61 votes

How to stop forgetting proofs - for a first course in Real Analysis?

Don't try to memorise the proofs: try to memorise the methods that are used in most analysis proofs. That way you only have to memorise a handful of methods instead of 30-50 proofs, and you can adapt ...
  • 5,968
51 votes

How to explain the formula for the sum of a geometric series without calculus?

This could be explained using algebraic transformation but i would rather show a very simple geometric proof for sum: 1 + 1/2 + 1/4 + ... = 2
50 votes
Accepted

Is it normal to treat Math Theorems as "Black Boxes"

I think it's important to remember the key ideas in long proofs, as these may serve you later. Trying to remember every single equation in a long proof is a loss of time and energy. However, it is a ...
  • 3,988
43 votes

Self-learning mathematics - help needed!

What you're talking about seems much less like a mathematical or academic complaint than a psychological one. Here's what I read in your post: You seem to be insecure about your understanding of ...
  • 17.1k
38 votes

Is it necessary for one to understand analysis?

To me, asking if you need to understand analysis is roughly* like asking "Is it necessary for one to understand how to operate a computer to pursue a career in mathematics?", in that the answer is ...
  • 12.5k
33 votes
Accepted

Is it necessary for one to understand analysis?

Eric's answer is quite thorough, but it is missing an important point: in mathematics, everything tends to be related and intertwined. You might think that doing abstract algebra will get you far from ...
  • 33.4k
31 votes
Accepted

How to stop forgetting proofs - for a first course in Real Analysis?

One thing that works for me when learning a theorem is to go through all the conditions and find corresponding counter-examples, as well as seeing exactly where the proof fails. Take for instance ...
  • 2,708
22 votes

I'm looking for some mathematics that will challenge me as a year $12$ student.

I have been asked for a hint. Gauss was thinking about roots of unity. The method of Gauss for dealing with these polynomials is in chapter 9 of Cox Galois Theory. In fact, as the author points out, ...
  • 132k
19 votes

I take it so long everytime I learn mathematics myself. What should I do?

I will try to give an idea of what I did when I was in your situation (It worked for me, and I hope it will work for you too). I had a teacher who used to say self learning is best learning. I was one ...
18 votes

Intuition on the Orbit-Stabilizer Theorem

Intro I don't think that the Orbit-Stabilizer is 'evident' or 'obvious' in any respect. Usual explanation that is on sale is this. We take a cube that has all faces painted with a different color. ...
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18 votes

Self-learning mathematics - help needed!

There's a quip due to von Neumann: "In mathematics we don't understand things; we just get used to them." He was famously insightful and also, apparently, somewhat introspective about his methods, so ...
  • 6,120
18 votes

How can I pick up analysis quickly?

Principles is an excellent text, but I don't think it's well-suited to self-study. There's nothing wrong with it, honestly, and you'd probably be fine reading it, but to me it's one of those many ...
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18 votes

How to stop forgetting proofs - for a first course in Real Analysis?

If you just remember individual proofs, and arrange them in a circle on your notepad, you will notice that it's $2\pi r$ to go around and remember all the proofs, but if you start at the centre (the ...
18 votes

Is it necessary for one to understand analysis?

Yes. Analysis, as far as you have seen it, is training on mathematical reasoning, on the meaning and usage of quantifiers, on organizing more or less complex proofs, and such. You will not get far ...
18 votes
Accepted

Why does $0.888888888889 \times 9 = 8$?

You have discovered a calculator truth, not a mathematical truth. Calculators and computers(when using the standard floating point numbers) only store numbers with a fixed number of significant ...
17 votes

I'm looking for some mathematics that will challenge me as a year $12$ student.

Project Euler is a great source of interesting problems. Many of them require you to learn a little computer programming, which I highly recommend you try if you haven't before. (And if you don't have ...
  • 24.8k
16 votes

Best Sets of Lecture Notes and Articles

Undergraduates in Cambridge often refer to the notes of a student that typeset a lot of the lectures when he was there: https://dec41.user.srcf.net This includes detailed notes for the courses from ...
16 votes

How to stop forgetting proofs - for a first course in Real Analysis?

Do not memorize proofs. Just become comfortable with how to think about certain proofs and the basic framework of proving certain ideas. That is, you do not need to memorize how to prove that $x^3$ is ...
  • 1,638
16 votes

How to explain the formula for the sum of a geometric series without calculus?

I think a 14 year old can grasp the fact that $$\frac 12 + \frac 14 + \frac 18 + \frac 1{16} + \cdots = 1$$ rather intuitively. (Go halfway there, then half the remaining distance, then halfway again, ...
  • 50.4k
16 votes

I'm looking for some mathematics that will challenge me as a year $12$ student.

If you are really seeking for recomandation I can suggest you some books: $1$.Putnam and beyond: Book by Razvan Gelca and Titu Andreescu. $2$. Elementary Number Theory: Primes, Congruences, and ...
16 votes

Do Math Professors still do textbook problems to learn subjects foreign to them?

This is just an opinion, based on my own experience as a working mathematician. Let me say honestly that a professor seldom studies a subject as a student would do. In our times, publishing has become ...
  • 34k
16 votes
Accepted

Is There Anything to be Learned from Very Long Computations?

One of the most extraordinary pieces of mathematical work in the history of humankind is Gauss's Disquisitiones arithmeticae. In that book he sets the groundwork for much of modern number theory — for ...
15 votes

Vector spaces - Multiplying by zero scalar yields zero vector

To shorten the proof, we may write as suggested by André Nicolas, Proof. Let $(V,+,\cdot)_F$ be a vector space over the field $F$. We wish to show that $\forall v\in V$ one has $0\cdot v=\mathbf{0}...
  • 6,603
15 votes

Collecting math websites

I have found these extremely useful, from time to time: Online Encyclopedia of Integer Sequences NIST Digital Library of Mathematical Functions They both provide facts of a specialized nature, rather ...
13 votes

How to study for analysis?

While this is doubtless too late for the OP, it may help others studying analysis. Lara Alcock, who does research on how people understand abstract mathematics, has recently written a book, How to ...
  • 2,196
13 votes

Best Sets of Lecture Notes and Articles

The Oxford University Maths department seem to have their full set of lecture notes with example sheets on here: https://www0.maths.ox.ac.uk/courses/material/
13 votes

Strategies to study apart from "books cover to cover."

Reading math books to learn math does, at least at times, feel a bit like reading a dictionary to learn a language. Linguists and polyglots will affirm that this is a bad strategy. Instead, they will ...
  • 26.2k

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