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20h
comment Refuting the Anti-Cantor Cranks
@DaveL.Renfro That in't a proof by contradiction, it's a proof by contrapositive. "If I am the murderer, then I was not at the party", to which the contrapositive is "If I was at the party, I am not the murderer." But even in mathematics, contradiction is often confused with contrapositive.
2d
comment Number of points of accumulation of a sequence
There's a classic theorem that if you walk around a circle in discrete steps of $a$ radians, where $a/\pi$ is irrational, then the set of points that you visit is dense in the circle. This implies that $(\sin(an))_{n\in\mathbb N}$ is dense in $[-1, 1]$, and you can take $a=1$ for a cute example of a sequence with infinitely many accumulation points.
2d
comment Irrational numbers generated by a deterministic cellular automaton?
I'm not sure why you're so pessimistic about proving that a cellular automata-generated number is transcendental, when you then immediately cite a very similar (and marvelous) result about how finite automata-generated numbers are transcendental.
Apr
24
comment Is there a way to write an infinite set that contains only irrational numbers without integer multiples?
More generally, you're relying on the lemma that for any finite set of irrationals, the set of the their integer multiples does not cover the set of irrationals. But this is obvious: the set of integer multiples of any finite set is discrete, but the irrationals are dense.
Apr
21
comment Injective: In what circumstances would there be less than one pre-image of an image?
No, no - of course an "image with less than one (ie, zero) pre-image" is a contradiction, that can never happen. An injective function has no more than one pre-image for each image, but the opposite of "no more than one" isn't "less than one", it's "only one".
Apr
17
comment Is PA the first axiomatization of arithmetic to be discovered?
Euclid sort of axiomatized arithmetic, or tried to.
Apr
14
comment Does this equation my professor wrote actually work?
@hexomino No no, that's not the problem - $(\frac 1 e)^0$ is well-defined. The problem is that for some reason the OP thinks $(\frac 1 e)^0$ should give $0.5$, which it shouldn't - $0.9$ (ish) is correct. How do you figure that's 0.5, OP?
Apr
12
comment How Are The Graphs Related?
@JebusCrust Then it moves to the left by $c$ units.
Apr
12
comment Finding a Mathematical definition of a Discrete Time Game
@MathNerd So if it's in discrete steps, why are you modelling time by a real number? Just use an integer.
Apr
12
comment Finding a Mathematical definition of a Discrete Time Game
I feel you've left out some details as to how the game works. Do the players move in discrete steps? If not, do they move at constant speed, or can they accelerate and decelerate? If motion is continuous, then when is a tank considered to leave one square and enter the next? If a square is shot at and a tank is partially on that square, is that a hit?
Apr
8
comment Why does the Elo rating system work?
Great, thanks. So we basically model chess players with a weighted directed graph (the weight on the edge A -> B being the odds that A beats B), with the transitivity assumption you described. That seems like quite a non trivial assumption. Do you know a reference with more discussion on the details and the validity of the model?
Apr
7
comment Why does the Elo rating system work?
I haven't read the papers in full yet, but the one which gets closest to answering the question still seems to miss the mark. It helpfully describes a model in which we think of chess players as generating normally distributed random numbers, and the winner of a game as being the one that generates the larger number, but then when it gets to the formula for the probability of one player winning the game (page 10) it just calls it an assumption of the model. I find it hard to swallow that such an arbitrary formula is literally just assumed with no justification.
Apr
6
comment Area of a circle $\pi r^2$
The point is, you are the one imposing the interpretation on $r^2$ as being the area of a square. Really, $r$ is just a number. And $r^2$ is another number. And if you multiply that by $\pi$, you get yet another number, which happens to equal the area of the circle (which is also just a number).
Apr
6
comment Area of a circle $\pi r^2$
I have nine pens on my desk. Nine is $3^2$. This is the area of a square with side length $3$. How can a square become pens??
Feb
11
comment Comma placement inline math
@BrianO The first fragment can be read that way as well, imagine there's an implicit "where" or "with" after $f(x_j)$.
Feb
11
comment Comma placement inline math
@BrianO I believe the OP is saying that you can read the sentence as "Given $f(x_j)$, where $j=1, ... N$, - our goal is to...". In this case the comma is clearly recommended.
Feb
10
comment Can an infinite sum of irrational numbers be rational?
And the linear independence of the terms follows from the transcendence of $\pi$, which is the cleverest part of the answer. In fact any power series with rational coefficients, evaluated at a transcendental value but having a rational sum, would have done. It makes me wonder if there are "super-transcendental" numbers, which are not roots of any power series with rational coefficients.
Feb
9
comment How limiting/ heavy is the “triangle inequality” assumption?
If you drop the triangle inequality, basically all you have left is a symmetric function from $X^2$ into the positive reals. The triangle inequality is responsible for every important property of a metric.
Feb
8
comment Is it accurate to say that multiplication of two integers yields an integer?
@user7530 The important issue that needs to be dealt with here is that the OP thinks they can trust a computer's floating point circuits to tell them about math. This will just be over their head.
Feb
8
comment Is it accurate to say that multiplication of two integers yields an integer?
I just put $\sqrt 5^2$ into my computer and got $5.000000001$. Computers only do arithmetic approximately.