mar 13 at 10:49
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@user18921 For an engineering curriculum, yes. For a math curriculum, no, especially for undergraduates. If you're aren't going to introduce students to proofs there, when will you? –  fgp Mar 13 at 14:32
    
@fgp I'd +1 just for "don't take notation too seriously. It's just a tool used to communicate ideas, just as any language is. And just as any other language, it's sometimes ambiguous, and sometimes a bit confusing. The trick is to try to see beneath that." It took me some time to understand that! –  Mon Kee Poo Mar 14 at 6:11
    
@fgp I get your point, and it is a real insight for me. But, on the other hand, is it not the key difference between language and mathematics, that mathematics is supposed to be unambiguous ? I read now a little bit about structural set theory, because I heard that this would somehow solve problems like $\mathbb{R}$ being a subset of $\mathbb{C}$ or not. But in the meantime, I will remember what you said. –  Kasper Mar 14 at 17:34
    
@Kasper The math is unambiguous, but mathematical notation is not. That's my key point. And making the notation totally unambiguous usually isn't even a goal - if you did so, notation would become so unwieldly that it'd obfuscate, rather than clarify, the actual math behind a question. Your question is the perfect example - complex analysis isn't about how you model the complex numbers, it's about their properties, and about properties of functions from $\mathbb{C}$ to $\mathbb{C}$. –  fgp Mar 15 at 13:54

Aaaaauaauaghaguaghaguaghaguahguaghaughaguahguahguahgh!

Sorry.

I don't mean this as an insult, so please don't take this the wrong way, but your entire approach is precisely backwards. To be fair, just from the way you express yourself, it sounds like you do actually have a pretty decent understanding of what you're studying, and I'm thrilled to hear that you think it's "fun" to think about the underlying concepts. (Yes! It is fun! That's why we do it! I mean, yeah, it's also useful for science and stuff, but pure math is art, not just a means to an end.) So while you may have exaggerated your inclination to just "gloss over" the core of your studies rather than plumbing their depths, I'm still going to put forth my argument for why I disagree with the approach you described.

As a student, even though you're paying for a degree, your goal should be to understand mathematics--and therefore your attempt to understand the underlying theory should never be a last-minute "if I have time" activity. This is true regardless of whether you're an undergraduate or not. It seems both dangerous and unfulfilling to assume that the "tricks" are they key to "how mathematics is done," while true understanding is only in the purview of "advanced" mathematicians (whether that means graduate students, post-docs, professors, or whatever).

Instead of remembering "tricks" and procedures to solve particular problems, try to figure out why each trick works, then try to re-derive the trick yourself. If you really understand the core concepts, then in theory you should be able to take a test without having memorized anything; you can simply re-derive every problem-solving method you need. (In fact, when I was an undergraduate in mathematics I used to tell people that I'd chosen that major because I hated memorization, and memorization was generally useless for the way I took tests.) In practice, you'll probably find that the "tricks" are actually easier to remember once you understand them in this way, and if you "mostly" remember a trick but don't quite remember some details (for instance, the order of a couple terms, or whether an operation is an addition or a subtraction), you can quickly figure out the missing parts and then proceed with confidence.

Regarding your particular question about the real and complex numbers considered as formal sets, I hope you've found your discussion in the comments with Malice Vidrine enlightening; I don't know that I have much to add. (Though personally I do consider $\mathbb{R}$ to be a subset of $\mathbb{C}$, which should add a bit of weight to Vidrine's statement that the structure of the thing matters more than the formal definition.)

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My first year of mathematics, I actually did it the way you are talking about. Really trying to understand it. It was kind of fun that way. I was much more creative and sophisticated back then, but I almost didn't pass any exam. For example, the first day of university I got the epsilon definition of convergence. (my university skipped calculus, and began straight with analysis). I was completely blown away with this definition. I wanted to be able to understand how this definition made sense, how I could say that is the same as my intuitive idea of convergence. (cont.) –  Kasper Mar 13 at 20:02
    
But it took me 2 weeks, before I understood that. I think it was that I just never really thought about a logical statement that way. And this logical statement was kind of complex: $$\exists L,\forall \epsilon, \exists N, \forall n>N: |s_n -L|<\epsilon$$ Besides that, absolute value and inequalities where very new for me. So I did some self study, I started study logic on my own, and I actually made for myself an equivalent definition that maked much more intuitive sense to me. But you know, this way of studying, is not what gives you good grade. –  Kasper Mar 13 at 20:03
    
It gives some more satisfaction in some way, but now I'm trying for some more balance, I'm allowing myself only to go deep in stuff like this, if I know I will get a good grade. –  Kasper Mar 13 at 20:06
    
@Kasper Hm, I definitely understand your concern and your dilemma. I think all I can recommend in this case is talking to your professors more. Yes, writing out the formal definition of convergence in logical symbols makes it quite difficult to decipher intuitively; a competent professor, however, should be able to help you gain a solid understanding of the real "meaning" behind the definition as well as the specifics of the definition itself (and the reasons for even writing it in that way). –  Kyle Strand Mar 13 at 21:02
    
Of course, you won't always have good professors; some of them may just be poor teachers, or may not like talking to students even though that's part of their job, or (in some cases) may not even really understand the material themselves. (I like to think this is pretty rare, but....) And even when your professors are quite good, you have so little free time as a (diligent/conscientious) student that you may think it's not worth it to go talk to your professors. –  Kyle Strand Mar 13 at 21:06

You (presuambly) have a prior notion that allows you to interpret the elements of $\mathbb{R}$ as expressing the abstract concept of a "real number".

What this construction is saying is that we also want to interpret the elements of $\mathbb{R} \times \{ 0 \}$ as expressing the abstract concept of "a real number" (but only in the context of interpreting elements of $\mathbb{R} \times \mathbb{R}$ as expressing the abstract concept of "a complex number" in the way you describe). And we want these two interpretations to be related in that we want to interpret $x$ and $(x,0)$ as expressing the same thing.

Different expressions for the same thing is a familiar concept: e.g. $1$, $2-1$, and $3/3$.

The problem comes when you become too invested in the idea that some particular representation is what something "really is" -- e.g. when you simultaneously hold the notions that the abstract concept of "a real number" really is the same thing as "an element of $\mathbb{R}$" and "when I'm talking about elements of $\mathbb{C}$, I'm talking about what they really are", then it becomes very difficult to digest the idea that an element of $\mathbb{C}$ can be a real number.


Sometimes, you see other approaches used to allow you to work around this dissonance rather than trying to work through it, which have their advantages and disadvantages: e.g.

  • There is an injection map $i : \mathbb{R} \to \mathbb{C}$, but we usually won't write it explicitly. e.g. when I say "$x + \bar{x}$ is a real number", I really mean "there is a real number $r$ such that $i(r) = x + \bar{x}$".

  • That definition of $\mathbb{R}$ you learned in analysis was just a temporary definition of the real numbers that we used so that we could define $\mathbb{C}$ properly. Now that we have our definition of $\mathbb{C}$, we will start using the "correct" definition of the real numbers which is the corresponding subset of $\mathbb{C}$

  • We could define $\mathbb{C} = \mathbb{R} \times \mathbb{R}$ this way, but we really want it to contain $\mathbb{R}$. So what we'll actually do is define $\mathbb{C} = (\mathbb{R} \times (\mathbb{R} \setminus \{ 0 \})) \cup \mathbb{R}$.

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I understand what you are saying except the last sentence. –  Kasper Mar 13 at 0:40
    
@Hurkyl The relationship between $x$ and $(x, 0)$ doesn't really correspond to that between $1$ and $2 - 1$. Given any particular $x$ and any ordinary way of interpreting ordered pairing, $x$ will not actually equal $(x, 0)$. Instead, $x$ and $(x, 0)$ are thought of as the same because $x$'s purpose with respect to some structure corresponds to $(x, 0)$'s with respect to some (other, related) structure. In contrast, $1$ really does equal $2 - 1$. Maybe you mean the relationship between $x$ and $(x, 0)$ corresponds to that between the texts "1" and "2 - 1"? –  Eliah Kagan Mar 13 at 6:37
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@Eliah: Just as we might choose to distinguish the texts 1 and 2-1 or simply interpret them as expressing the same integer depending on our needs, we might choose to distinguish the sets (notated by the texts) x and (x,0) or simply interpret them as expressing the same real number depending on our needs. Any two things that are obviously the same can be seen to be different after a shift of perspective. 3 and 3 identical? Not quite: one of them is an inch to the left of the other one; they have different physical position, so they can't be the same! –  Hurkyl Mar 13 at 10:52
    
@Hurkyl One way $=$ differs from some other equivalence relations is that, when $x = y$, any true sentence about $x$ remains true with $y$ substituted. As it's fundamental to the mathematical notion of a text (some prefer the term string) to have well-defined length (and contents), "1" $=$ "2 - 1" is never true. $1 = 2 - 1$ does not express the same idea as "1" = "2 - 1"; the first is about numbers, the other about sequences of symbols. Unless you want students to think of math as being about meaningless symbol manipulation, I suggest not glossing over this distinction. –  Eliah Kagan Mar 13 at 14:47
    
@Hurkyl What do you mean with this: $$\mathbb{C} = (\mathbb{R} \times (\mathbb{R} \setminus \{ 0 \})) \cup \mathbb{R}$$ ? –  Kasper Mar 13 at 15:14

I too believe that an exam mostly needs knowledge of so called 'tricks' instead of a complete understanding of the material. To understand some subject completely takes much more time, and is probably not reachable by an undergraduate student, since solving one question probably only brings more new questions. However, digging further into the material than your course does is certainly possible (and I think very enjoyable).

In your example: There are many ways to define the complex numbers, but probably the most common is that it is an extension of $\mathbb{R}$ which contains all the roots of all polynomials with coefficients in $\mathbb{R}$. It is called the algebraic closure of $\mathbb{R}$ and in algebra it can be formally defined as $\mathbb{R}[X]/(X^2+1)$. Of course this is not really understandable if you haven't had any group and ring/field theory, but these subjects are almost certainly handled in undergraduate courses (at my university in the second semester of the second year).

So just keep digging deeper step by step and you will learn more and more, even as an undergraduate! Also just ask people when your stuck!;)

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Also frustration is sometimes a part of the journey! –  Marc Mar 12 at 20:58

My strategy is to keep a journal - write down everything that doesn't make sense right now, study what you have to to get a good grade so you can move on, and keep moving forward. Of course, also constantly ask professors, friends and older students if they have any insights, but don't get too stuck on one idea because you'll always be able to come back to it later!

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I recognize certain things in your story here. When I was in High School, there were a lot of things in Math that did not really make sense to me, but I knew the rules and formulas (didn't care how to derive them) and I was pretty ok in math. Then in my undergrad, a lot of math topics from High school start to make sense because I learned more math to view High school from a different angle and I did a lot of high school tutoring. I was also getting more mature (at home they still disagree to this day...) This all continued into grad studies where a lot of undergrad topics started to become clear why it worked the way I learned them. Now I am in the teaching field and still take grad classes myself off and on and believe me, by the passage of time and experience, things do clear up. I don't know how old you are (don't need to reveal either), but I can guarantee you that if you continue to use math higher up or go into teaching, you will have your questions answered! Right now ,pass those exams by knowing the rules and get back to the material at a later stage of life. It certainly will become clear.

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I can't speak to your exact questions about the reals and complex numbers, but I can definitely relate to your desire to understand the fundamentals rather than taking things on faith. In fact, this was one of my primary stumbling blocks to enjoying math, and it really wasn't until my very unusual calculus teacher taught calculus very rigorously from the ground-up (with deltas and epsilon proofs, which I understand is not how it's usually taught), and encouraged us never to move on until we thoroughly understood something, that I finally began to really fall in love with math.

I think it's insane and pointless to move on without understanding the fundamentals. To do otherwise is just to turn us into number-crunchers, which, to be honest, it seems like all school is really trying to do. "Actually learn the material? Pfft! Just get an A on the test!" (Though, of course, truly learning the material can very much help with acing the test.)

For example, I got an A in Algebra II without understanding logs and matrices one bit! Okay, so I had a general sense that logs were inverse exponents, but I had no clue as to why they were invented, what they were good for, and how, really, they worked. It wasn't until literally just a couple weeks ago when I finally decided to wrestle the question to the ground. I went alllll the way back to John Napier and even read his and Briggs' original papers until I got it. And now, finally, I understand. It's wonderful!! When I just crunched the numbers without understanding, I felt an uncomfortable sort of cognitive dissonance, but after understanding something, I'm like, well yes, of course.

That's my two cents. I know a friend of mine who's very advanced in math had the opposite view--that sometimes you just have to take something on faith and move on--but in my opinion, that takes all the fun out of it. I'd say that's good to do really only when necessary--when you're coming up to a deadline or a test, or perhaps when you've gone down a side tangent that's actually not all that necessary to your primary focus (for instance, Napier logs, which aren't even used anymore).

Otherwise, I heartily encourage you to understand the fundamentals as much as you can, even when you're not "ready" for them. Personally, I think the way math is parsed out and incremented over the years is unnecessary and ridiculous. (And I'm not the only one.) I think algebra and calculus, for instance, should be taught MUCH earlier than they are. I say go as far as you can, and when you hit that wall of incomprehensibility, okay, stop then. But I think you'll be surprised by how far you can go -- and how much going that far will help you. :)

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I guess you have problem with formal language of math which may hide the beauty of the idea behind the theorem.

Suppose you have an set with elements apple,orrange and lemon.If I say that I can see these set as subset of $Z_6$,does it make sense?

Actually,answer is yes since with defining suitable operetion,I can see apple,orrange and lemon as $\{0,2,4 \}$ which is subset of $Z_6$.So,$R$ can be seen 'same' with $0\times R$ or $15\times R$ .

Thus,My suggestion is

$*$ try to understand the idea behind the theorems written in formal language

$*$ Avoid to make rigid defination of objects (you can see everything as an element of set,an elemet as function,a function as a morphism ....a letter as number,a number as a letter...)

$*$Do not just try to understand the theorem,also try to understand the defined concept.( why is it defined in that way.)

$*$Be patient to see big picture,do not expect yourself to grasp every consept deeply.

When you grasp the idea behind theorems and concepts,you can feel yourself in 'safe'.

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If you want to do research in mathematics or in other subjects with a heavy mathematical content, you should definitely try to really understand what you are doing starting at the undergraduate level.

Note that I said try, in some cases you will manage to, in some other you will not and, curiously, in some cases you will think you really have understood but you will have not.

In order to better understand mathematics you need experience and so called mathematical maturity. But you build very little of those if you just learn tricks and apply formulas, the more you try to understand things the more you will eventually understand things.

Don't get discouraged if you feel that, despite your best efforts, you still cannot understand something, it is normal. In many cases, things will become clearer later - not for free, but because all the thinking you have done will give its fruits.

Don't go overboard, you should not risk to compromise your marks. If you get stuck, keep thinking for a while but at some point move on, there will surely be some things which you cannot make sense for the moment. Sometimes it is not because the topic itself is very difficult, but it requires a level of abstraction, or mathematical maturity, or confidence with the material which you do not have.

In a line: as you are an undergraduate student, don't get stuck for too long, but by all means keep asking yourself questions and try to understand what you are doing.

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I think a natural part of learning mathematics is growing comfortable with the fact that there can be (and usually are) many different ways to describe a single idea. I think this fact of life is essential for our sanity actually, if for example I had to start talking about Dedekind cuts whenever I use real numbers in even a basic calculation I would switch jobs.

At some point the "ordered pair" approach to complex numbers may make more sense to you, after you have some more context to back up that definition and some experience in moving between different representations in other contexts as well.

So I would say: Don't worry too much if something like this doesn't make perfect sense immediately. If it is a math topic that you intend to apply often, then experience will naturally improve your understanding. It takes a very rare type of individual who will genuinely improve their understanding by trying to work "from the ground up." Remember that historically mathematicians did not work this way. It was only in hindsight that we have come up with the definitions which are so efficient in wordcounts but perhaps not in pedagogy.

To gain more understanding, just keep doing math and don't stop. Read good math books (especially the introduction sections by respected mathematicians), ask plenty of questions when something doesn't make sense. But most of all just don't stop doing math.

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This learning problem is a human trait, and occurs in almost all subjects. It is common in the science, technology and engineering subjects in the same way. Do not be worried by an open and inquisitive mind.

It is worth getting to the core of the understanding as from there you can get to any of the derived results in one step, rather than taking 2 pi steps to walk around the periphery of rote learning (the usual exam revision problem).

There are two techniques I've used over the years.

One is to skim read text books 'back to front' so that you know what the good bits and conclusions will be, and you already know you won't fully understand the end sections yet. So you don't worry. And you know where a technique will lead.

The other technique is to go back and read the introduction to the good text books (that's the Preface and Chapter 1), because that's were they hide those little nuggets of 'obvious' information and understanding that we all struggle with when researching new areas of interest (you should think of your course as guided research).

As an Engineer I have to drop into all sorts of obtuse maths to get an understanding of how some calculation is completed (e.g. currently Radar Cross Section calculations led to Fast Multipole Methods and then onto Gegenbauer series and glazed eyes, and then some comprehension & enlightenment...).

Even Astrologers have to build up confidence in what they understand, having take the axiom of choice one step further than most ;-)

Once you have conquered complex numbers have a look at quaternions (or are they bi-complex?) to see more roots of -1 if you want a little fun.

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In mathematics, proof part is the only main thing to learn. If every immediate logical implications are written down, then math would become the most easiest and smallest subject in a university course. Theorems are democratic in the sense that every one is entitled to know it. They should be presented in the actual length. To test the ingenuity of a student, there is a separate section of Exercises or Problems. Textbook authors should not intermingle Theorems with Exercises. How many people will believe that the actual size of a proof without any missing steps of Implicit Function Theorem takes nearly twenty pages. On the other hand, no book on Analysis allows more than three pages for it. Authors and publishers supply various excuses. Such a bad situation does not prevail in chemistry or biology or history, etc. All hue and cry for being stuck in math has this one as the main reason. Fortunately, it is not an insurmountable difficulty. First of all, authors should discover a way for how to write every step in a proof while consuming the least amount of space on paper? Authors should avoid the temptation of writing the phrase "clearly", because there is nothing in math that is clear. One such author is Rajnikant Sinha. Check his book "Calculus of tensors and differential forms" in this connection.

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From my experience, future job prospects will value an applicant for both high undergraduate scores and depth of knowledge. But, if you want to excel in an environment where your prospective employer wants you to do problems that can be solved by knowing the 'tricks', be prepared for that inquisitive mind to be wasted. If this is how you study in your undergrad, this is what you will be prepared for in the start of your career.

Educators have a responsibility to educate, but not all of them are preparing you for YOUR dream position (and not all of them are good at their jobs). Start preparing yourself for YOUR dream position by studying material in a way that will land you that job. Bottom line is: professors will not walk you to your definition of success, that is up to you. The OP's question was proposed in a binary fashion, but I believe the answer is not: Do what is necessary to excel in the scholastic arena because GPA will be important AND learn theory, foundation, accessory information, etc. while you do it. The most ideal candidate will have all of these things. Find your own balance with the time that you have. GPA will help you get the interview, but depth of knowledge, an ability to self-educate to gain that depth of knowledge, and an inquisitive mentality will gain you tons as an employee. That mentality is learned NOW, not after you graduate. The requirements of success in the class room (GPA) should be considered a minimum, and if you are inspired to understand the depth of the material, do so in your own time, and it will be a great investment to your future.

I was fortunate during my undergraduate education to be taught by a professor who was very interested in the science of education. His requirements where very similar to the process outlined by Philip Oakley's response: he would require us to take a high level quiz over the material before he started lecturing on it, which would require us to have already been exposed to the knowledge before even starting to teach it. It was a tremendous approach that created many more logical connections to materials than other approaches had, and it helped bridge the gap between concepts.

As a final point: read Euclid's Elements. There after, I think you will appreciate why professors must encapsulate material, it would be impossible to grow some of those concepts from axiom up with the time frame allotted. And most people just don't care to know. If you do, however, learn what you can outside of the constraints the classroom puts on the educational process, and become great because of it!

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