tag:blogger.com,1999:blog-11295132.post112466233538266118..comments2016-10-18T06:55:32.586-07:00Comments on A Neighborhood of Infinity: The Alternative NumbersDan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-11295132.post-60436983201044113902011-02-04T21:38:46.985-08:002011-02-04T21:38:46.985-08:00This confusion is why I've settled on the nota...This confusion is why I've settled on the notation \(\mathbb Z_{\ge 0}\) for \(\{0, 1, 2, \dotsc\}\) and \(\mathbb Z_{> 0}\) for \(\{1, 2, \dotsc\}\). It's so ugly that everyone can be offended!<br /><br />By the way, the issue with the 10-adics is slightly more extreme than you suggest. First, I note that, when you refer to the \(p\)-adics but write \(\mathbb Z_p\), you are denoting not the full <em>field</em> \(\mathbb Q_p\) of <i>p</i>-adic numbers but only its sub<em>ring</em> of <i>p</i>-adic integers (just as in the real case, those with only 0's to the right of the decimal point)—in which, just as in the ordinary integers, it is not always possible to divide. The remedy here is that one can set up formal fractions of <i>p</i>-adic integers, and manipulate them just as one does ordinary fractions of integers; and in this way one gets the field of <i>p</i>-adic numbers. (This may seem a backwards way to do it if you think of \(\mathbb Z_p\) in terms of \(p\)-adic expansions, where it seems like we start with \(\mathbb Q_p\) and cut it down; but it becomes much more natural when we think of \(\mathbb Z_p\) as what it ‘really’ is, namely, a projective limit.)<br /><br />Anyway, the point is that the 10-adic integers are even worse; not only can you not <em>actually</em> divide, you can't even <em>formally</em> divide—because there are 10-adic 0-divisors.Lorenhttp://www.blogger.com/profile/10866289941226429119noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1125270031703982362005-08-28T16:00:00.000-07:002005-08-28T16:00:00.000-07:00I always forget that some people are prejudiced ag...I always forget that some people are prejudiced against considering 0 to be a natural number. But that's probably just because I mainly hang out with set theorists and logicians, who are familiar with N only through the Peano axioms and the von Neumann ordinals, both of which start with 0. I do dimly remember hearing of this distinction between the "natural numbers" and "whole numbers", but we could just as easily reverse those two designations.Kennyhttp://www.blogger.com/profile/12226268498253877151noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1124841696190494262005-08-23T17:01:00.000-07:002005-08-23T17:01:00.000-07:00That explains a bit, although I think that, in the...That explains a bit, although I think that, in the US at least, the consensus is against Bourbaki.Vitohttp://www.blogger.com/profile/01844498756899478213noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1124742250062760802005-08-22T13:24:00.000-07:002005-08-22T13:24:00.000-07:00I'm following BourbakiI'm following <A HREF="http://mathworld.wolfram.com/NaturalNumber.html" REL="nofollow">Bourbaki</A>sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1124725105657930732005-08-22T08:38:00.000-07:002005-08-22T08:38:00.000-07:00Umm, N is {1,2,3,...} The set {0,1,2,3,...} is the...Umm, N is {1,2,3,...} The set {0,1,2,3,...} is the <EM>whole</EM> numbers.Vitohttp://www.blogger.com/profile/01844498756899478213noreply@blogger.com