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 => 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 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 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