tag:blogger.com,1999:blog-3722233.post8824991713370923749..comments2023-09-30T21:44:03.907-05:00Comments on Computational Complexity: (1/2)! = sqrt(pi) /2 and other conventionsLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-3722233.post-9387023297031486222015-03-05T21:49:19.530-06:002015-03-05T21:49:19.530-06:00Hi, Dave---I gave further argument for your instan...Hi, Dave---I gave further argument for your instances at https://rjlipton.wordpress.com/2015/02/23/the-right-stuff-of-emptiness/Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-8803672964984040202015-03-05T18:59:37.456-06:002015-03-05T18:59:37.456-06:00Actually, neg * pos = neg and neg*neg=pos are not ...Actually, neg * pos = neg and neg*neg=pos are not conventions and can be derived easily from the basic laws of arithmetic, eg.; (a-a)*b = 0 and then rearranging and simplifying as needed. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-66015092871121803822015-03-05T15:33:47.346-06:002015-03-05T15:33:47.346-06:00neg*neg = pos is not a convention, it is true sinc...neg*neg = pos is not a convention, it is true since it follows the laws of arithmetic, essentially following cancellation law. Consider (-a)*(b-b) = 0 => (-a)* b+ (-a)*(-b) = 0, then adding a*b to both sides yields (-a)*(-b)= ab. Did I miss something there?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-14995670821737849192015-03-05T12:52:23.540-06:002015-03-05T12:52:23.540-06:00Excellent, this is JUST the kind of thing I could ...Excellent, this is JUST the kind of thing I could tell STUDENT.<br />For those who don't know, the volume of an n--ball of radius R is<br /><br />( pi^{n/2}/Gamma(n/2 + 1))R<br /><br />for n even you would use factorial on something- I won't say to avoid being off-by-one again, but for n odd you need to use the actual Gamma Function!<br />actual Gamma function.GASARCHhttps://www.blogger.com/profile/06134382469361359081noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-13765737038123274482015-03-05T10:08:16.501-06:002015-03-05T10:08:16.501-06:00I definitely think that 0^0 is 1, not undefined.
...I definitely think that 0^0 is 1, not undefined.<br /><br />What is the sum of the weights of all unicorns? Most people would agree that it is 0.<br /><br />What is the product of the weights of all unicorns? It should be 1, to make the identity<br /><br />(product of unicorns)(product of people) = (product of people and unicorns)<br /><br />correct. Why would this argument change if the unicorns were individually weightless?<br /><br />For that matter, the concatenation of the names of all the unicorns is the empty string.DaveMBhttps://www.blogger.com/profile/00779581893863396042noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-4030497454489784142015-03-05T10:05:31.787-06:002015-03-05T10:05:31.787-06:00While we are onto conventions making things work o...While we are onto conventions making things work out...<br /><br />What about 1+2+3+4 + ....= -1/12?<br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-75772536779300592322015-03-05T09:59:41.367-06:002015-03-05T09:59:41.367-06:00Thanks for the correction. on the off-by-one thing...Thanks for the correction. on the off-by-one thing.<br />As for viloating assoc las- YES, and that is indeed why we have NEG*NEG=POS and<br />I don't disagree with it as a convention.<br /><br />But realize that when the Quaternions wehre invnted they were NOT COMMUATIVE!<br />but they were USEFUL! <br /><br />I wonder if the same might happen for some system with NEG*NEG=POS.GASARCHhttps://www.blogger.com/profile/06134382469361359081noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-9470952898918929192015-03-05T09:56:34.540-06:002015-03-05T09:56:34.540-06:00The volume of n-balls gives a pretty good geometri...The volume of n-balls gives a pretty good geometric reason why (m+1/2)! is defined the way it is. Wim van Damhttps://www.blogger.com/profile/14484831637730978511noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-80096381901707701202015-03-05T09:55:02.728-06:002015-03-05T09:55:02.728-06:00gamma(3/2) = "(1/2)!" = sqrt(pi)/2, not ...gamma(3/2) = "(1/2)!" = sqrt(pi)/2, not sqrt(pi)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-78566190924112508602015-03-05T09:51:49.219-06:002015-03-05T09:51:49.219-06:00I understand your student: (1/2)! is NOT sqrt(pi)....I understand your student: (1/2)! is NOT sqrt(pi). It is sqrt(pi)/2... OK, it is maybe not much more intuitive, but still!<br /><br />The convention 0^0 = 1 is MUCH more used than 0^0 = undefined I think.<br /><br />For NEG * NEG, I think the point is that we like to have associative laws: If you take the convention NEG * POS = NEG and associativity, you have (1 + (-3)) * (2 + (-6)) = (-2)*(-4) = ± 8, and it is also equal to 1*2 + (-3)*2 + 1*(-6) + (-3)*(-6) = ± 18 - 10. Then only possibility here is to get NEG * NEG = POS as convention. <br /><br />More generally, of course you could define an operation which is as the multiplication but with NEG * NEG = NEG. The point is that is does not have good properties. So to my mind, it is NOT ONLY a question of applications, and it wouldn't be a NICE alternative system.<br /><br />P.S.: I hope I respect GASARCH's typographic rules! ;-)B.noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-10017671585865326962015-03-05T09:47:25.531-06:002015-03-05T09:47:25.531-06:00Actually, it's Gamma(1/2) = sqrt(pi), so (-1/2...Actually, it's Gamma(1/2) = sqrt(pi), so (-1/2)! = sqrt(pi). Remember that (x)!=Gamma(x+1)=x Gamma(x). Off by one error... ;)<br /><br />So we have (1/2)! = Gamma(3/2) = sqrt(pi) / 2.NP Slaglehttps://www.blogger.com/profile/06322388966706601689noreply@blogger.com