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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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Steven D'Aprano

Posts: 11
Registered: 3/22/11
Re: Is there a name for this notation?
Posted: Oct 9, 2011 8:19 AM
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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





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