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Dec
20
reviewed Reviewed How do I implement an induction argument to show that we may assume that a partition $Q$ of $[a,b]\in{R}$ which refines $P$ has only one more point?
Dec
19
comment Axiomatic characterization of the rational numbers
Somehow, it seems more elegant to define it as having characteristic 0 and deriving that it is ordered rather than the other way around.
Dec
19
comment Axiomatic characterization of the rational numbers
Saying that a field is not complete is not sufficient. There are lots of fields that are not complete, whose completion is $\mathbb{R}$. $\mathbb{Q}$ and $\mathbb{Q}(\sqrt{2})$ are two trivial examples.
Dec
18
comment Working out digits of Pi.
Which mathematician?
Dec
18
comment Is there a fundamental reason that $\int_b^a = -\int_a^b$
In case anyone is wondering, we are going to go with the Karr convention. The reason is that it's the only one that's consistant with computing a summation with symbolic limits and then substituting a numerical value vs. computing the summation with the numerical value to begin with, where the numerical value gives reversed limits. This also provides a good argument for integrals (in terms of the Fundamental Theorem of Calculus).
Dec
18
revised Working out digits of Pi.
Give the series explicitly
Dec
18
comment Working out digits of Pi.
I took the liberty of editing your answer to give the series explicitly, and to point out how slow it is.
Dec
18
suggested approved edit on Working out digits of Pi.
Dec
18
comment Working out digits of Pi.
One thing I'm not sure about: the mpmath source says that each term adds roughly 14 digits, but it's not clear if it's 14 more digits of $\pi$ or 14 more digits of $\frac{1}{\pi}$.
Dec
18
comment Working out digits of Pi.
This is an insanely inefficient way to calculate $\pi$. Even if you have a calculator and only want to calculate 10 digits or so, it will take you forever.
Dec
18
answered Working out digits of Pi.
Dec
18
comment Which one result in mathematics has surprised you the most?
I'm not familiar with the wrapping proof. Is that the same as this?
Dec
18
comment Which one result in mathematics has surprised you the most?
${}{}{}{}{}{}x^3$?
Dec
18
comment Which one result in mathematics has surprised you the most?
Also known as $\sqrt{\pi}$.
Dec
18
accepted Is there a fundamental reason that $\int_b^a = -\int_a^b$
Dec
18
comment Is there a fundamental reason that $\int_b^a = -\int_a^b$
"The fundamental theorem of calculus...is based on this interpretation of the integral." This, I think, is the key insight here.
Dec
18
comment Is there a fundamental reason that $\int_b^a = -\int_a^b$
But what about signed measures?
Dec
18
comment Is there a fundamental reason that $\int_b^a = -\int_a^b$
Your very last sentence also gives a very nice motivation (it's ultimately the same as the accepted answer from math.stackexchange.com/questions/232455/…). I need to think on how this can apply to summations.
Dec
18
comment Is there a fundamental reason that $\int_b^a = -\int_a^b$
"...and $\int_A f+\bar\int_A\,f=0$." As you've defined it, how could this not hold?
Dec
18
awarded  Nice Answer