New answers tagged irrational-numbers
7
votes
Accepted
How well can $e$ be approximated?
It turns out that we've basically solved this problem completely, because $e$ has a known continued fraction. The $q^{-2}$ can be replaced by
$$
\frac{(\tfrac12+\varepsilon)\log\log q}{q^2\log q}
$$
...
- 84.9k
0
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Remarkable/unexpected rational numbers
A problem posed here: Take an unpainted circular disk. Repeatedly paint the interior of a randomly-chosen semicircle of it (bounded by one of its diameters), stopping once the whole disk is painted. ...
- 2,996
0
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Bijection between rational and irrational numbers?
I suppose you want something like this. Knowing the rationals are countable, we can list them in an orderly fashion as:
$$\mathbb{Q}= \{q_1, q_2, \dots\}.$$
Now define a function $f:\mathbb{Q}\to\...
- 3,304
3
votes
Bijection between rational and irrational numbers?
One issue is that the (ir)rationals involved need not be unique.
You suppose there exists a rational between $-\pi$ and $\pi$; well, $1$ meets that criterion.
And likewise a rational between $-\sqrt 2$...
- 47.4k
0
votes
Accepted
A monotone function has at most countably many discontinuities
Perhaps this is what you mean:
Since there are uncountable irrational numbers, and one irrational number can be chosen from each interval, there is a one-to-one correspondence between such intervals ...
- 1,993
0
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Intuitive explanation why Lebesgue measure of irrationals in [0,1] equals 1
While your interpretation is in the right track, I think you need to be a bit clearer on what you mean in certain passages.
the Lebesgue measure requires a countable union of disjoint intervals to ...
- 186
5
votes
Prove that $\log_{10} (\phi)$ is irrational
Theincredibleidiot gave a nice answer. Adding the following different argument relying on a little bit extra machinery. It should be generalizable.
Let $\tilde{\phi}=(1-\sqrt5)/2=-1/\phi$ be the ...
- 137k
8
votes
Accepted
Prove that $\log_{10} (\phi)$ is irrational
Note that $\phi^2=\phi+1$. So for every positive integer $n$ one can write $$\phi^n=a\phi+b,$$ for some suitable integers $a(\neq 0),b$. From your work it follows that if $\log_{10}(\phi)$ is rational,...
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