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Re: Is there a name for this notation?
Posted:
Oct 9, 2011 8:19 AM


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 => A 2^11 => B 2^12 => 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 nonnegative powers of two. > No integer other than two has that property.
I believe you are missing the word "unique" in that sentence. If you allow repeated 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



