# Clearly, this proves that I'm stupidJanuary 31, 2013 8:35 PM   Subscribe

I can't seem to wrap my mind around the language of higher level math. Definitions, theorems, and proofs make me fall asleep, but I really do want to understand. Do you know of any good resources that can help me out?

Back in high school (Canadian) I was a math whiz. I had no trouble with any of the concepts, including calculus in Gr 12. I can "do the math" and am good at problem solving. However when I got to university and started taking Number Theories, Abstract Algebra, etc., my logic suddenly disappeared on seemingly simple concepts. I've obtained my degree in CS already, somehow memorizing or handwaving my way through on my assignments and exams, but now I actually need to understand higher math for my current job.

For example, these are excerpts from one of my textbooks (A Concrete Introduction to Higher Algebra by Lindsay Childs):

[on real numbers] "The real numbers form a complete Archimedean ordered field...Archimedean means that for every positive real number r there is a natural number n with n > r."
[division theorem] "Given nonnegative integers a > 0 and b, there exist integers q> 0 and r with 0 <= r < a such that b = aq +r

I thought I understood real numbers and division until I read these definitions and was also asked to prove them. First of all, my brain have difficulty just parsing out what's written; it's like a foreign language to me. Secondly I never know how to begin when proving things. I can somewhat follow along if there's an example in the textbook, but on my own I would be completely lost.

I'm not looking to take any classes in math or hire a tutor. Books or online resources would be good. At minimum I'm hoping to at least be able to read through a math textbook and actually understand the explanation.
posted by lucia_engel to Education (19 answers total) 17 users marked this as a favorite

There are some common proof "templates" and "strategies": direct, by contradiction, induction, etc. Most upper-level math textbooks will have bits on proof technique in one of the appendices. Typically, an undergrad's first real analysis or abstract algebra textbook will introduce proof techniques in the abstract. Or you could buy a book like How to Prove It (but there are many others).

As far as tips, think of "terms" as a shorthand form of "definitions." For example, take "The real numbers form a complete Archimedean ordered field." In place of "Archimedean" you can substitute its definition: "The real numbers form a complete ordered field in which for every positive real number r there is a natural number n with n > r." Or you can find and plug in the definition of "field" ("a set equipped with the operations + and · that obeys the following laws: …"). Proving is the exercise in which, given two statements A and B, you substitute definitions and parts of definitions until statement A turns into statement B. This is identical to working with equations in high school math, where you continually rewrite strings of symbols by implicitly relying on the laws of arithmetic.
posted by Nomyte at 8:46 PM on January 31, 2013 [1 favorite]

When I'm trying to prove something, I often first go through some examples of the theorem in question and try to convince myself that it's actually true. (For example, for the division theorem, if you take 7 and 3, there's only one way to write 7 = 2 * 3 + 1 where 0 <= 1 < 3

It can be helpful to think about it as "Why is it true for 7 and 3? Well, what about 100 and 3? What about 100000000 and 3? What about EVEN BIGGER NUMBERS?" and eventually come to a general proof.
posted by oranger at 8:59 PM on January 31, 2013 [2 favorites]

Higher level math isn't like high school math. There's more emphasis on proofs and rigor, and you have to be able to unpack all the terminology (e.g., what's a field? what does the modifier Archimedean mean? what does the modifier 'ordered' mean?). Doing proofs is a learned skill as well, it comes down somewhat to individual tendencies (many proofs are obtainable in more than one way, so some ways will be more natural to you than others).

I always started by writing out what I knew (usually the starting points are given in a proof, or you know what the subject matter is), and then trying to figure out where to go (problems like "Prove X" don't have that step, you know X, but sometimes it's "Prove or disprove: X" so you need to figure out if X is likely true or not -- sometimes you'll have an idea, sometimes you can just try some examples). Then I would add to the starting/ending point as I could, by adding lemmas or introducing facts learned from other proved things (e.g., due to the Prime Number Theorem, we know an upper bound on the number of primes less than x is blah blah blah). You then combine those to advance the proof toward a conclusion. Your methodology might vary, but it's something you have to teach yourself, and that you didn't learn in high school (or at least, I didn't).
posted by axiom at 9:07 PM on January 31, 2013

I thought I understood real numbers and division until I read these definitions and was also asked to prove them. First of all, my brain have difficulty just parsing out what's written; it's like a foreign language to me.

It is a foreign language. Never assume that any of those words mean what they mean in ordinary language.

A field, for example has a very precise definition which has very little to do with any way you've used the world 'field' before.
In abstract algebra, a field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division satisfying the appropriate abelian group equations and distributive law. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth. To avoid confusion with other uses of the word "field", the term "corpus" may also be used.
If you don't know what any of those words mean, try watching these harvard courses on Abstract Algebra if you have time.

The Khan Academy videos on Linear Algebra would probably also help, as would these videos on permutations.

Also, this is really advanced stuff, so don't feel bad if you don't understand it easily. The vast majority of people don't understand it and never will. Once you start getting into abstract math, intuition becomes very unreliable.
posted by empath at 9:18 PM on January 31, 2013 [1 favorite]

now I actually need to understand higher math for my current job.

Also, if you tell us what exactly you need to understand, it would be easier for us to lay out all the pre-requisites for you.
posted by empath at 9:22 PM on January 31, 2013 [2 favorites]

Once you start getting into abstract math, intuition becomes very unreliable.

A number of the recent major discoveries in mathematics have been based on intuitive leaps, often framed in conjectures that take a long time to be proven, and which can form the building blocks of proofs for larger theorems.
posted by Blazecock Pileon at 9:46 PM on January 31, 2013

Yeah, and a lot of people have written really cranky papers based on intuition when they don't really understand what they're talking about.

In any case, what I mean is that you can't just visualize the kinds of spaces and fields that you study in abstract algebra.
posted by empath at 10:05 PM on January 31, 2013

Mod note: Answers should focus on what will help OP get up to speed; thanks.
posted by LobsterMitten (staff) at 10:14 PM on January 31, 2013

I have a very hard time translating math into words. Things I understand perfectly and can work with alone very happily I am at a loss to explain to other people in, like, words. I also have to work through a problem to understand a paragraph like the one above, laboriously going back and forth from equations to words like I'm translating from a language I don't know. I think I'm math dyslexic or something. It's a pita to be honest. Doing it a lot is the only thing that helps.
posted by fshgrl at 11:25 PM on January 31, 2013

Your problem is that you don't realize that it takes time to understand things. You were spoiled by having too easy a time. You need to put in the work to understand these things. They are more like legal documents than like prose and even a lawyer will have to spend time reading a contract. Try to tolerate the frustration and the feelings of helplessness and anger as side effects and soldier on.
posted by Obscure Reference at 4:39 AM on February 1, 2013 [2 favorites]

Response by poster: Thanks for all the answers so far. To clarify a few things:

Also, if you tell us what exactly you need to understand, it would be easier for us to lay out all the pre-requisites for you.
The concepts in the textbook I'd mentioned is what I want to understand. Here's the Table of Contents. I do welcome general tips on learning how to learn as well.

I do understand what is involved to prove things. I did well in my advanced algorithm class during my CS studies. It involved designing an algorithm to solve a specific problem (ie. A man needs to travel from A to B, design the shortest route...) and proving its complexity (...and prove that your algorithm will always produce the shortest route). I have problem transferring this skill to math proofs though.

Yeah, and a lot of people have written really cranky papers based on intuition when they don't really understand what they're talking about.

Sometimes when I read example proofs, it feels like it's all circular logic (even though it probably isn't the case) or they take giant leaps without explaining how they arrived there.

Silly examples (completely made up):
Prove division theorem
"Division theorem works because ham sandwich theorem is true."
"But we haven't proved the ham sandiwch theorem yet."
"That's a given, you just have to understand it's true."
"But to me the division theorem is a given. How do I know which theorem is a given when I prove something?"

or

"Division theorem works because real numbers are Archimedean duh, clearly that proves it's true"
"??? I think I missed 10 pages right there."

I feel like there's a hierarchy of theorems and definitions I need to really understand first.
posted by lucia_engel at 5:18 AM on February 1, 2013 [1 favorite]

I don't know if this is a helpful analogy but you can think of abstract algebra as being like a really high level programming language. in that context, lots of elementary problems can be proven simply by turning yourself into a parser who replaces terms like "Archimedean" with their low-level representations. lot's of exercises in abstract algebra are really just sanity checks: if i translate this abstract statement into low-level terms will what i already know to be true be obtained and many of the problems students have with "higher math" are analogous to ones beginning programmers have had with say, "object oriented" programming.

Sometimes when I read example proofs, it feels like it's all circular logic (even though it probably isn't the case) or they take giant leaps without explaining how they arrived there.

but the thing is, you know that isn't true or at least you have to constantly remind yourself. i think that's a pretty standard reaction to abstract algebra but it's something you have to get past and getting past it isn't necessarily easy. many proofs are written to make the reader feel like an idiot; it's one of the hazards of the trade and I think algebra is one the worst for this sort of thing. one of the things you have to do learn to do is rewrite things in your own terms if you don't understand it... but that takes time.
posted by ennui.bz at 6:32 AM on February 1, 2013

I majored in math in college and did my senior seminar on Galois Theory (a topic in abstract algebra), but I was never very good at math. What I mean is that I had to work very hard to understand these kinds of things, and I think you might need what I needed: practice.

I totally understand what you're saying about reading a proof and then feeling like you did not get it or that it's somehow circular. The way around this is to prove lots and lots of things yourself so that you get a better feel and intuition for the language of algebra. The prof in the Harvard video linked above even talks about intro algebra as being basically a language class for math, which I think makes a ton of sense.

So I'd recommend going back to your Abstract Algebra textbook and just working through the problems. If you can't, maybe take a class at a local community college or do something else where you can get help from a live human on the particular proof you're studying. Or, since you're doing this for work, ask for help there and see if there are any internal resources you can call on.
posted by Aizkolari at 6:50 AM on February 1, 2013 [1 favorite]

Best answer: I have a math degree from MIT, though it was a struggle. I feel you on this. A few things to think about:

1) Math notation is very, very dense. Allow a long time to get anywhere.

Now, strategies for unpacking statements:

Given nonnegative integers a > 0 and b, there exist integers q> 0 and r with 0 <= r < a such that b = aq +r

2) Translate statements into synonyms. In order for a statement to make any sense, you have to hold the whole thing in memory at once. Repeating it in different ways helps me. "Okay, a is an integer > 0, so a is at least one. A is a positive number." "b = aq + r, okay, so if you take aq and add r to it you get b (I know, I know, this sounds dumb but I literally do this and it helps me, YMMV.) That is, b is r bigger than aq. b - r is equal to aq."

3) Tease apart little statements hidden in the dense notation. "nonnegative integers a > 0 and b -- okay, so a and b are both non-negative, ah except that a isn't allowed to be zero. What's the significance of that? I guess it will have something to do with division, and a will end up in the denominator somehow."

So now you can kinda repeat the statment again. "Okay, so a and b are positive (except a can't be zero). r is between zero and a (ah, so I guess that's probably part of why a can't be zero). Looking at all the numbers here, they're all at least zero, with a and q at least one. And r is at most a. The statement begins _given_ a and b, so those can be freely chosen, and the proofy part is that _there exists_ q and r. So looking at b = aq + r we can choose any a and b (except a is > 0), and find a q and r to fill that in."

Now I like to fill in concrete numbers to help solidify the concept.

"So if I choose b = 20, and a = 3, then that's saying there's an integer I can multiply times three to get close to 20 (where close, i.e. r, is within 3). Sure that makes sense. If I pick 100 and 30, I can get to 90 by multiplying by 3, and then adding on 10, which is less than 30."

"Okay, so I think I've got what the statement is saying -- what's the significance of it?" [And here, I'm at a loss, without knowing the context this has come up in]. What if this statement was false? What if we couldn't always do this? I've never thought of this before, but I guess that means modulo wouldn't always work. In programming, I've always taken it for granted that I could do something like

(int a, b)
int c = a / b
int remainder = a % b
c * b + remainder == a

But this kind of assumes that if you divide an integer by an integer, you'll only be off by the remainder you get when you take the modulo. That is, a = c*b + remainder, with the remainder being less than b.

So that was kind of long winded stream of consciousness answer, but I thought since I didn't recall the "division theorem", and the statement of the theorem was confusing to me, too, bringing you along for the ride while I tried to understand it for myself would be helpful. Hopefully it wasn't too crazy.

In summary, the strategies that work for me are: going very slowly through a proof statement, and talking it out to myself, using synonyms, and trying to find little things the statement is implying without saying directly.
posted by losvedir at 6:54 AM on February 1, 2013 [4 favorites]

You may want to pick up a terse mathematical dictionary, like that of Collins Press. (I think wikipedia is good for wordy mathematical defintions.)

You may also want to build up a glossary of terse definitions, aka "notes", copying stuff out from short term memory.

"But to me the division theorem is a given. How do I know which theorem is a given when I prove something?"

Informally, everything that came earlier in the book is given. Unless the question specifically tells you to rederive it.

Formally, when solving a question, other theorems are probably given unless they're consequences of what you're trying to prove.

Secondly I never know how to begin when proving things. I can somewhat follow along if there's an example in the textbook, but on my own I would be completely lost.

Much of the time, I start by looking at the contrapositive. "Prove, if A, then B." "Huh, then not-B would force not-A...", or I try to informally prove the inverse (and fail), to get a feel for it.
posted by sebastienbailard at 8:24 AM on February 1, 2013

For samples of proof forms, try these: http://cs.brown.edu/courses/cs022/docs.html

That is the course that enabled me to go into higher math (well, 'higher' math for the middling undergrad).
posted by batter_my_heart at 1:08 AM on February 2, 2013

I always found it a lot easier, for some reason, to prove something by contradiction than just to flat-out prove it. Just thinking about what the consequences would be should it not be true can often help.

It's normal to struggle at first, I think. I had a two-semester-long proof sequence in the final year of my Math undergrad. We started with the barest building blocks (natural numbers, addition, etc) and painstakingly recreated the undergrad calculus sequences theorem by theorem. I spent most of my time for that class not writing proofs but staring at the walls in the library just thinking about math.
posted by ZeroDivides at 5:58 AM on February 2, 2013

One thing that helps me when I read math or physics texts is to write down every equation.

When I just see an equation it is easy for me to say "OK, sure, I guess", and move on, and not internalize any of it, and then when I get to a point 5 pages later when I'm expected to use all of that earlier stuff I realize that I never really learned it.

Copying things down helps a lot because for some reason it is very uncomfortable for me to write down something I don't understand fully. So that slows me down enough to ensure that I've really understood and internalized each line before I move on to the next.
posted by dfan at 10:33 AM on February 2, 2013

Response by poster: Thanks for all the answers. I found a course on Coursera (Introduction to Mathematical Thinking) which is suppose to help transition highschool students to post-secondary math. Hopefully it'll help me out!
posted by lucia_engel at 8:47 AM on March 8, 2013

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