Wikipedia:Reference desk/Archives/Mathematics/2007 April 6

= April 6 =

Sweeping the floor
A child is instructed to sweep a floor. He concludes that to sweep every single bit of the floor- he must sweep it in a systematical way, going up and down the floor, then moving a broom's length to the left or right. By doing this, it is deduced that the chance of sweeping every part of the floor is 100%. Therefore, it is deduced that if the child did not sweep it systematically, there would be a much less chance of sweeping 100% of the floor.

Would this be true, or is it a flaw in our logic?

p.s i have tried the humanities desk but have been referred here.--Howzat11 01:38, 6 April 2007 (UTC)


 * Not a great question. "He concludes that … he must …." (Emphasis added.) If the conclusion is correct, then there is no choice. If the conclusion is incorrect, it is rather confusing to state it. Without the "conclusion", we have the following parallel example: "A child has a piece of cake, and finds it a satisfying dessert. If ice cream is substituted for the cake, dessert will not be satisfying. True or false?" --KSmrqT 01:52, 6 April 2007 (UTC)


 * If it is true that withered flowers come only from not watering them, then not not watering them cannot result in (as opposed to give "a much less chance of") withering them. If on the other hand withering can come by several means (e.g. age), (not not) watering them gives no guarantee of unwitheredness.


 * Reverting to the floor, it seems reasonable anyway that a systematic way is not necessary for 100% sweeping - a sufficiently long random approach would eventually cover every part.81.153.220.170 12:15, 6 April 2007 (UTC)
 * Wouldn't the area not sweeped just become infinitely small? --YbborTalk Survey! 13:30, 6 April 2007 (UTC)
 * No, the expected area sweeped would never reach zero, but the actual area sweeped will. The probability of the floor being completely sweeped will approach, but never reach, 100%. As to answer the question, assuming not sweeping systematically makes sweeping the whole floor highly probable, but not certain, the probability will be slightly less, but the odds will be infinitely less. For example, if there is a 99% (0.99) probability of sweeping the whole floor randomly, then the probability is 1% (0.01) less, but the odds of doing it randomly are 99:1 (99) and the odds of doing it systematically are 1:0 (infinite) making the odds infinitely less. — Daniel 17:32, 6 April 2007 (UTC)


 * These are sweeping statements. For a mathematically and logically rigorous treatment of the original question, we need a definition of "systematic". The method of sweeping only along the edges of the area is a system and so could be considered systematic under at least one reasonable definition of the term. It is not clear why the age or developmental phase of the sweeper is introduced; can we abstract from that? --Lambiam Talk  20:06, 6 April 2007 (UTC)


 * One reasonable interpretation of the question is, "I have to sweep a floor. I'll start at a corner. I'll clear the floor in parallel lines, starting from one wall and moving to the opposite wall. Doing this, I'm guaranteed to clean the whole floor. However, if I swept in a less systematic way, I'd risk missing a spot. Am I right?" It's hard to answer that fully without knowing why you're asking the question, but I'd say, yes and no. It's true that that would clean every part of a normal floor, but there are other systematic ways of doing it. For instance, you could spiral inward and sweep the dust into a pile, then sweep that pile away. Or you could start at the center and sweep outwards in radiating lines, then sweep around the edge. Or (to steal a maids' trick) you could sprinkle coffee grounds over the whole floor, then sweep it any way you please until all the grounds are gone. You're right, however, that if you go at it blind, with no plan and no way of telling when you're done, you'll probably miss a spot. Black Carrot 08:14, 9 April 2007 (UTC)

Numbers not definable
Is there any name for numbers not definable in a finite amount of data? e would not be one because in could be defined as $$e = \sum_{n = 0}^\infty \frac{1}{n!}$$. They can be proven to exist because each definition can be given a unique Gödel number which will be a natural number and therefore there will be at most $$\aleph_0$$ numbers that can be defined and there are Cardinality of the continuum real numbers, which has been proven to be more. An interesting paradox arises because, although they exist, it is impossible to find an example. — Daniel 21:18, 6 April 2007 (UTC)


 * You might call them "undefinable numbers". Precisely because a counterexample cannot be constructed, constructivists will deny that such numbers exist – and escape a contradiction because they don't believe either that a bijection can be constructed between the natural numbers and those Gödel numbers that effectively define a real number. --Lambiam Talk  22:06, 6 April 2007 (UTC)

Excuse my ignorance, but in the original question isn't Euler's number being defined by a "non-finite" amount of data, in that n has to go to infinity? Icthyos 22:11, 6 April 2007 (UTC)
 * Depends on what you mean (BTW, no one calls it "Euler's number"; took me a minute to figure out what you were talking about -- it's just e; that's the standard name). You have kind of hit on the key issue, though. For a given fixed, precise notion of definability, we can define a specific real that isn't definable according to that fixed scheme. "Definable" itself, it seems, is not definable. --Trovatore 22:36, 6 April 2007 (UTC)
 * I'll just edit the phrase "Euler's number" out of the second sentence of its article, shall I? :P~ So what you're saying is, for any interval of real numbers there will always be an irrational number in there? I know irrational numbers aren't undefined, but I think I've come across a proof of that (my case) before. What precisely do you mean by "notion of definability"? Icthyos 22:44, 6 April 2007 (UTC)
 * I think the "specific real" refers to Cantor's diagonal argument; look for s0 there. --Lambiam Talk  23:02, 6 April 2007 (UTC)


 * See also Definable real number. PrimeHunter 23:43, 6 April 2007 (UTC)
 * But take it with a grain of salt. It's not as bad an article as it once was, but it's still a bit shaky. --Trovatore 23:47, 6 April 2007 (UTC)