Wikipedia:Reference desk/Archives/Mathematics/2018 May 26

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May 26

e is a rational number

Here is proof.

$e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots$

even number = even number

even number + even number = even number

even number + even number + ... = even number

so

rational number = rational number

rational number + rational number = rational number

rational number + rational number + ... = rational number

so e must be a rational number. 49.177.234.140 (talk) 03:38, 26 May 2018 (UTC)

What is true is that every $N$, $\sum _{k=0}^{N}{\frac {1}{k!}}$ is a rational number. Count Iblis (talk) 07:04, 26 May 2018 (UTC)

A proof that requires an infinite number of steps or a vague "and so on, to infinity" is not a valid proof, though Mathematical induction is valid. Dbfirs 08:38, 26 May 2018 (UTC)

You could simplify that to saying that since 2, 2.7, 2.71, 2.718, 2.7182, 2.71828 etc are all rational then e is rational. Yes they are all rational, but none of them is e, they are just approximations to it. Think of e as the ground and each of those numbers being you as you fall down to it. That you are safe and unharmed for every single point above the ground doesn't mean hitting the ground will not be painful Dmcq (talk) 11:36, 26 May 2018 (UTC)

Every irrational number can be expressed as an infinite decimal. So if the OP's reasoning were valid, this would have proven that all numbers are rational. CodeTalker (talk) 23:27, 26 May 2018 (UTC)

If we deny the existence of infinite quantities, then the OP has a valid point, although the number e the OP refers to then doesn't exist. So, one can say that only finite objects sets exist, and infinite means "not finite" and that then implies that quantity in question doesn't exist. Count Iblis (talk) 08:49, 27 May 2018 (UTC)

As, in general, if we take ridiculous positions, we get ridiculous answers. --Trovatore (talk) 08:55, 27 May 2018 (UTC)

Indeed. Count Iblis (talk) 09:41, 27 May 2018 (UTC)

The Hilbert Hotel is just what would really happen, if its conditions were met. Of course, they can't be, physically, but it makes perfect sense as a thought experiment. It's not ridiculous at all. It's just true. --Trovatore (talk) 17:56, 27 May 2018 (UTC)

Under various definitions of true. How about the Hilbert Hotel with rooms 1 to infinity with a microbe in room 1. The microbe keeps dividing such that a microbe in room N leaves that room and divides to occupy rooms 2N and 2N+1. How many rooms are inhabited after infinite time, 0 or $2^{\aleph _{0}}$? How about if the microbe starts off in room 0? Dmcq (talk) 18:15, 27 May 2018 (UTC)

It's true, you can come up with supertasks where the appropriate limit to take at limit ordinals is not clear. However, in the case of the Hilbert Hotel, it is clear. --Trovatore (talk) 20:02, 27 May 2018 (UTC)

And then there is Hilbert's computer which the hotel employees use. It has an infinite memory and each clock cycle takes half the time of the previous one. So, it can do an infinite amount of computations within twice the time it takes for the first clock cycle. Hilbert knows that the Riemann hypothesis is true because his computer has checked that all the nontrivial zeros are exactly on the critical line, one by one ad infinitum. He has also verified that there is no finite proof of the theorem because he has run his computer through all proofs of all theorems and no proof of the Riemann hypothesis was encountered. Therefore he proposes that his computer's result should count as a proof of infinite size. Count Iblis (talk) 18:42, 27 May 2018 (UTC)

Sure, why not? It's not a proof in the sense of first-order logic, but it's a proof in the sense of establishing the proposition. --Trovatore (talk) 20:02, 27 May 2018 (UTC)

No, it needn't. Many apparently 'obvious' properties of objects are not preserved in case of infinite collections. For example, a sum of any two natural numbers, and, by induction, a sum over any finite set of natural numbers is natural – however, the 'sum' of infinite natural sequence is not; a sum of any two continuous or differentiable real functions is continuous or differentiable, respectively, hence also a sum of any finite set of continuous or differentiable functions is continuous or differentiable – but the sum over an infinite set of functions may be discontinuous or not differentiable (see illustrations for a rectangular or sawtooth wave at Fourier series or a Weierstrass function which is defined as a sum of a series of cosines, but is itself nowhere differentiable); any iteration of the dragon curve generated by the Lindenmayer system has finite length, however, the 'limit curve' has the length infinite and it fills the area of the resulting fractal. --CiaPan (talk) 14:04, 28 May 2018 (UTC)

Two identical jigsaws

You have two identical jigsaws (same cut, same picture, so the equivalent piece from each would fit in the other). You empty both into a box and shake them about, then take exactly half the pieces and try to do the jigsaw. What proportion of the pieces are you likely to be missing? My initial intuition, and investigation for small numbers of pieces, indicates that you would expect to be missing 1/3 of the pieces, but I'm wondering how to prove this. -mattbuck (Talk) 08:51, 26 May 2018 (UTC)

If there are n pieces in each jigsaw, then the number of ways of missing exactly k pieces is

${\frac {n!}{k!^{2}(n-2k)!}}2^{n-2k}.$

The reasoning here is if k pieces are missing, then you also have k duplicates and n-2k pieces where you have exactly 1. There are

${\frac {n!}{k!^{2}(n-2k)!}}$

ways to divide up the set of n pieces into subsets with these sizes, and once these subsets are fixed there are 2^n-2k ways to chose the n-2k 'singles'. The total number of ways to pick n pieces from the 2n pieces in the box is

${\frac {(2n)!}{n!^{2}}}$

so the probability of getting k missing is

$P(k)={\frac {{\frac {n!}{k!^{2}(n-2k)!}}2^{n-2k}}{\frac {(2n)!}{n!^{2}}}}.$

That gives the probability distribution (unless my algebra is faulty), but you still need to do more analysis to get the expected value. It's a start anyway. --RDBury (talk) 09:58, 26 May 2018 (UTC)

PS. It's not the expected value, but the k with the largest value (the mode) is easier to compute and works as a ball-park estimate. I'm getting that if n is large then the mode occurs at about 40% of n. --RDBury (talk) 10:11, 26 May 2018 (UTC)

PPS. Actually it should have been 25% of n for the mode. I worked out the first few values and there does seem to be a closed form expression for the expected value, namely

${\frac {n(n-1)}{2(2n-1)}}.$

This is from inductive evidence only, I don't have a proof. --RDBury (talk) 13:10, 26 May 2018 (UTC)

If there are n pieces in the puzzle, then for a given piece, the probability that neither of its copies will be included is approximately 1/4. So by linearity of expectation, the expected number of missing pieces is $n/4$ (for large $n$). (To forestall the objection that someone always manages to raise: linearity of expectation does not require independence.)--108.36.85.137 (talk) 16:56, 26 May 2018 (UTC)

Ah yes, actually the probability that neither copy of a given piece being included is

${\frac {\frac {(2n-2)!}{n!(n-2)!}}{\frac {(2n)!}{n!^{2}}}}={\frac {n-1}{2(2n-1)}}.$

So, by your idea of linearity, the expected number of missing pieces is n times this or the value above. I was going to spend the morning trying to prove the value using some sort of hypergeometric identity but no need now; nice insight! --RDBury (talk) 21:25, 26 May 2018 (UTC)