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Jul
30
comment Continuity Must Hold in an Entire Open Set?
Would be cool if you could show the second example too!
Jul
29
comment When I was teaching absolute function properties, I suddenly made this question …
I was confused what you meant by the "absolute" function. I think you mean the "absolute value" function...
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@IttayWeiss: OK, I see what you're saying, but while the proof you mentioned is trivial to justify, I think it's not at all trivial to to derive in the first place. How is a student supposed to know that he should go through the extra step of decomposing the vector with respect to an entire orthonormal basis merely to show that the weight of one vector in that orthonormal basis is the desired dot product?
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@DavidZ: Well if you define dot product that way then I guess my problem is that it's not obvious to me why the dot product is the sum of the componentwise products.
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@DavidZ: Nope, read my previous comments again. I literally said "the hard part is understanding why dot product has anything to do with projection in the first place". i.e., no, I don't already "know" that the dot product has to do anything whatsoever with the projection or with the angle. That fact is precisely what I'm calling out as non-obvious here.
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
@DavidZ: That's not quite what I meant. I meant that (geometrically and intuitively) projection has to do with the (cosine of) the angle between the vectors. Why should the (cosine of) the angle have anything whatsoever to do with the dot product of the two vectors? It's not at all obvious to me that the two might have any significant relationship, yet they do.
Jul
26
comment Intuition for the Cauchy-Schwarz inequality
I feel like the hard part is understanding why dot product has anything to do with projection in the first place. Why does the sum of a componentwise product tell you something about the vectors' projection? Is it just me, or is this a completely non-obvious fact that everyone takes for granted?
Jul
26
revised What do sine, tan, cos actually mean?
added 70 characters in body
Jul
26
revised What do sine, tan, cos actually mean?
added 70 characters in body
Jul
25
revised What do sine, tan, cos actually mean?
deleted 3 characters in body
Jul
25
comment What do sine, tan, cos actually mean?
@Karl: Thanks! I added a couple more too :)
Jul
25
revised What do sine, tan, cos actually mean?
added 343 characters in body
Jul
25
answered What do sine, tan, cos actually mean?
Jul
24
revised Sum of an infinite series $(1 - \frac 12) + (\frac 12 - \frac 13) + \cdots$ - not geometric series?
deleted 1 character in body
Jul
24
comment Must an algorithm that decides a problem in NP also produce a solution?
Regarding the crypto part: just because an algorithm exists doesn't mean you can find it. If the proof is nonconstructive then it will have zero impact on modern cryptography. And even if it's constructive, if the algorithm itself is practically slow then that will also have no impact on modern cryptography. etc.
Jul
23
comment Why there is no sign of logic symbols in mathematical texts?
@AlphaE: Just because two sentences mean the same thing and just because you can understand both of their meanings, that doesn't mean they require the same amount of brainpower to parse.
Jul
22
comment Why there is no sign of logic symbols in mathematical texts?
@FireGarden: Er, don't get too excited with the strawman... I was referring to mathematical symbols under the (obvious) consideration of this question, not to all symbols that have ever existed in all of mathematics as a whole. I'm asking why no one wants to admit that the use of symbols in such contexts actually make things harder to read and understand. Everyone seems to go on tangents with reasons like elegance or difficulty of printing or whatever, when in fact it's clearly about ease of parsing for the human reader.
Jul
22
comment Why there is no sign of logic symbols in mathematical texts?
Why does no one want to admit that it's because mathematical symbols are harder to read and don't necessarily convey any more information than plain English?
Jul
19
revised If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?
I accidentally a word
Jul
19
comment Does the concept of a derivative a rate of change work for n dimensions?
While you're pondering this, make absolutely sure you understand the difference between partial and total derivatives. There are situations in which both can be taken but result in different values! (e.g. consider differentiating $y(x, t)$ with respect to $t$ but where $x$ also depends on $t$.)