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13h
comment a question about rational power irrational is irrational?
According to MathWorld this is an open problem.
14h
comment a question about rational power irrational is irrational?
@Asterrovels If $a$ is irrational then $2^a$ can be rational. In fact, for most rationals $x$, $\log_2(x)$ is an irrational number. For instance, you can easily prove by contradiction that $\log_2(3)$ is irrational, since if it wasn't you'd end up with a power of two being equal to a power of three.
20h
comment a question about rational power irrational is irrational?
@Asterrovels Okay, but in that case, since $a$ is irrational you can no longer claim that $(2^a)^{\frac 1 b}$ is irrational.
20h
comment a question about rational power irrational is irrational?
When you say "now assuming that $\log_2 p = \frac a b$", this is tantamount to saying, "let's assume $e$ is rational".
1d
awarded  Yearling
1d
answered Please Explain Kuratowski Definition of Ordered Pairs
May
1
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.
Apr
30
asked Does the PNT establish a connection between primes and the logarithm?
Apr
29
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.
Apr
29
answered Prove that the Gaussian Integer's ring is a Euclidean domain
Apr
29
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
27
awarded  Nice Answer
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
20
answered why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
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
13
awarded  elementary-number-theory
Apr
12
comment How Are The Graphs Related?
@JebusCrust Then it moves to the left by $c$ units.
Apr
12
answered Concise proof that every common divisor divides GCD without Bezout's identity?