```Date: Oct 9, 2011 8:19 AM
Author: Steven D'Aprano
Subject: Re: Is there a name for this notation?

William Elliot wrote:> On Sun, 9 Oct 2011, Steven D'Aprano wrote:> >> Long ago, I came across a book that mentioned a particular notation for>> writing numbers in terms of sums of powers of some base, but *not* in the>> conventional form.>>>> For example, using 2 as the base and comparing to decimal:>>>> 1 = 2^0         => "0">> 2 = 2^1         => "1">> 3 = 2^1 + 2^0   => "10">> 4 = 2^2         => "2">> 5 = 2^2 + 2^0   => "20">> 6 = 2^2 + 2^1   => "21"> > > 7 = 2^2 + 2^1 + 2^0 = 210> 8 = 2^3 = 3> 9 = 2^3 + 2^0 = 30> ...> 33 = 50> > How do you write 2^100 + 2^10 + 2^50?You'd need either a digit for 100, or some notation for grouping digits.E.g.:2^10 => A2^11 => B2^12 => C...but since we can't realistically have an infinite number of unique symbols,a grouping notation might be better:2^100 + 2^50 + 2^10 + 2^2 = (100)(50)(10)2> Every positive integer is a sum of non-negative powers of two.> No integer other than two has that property.I believe you are missing the word "unique" in that sentence. If you allowrepeated powers, one can do this:17 = 3^2 + 3^1 + 3^1 + 3^0 + 3^0 = "21100" to "base 3". > We could write 1/2 = -1> 1/4 = -2;  1/8 = -3;  1/16 = -4;  0 = -oo> > 3/4 = -1 -2> 1/3 = -2 -4 ...;  an infinite series.> > Let's try adding.> 21 + 31 = 3211 = 322 = 33 = 4.  Check.  6 + 10 = 16.> > 210 + 210 = 221100 = 321> 2 * abcd = a+1 b+1 c+1 d+1> > Looks like fun.  If you can't remember the details, let's reinvent them.> >> and so forth. Obviously there is no way of writing zero, and the order of>> the digits is arbitrary: I could have written either "12" or "21" for>> decimal 6.>>>> Unfortunately I have forgotten all details about this except the basic>> notation, including the name of the book.>>>> Is there a name for this notation, is it useful for anything, and where>> might I find out more about it?-- Steven
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