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Topic: Is there a name for this notation?
Replies: 8   Last Post: Oct 9, 2011 9:17 PM

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William Elliot

Posts: 248
Registered: 10/7/08
Re: Is there a name for this notation?
Posted: Oct 9, 2011 1:30 AM
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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?

Every positive integer is a sum of non-negative powers of two.
No integer other than two has that property.

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?





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