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1d
comment “continuously differentiable $\subseteq$ Lipshitz continuous” with $f(x) = x^2$
"over a closed and bounded set"!!!
2d
comment Intuition for the Cauchy-Schwarz inequality
@Mehrdad that student can be shown this fact, or requested to clarify the intuition in an exercise.
2d
comment Intuition for the Cauchy-Schwarz inequality
beware a circular argument: it is the CS inequality that allows one to define $\theta$ in the first place.
2d
comment Intuition for the Cauchy-Schwarz inequality
@Mehrdad the relation to the cosine (law) may be not the best way to see that. I like to think of it in terms of orthonormal basis. Given an orthonormal basis, any vector $u$ is simply $\sum_i (u\cdot v_i) \cdot v_i$ by a trivial computation which has nothing to do with CW or the law of cosines. This trivially shows that the dot product gives the components of a vector. When teaching these things I like to use the law of cosines to motivate the definition of the standard inner product. Then CS is shown to relate to a law of cosines (kind of) for any inner product.
Jul
24
comment How can I make math joyful to learn?
The analogy with chess is not a very strong one, I find. Chess is a game. Mathematics is a scientific practice. Yes, both require careful analysis, and yes both often exhibit high degrees of elegance. But their aims are completely different. Chess only applies to real life in that it teaches one how to analyse things carefully. Maths applies to real life both in the techniques it employs and, far more importantly, in its direct applicability to model real life situations.
Jul
24
comment How can I make math joyful to learn?
As you say yourself, it's a problem of how math is typically taught, not with maths itself. If I were taught that mathematics is just the boring manipulation of formulas and numbers, I probably would not find it joyful at all. Luckily, I learned maths from good books and a few good professors. In any case, it is not so clear what you are asking as you seem to say "maths is not the way it's commonly taught, so how do you, who obviously don't represent the common maths student, find it joyful".
Jul
22
comment Finite free objects
what do you mean by trivial here? As the question stands it highly unclear what you are looking for.
Jul
22
comment How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?
yes, the common vernacular is circularity: it's random cause we assume it's random. We just try to wrap it using longer words.
Jul
22
comment How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?
"... is effectively random" is precisely the problem. It seems the only way to say "the coin flips are truly random" is to assume that the process produces true randomness. I can only strongly suggest the first few chapters of Jaynes' book where these deep issues are discussed in great detail.
Jul
22
comment How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?
The physical attributes of the coin are only one side of the coin (ahm, pardon the pun). Even if the coin is physically perfectly symmetric etc., how do you separate the act of flipping the coin from from questions of fairness? With a bit of practice, one can learn to flip a coin in such a way so as to change the outcome. One can thus adjust for physical imperfections and make the coin appear fair. Or one can take a perfectly fair coin and make it appear biased. Jaynes addresses this beautifully as well.
Jul
22
comment How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?
I would conjecture that to be the leading reason.
Jul
22
comment How can tossing a coin be a certain 50/50 chance? Isn't that short-sighted?
I wonder what the down votes are based on.
Jul
22
comment Can $1=0$ ever make sense?
@JoshuaLaJeunesse I edited your question further. I hope this is still in line with what you have/had in mind.
Jul
22
comment Can $1=0$ ever make sense?
@JoshuaLaJeunesse indeed, I was trying to help :)
Jul
22
comment Can $1=0$ ever make sense?
@JoshuaLaJeunesse based on the comments so far (some demonstrating a negative answer, some a positive answer) you may wish to edit your question. It may assist in your question being re-opened.
Jul
22
comment Can $1=0$ ever make sense?
@Rahul but we do not make up rules. Instead we very carefully choose and pick very specific ones. Mathematics is a practice in which we can change the rules anytime we want, but we hardly ever do, and when we do it is out of compelling reasons to do so.
Jul
22
comment Can $1=0$ ever make sense?
It's a bit harsh to close this question so quickly. True, some more details are perhaps in order, but the question does make sense. I think we should be a bit more accommodating to questions spawned by curiosity that seems genuine. Moreover, the question has interesting answers.
Jul
22
comment The trivial group, multiplication, 0 and 1
it's not clear what you are asking, particularly the remark about finiteness. Both groups you describe are finite, each having precisely one element. Generally, it would probably ease your understanding if you realize that you can construct a group structure on any set with single element. If that element is the number $1$ or the number $0$ is of no relevance. Continuity issues are also trivial for any single element group, as there is a unique topology on the set, and the group operations are automatically continuous.
Jul
20
comment What are some easy to understand applications of Banach Contraction Principle?
No offence @RobertLewis I'm just still puzzled by OP's comments here.
Jul
20
comment What are some easy to understand applications of Banach Contraction Principle?
@RobertLewis I suspect you are being sarcastic. Can you clarify on the Gauss business, in case I'm missing something?