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Jul
13
comment Is sine of one degree a real? If not, how is sine continuous?
Also thanks for pointing to Galois theory! This is something on my list of the courses I'd like to take in the future.
Jul
13
accepted Is sine of one degree a real? If not, how is sine continuous?
Jul
13
comment Is sine of one degree a real? If not, how is sine continuous?
@Shailesh ok, thanks, I see now.
Jul
13
comment Is sine of one degree a real? If not, how is sine continuous?
@Shailesh in light of this: en.wikipedia.org/wiki/Angle_trisection#Proof_of_impossibility (the proof says that the sine of one degree would be a root of third degree polynomial, which is not a real number).
Jul
13
asked Is sine of one degree a real? If not, how is sine continuous?
Jun
24
comment If a function is defined on the interval $(a, b)$, is the derivative necessarily defined at $a$ and $b$?
@TravisJ no need. It's just one of those commonly misused words, and one of those rare cases when it was actually used properly :) so it is a bit ironic, but it doesn't matter since whether it was assumed, subsumed or presumed, the result would be the same :)
Jun
24
comment If a function is defined on the interval $(a, b)$, is the derivative necessarily defined at $a$ and $b$?
@TravisJ Normally, I try to avoid fighting over edits that change nothing in the question, but just out of curiosity, why did you change "subsume" to "assume"? I mean, the thing that I'm given to prove doesn't assume that, it subsumes (as in, if proved to be true, then this statement would be a particular case of some larger claim).
Jun
24
accepted If a function is defined on the interval $(a, b)$, is the derivative necessarily defined at $a$ and $b$?
Jun
24
comment If a function is defined on the interval $(a, b)$, is the derivative necessarily defined at $a$ and $b$?
Thanks, although I'm going to accept Noah Schweber's answer (simply because he was first). All answers in my eye are good. I just have to pick one to make SO rules happy.
Jun
24
comment If a function is defined on the interval $(a, b)$, is the derivative necessarily defined at $a$ and $b$?
Thanks! I'm going to accept Noah Schweber's answer just because it was the first one. Now I see that all answers are very nice, so there would be no other reason to prefer one. I'm just doing it to comply with SO rules. :)
Jun
24
comment If a function is defined on the interval $(a, b)$, is the derivative necessarily defined at $a$ and $b$?
@GitGud the assignment doesn't say that the function isn't defined anywhere outside $(a, b)$, so it could be defined, I'm just not told that it is.
Jun
24
asked If a function is defined on the interval $(a, b)$, is the derivative necessarily defined at $a$ and $b$?
Jun
24
accepted When are monic polynomials of fourth degree divisible?
Jun
24
comment When are monic polynomials of fourth degree divisible?
OK, that and en.wikipedia.org/wiki/Intermediate_value_theorem. I think I've got it now. Thanks, much appreciated!
Jun
24
comment When are monic polynomials of fourth degree divisible?
Oh, so this would be basically the proof of en.wikipedia.org/wiki/Factor_theorem, wouldn't it?
Jun
24
comment When are monic polynomials of fourth degree divisible?
Why should there be a leading term of odd degree? What if its coefficient is zero?
Jun
24
comment When are monic polynomials of fourth degree divisible?
@MichaelBurr ouch, I was thinking of something else... yes, that's definitely true.
Jun
24
comment When are monic polynomials of fourth degree divisible?
@ajotatxe if the graph of quadric function touches X axis, the root will be zero, but that possibility is already excluded in the given.
Jun
24
comment When are monic polynomials of fourth degree divisible?
@MichaelBurr fair point... I've re-read the text of assignment, and it doesn't say explicitly that the roots must be distinct, but if they shouldn't be distinct, then it's an absolute no-brainer, just by using the fundamental theorem of algebra...
Jun
24
asked When are monic polynomials of fourth degree divisible?