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Apr
24
awarded  Yearling
Mar
26
awarded  Announcer
Mar
15
comment How to convince a layman that the $\pi = 4$ proof is wrong?
This explanation is actually simply incorrect. In any reasonable way, the sequence of curves absolutely do converge to the circle. It just so happens that the limit involved in calculating the perimeter does not commute with the limit of the curves. There are theorems which give sufficient conditions for these limits to commute, and you are correct this sequence violates any of those conditions: but violating the condition of an implication does not guarantee the conclusion fails. Your reasoning is backwards. I imagine you are confusing this example with something like the Koch snowflake.
Mar
15
comment How to convince a layman that the $\pi = 4$ proof is wrong?
"A layman would ask 'why is your strange 'polygonal approximation' method correct, but the π=4 proof's method incorrect?' and I have to admit I fail to see strong arguments to convince him here." The π=4 method isn't "incorrect", it just gives a different notion of distance than Euclidean distance. In fact, the distance it gives is perfectly valid: it's the taxicab metric perimeter of the circle. It's a perfectly reasonable notion of distance, it just isn't the Euclidean distance we talk about in every day life: the one that is invariant under rigid rotations/translations, etc.
Mar
1
awarded  Popular Question
Feb
18
comment Find inflection points of parametric equations
Where did you find this definition? Wikipedia provides no citation for it. By your definition, it would be easy to write a parameterization of $y=x^2$ with infinitely many inflection points (say $x(t)=t\sin(t)$). If you are going to call it a geometry property, you had better change your definition. Perhaps you meant to impose conditions on the parameterization.
Dec
1
comment Why not define the Conway base-5 function, instead of base-13?
Oh thank you, I think I get it now, the extra symbol is to code all the negative real numbers. As a logician, I often forget what the "reals" are e.g. that they aren't just $[0,1]$ or $\omega^\omega$ or $2^\omega$ etc.
Nov
25
comment A definition of Conway base-13 function
I don't understand you argument that the function is not computable. You seem to be arguing there is no algorithmic procedure turning an expansion of the input base-13 into an expansion of the output base-10. But if that's your definition of computable, not even multiplication by 3 base-10 to base-10 is computable.
Nov
25
comment Why not define the Conway base-5 function, instead of base-13?
Conway's base 13 always confused me, I never understood why +3 is used rather than +2. Surely he just needs one extra digit for the decimal point, and one extra digit to mark the end of the base-13 prefix? So why didn't we end up with Conway's base 12 function?
Oct
15
answered Calculate the integral of the function only with the immediate integrals
Oct
15
answered What are the elements of $2^A$ if $A$ is a set
Oct
6
comment A Theorem on Compactness By Munkres
By "totally trivial" I don't mean it's obvious. Your question is a good one. I just mean that the difference between this definition and the usual definition is not mathematically significant. It's just a matter of phrasing. Munkres chose this phrasing for the purpose of emphasizing the intrinsic nature of compactness.
Oct
6
comment A Theorem on Compactness By Munkres
This is totally trivial: "open in $X$" just means the intersection of an open set with $X$. "Every open cover has a finite subcover" is entirely equivalent. Munkres is just trying to get away from the habit of thinking of "compact" as being a property of how a set is situated in a larger set, rather than as an intrinsic property.
Oct
6
revised Prove that $GCD(a,b)=1$ if for all natural numbers $c, a|bc $ then $a|c$.
added 21 characters in body
Oct
6
revised Prove that $GCD(a,b)=1$ if for all natural numbers $c, a|bc $ then $a|c$.
added 118 characters in body
Oct
6
answered Prove that $GCD(a,b)=1$ if for all natural numbers $c, a|bc $ then $a|c$.
Oct
6
answered Getting two answers to: How many monoalphabetic substitution ciphers of $\{A,B,C,D\}$ are possible in which no letter is fixed?
Oct
6
answered Limit of the integrals of a $\left\{f_{k}\right\}_{k \in \mathbb{N}}$ decreasing succession of integrable measurable functions,
Oct
5
comment What is $0 \times \infty$?
@fred To the right of all the finite numbers.
Oct
5
awarded  Custodian