I don't think so. The original point (2500 years ago ?) was and still is to
have a compact notation for the natural numbers that avoids the problem of
naming all or at a least large number of them individually. In English we name
the first few numbers individually: zero, one, two, three ... ten, eleven,
twelve, at which point we begin to notice that this is becoming cumbersome. We
wobble on to the equivalent , three-teen, four-teen, five-teen, ... nine-teen,
and then get more and more rational and closer and closer to a positional
system as the numbers get ever larger. We can at least always figure the value
of our names once we have mastered the art of combining a small number of basic
words:
one two three ...nineteen
twenty thirty .. .ninety (we are getting there: twenty = two-tens, thirty =
three-tens, etc)
hundred, thousand, million, billion, trillion, quadrillion ...
Quick now: express 75853729915229585876325067 in "words"
The point of positional notation is to use a very small set of atomic symbols,
the "numerals", say N of them, which form an ordered sequence that "names" the
first N-1 natural numbers 0, 1, 2, ... , N-1 (N some definite natural number,
"two", "ten", sixteen", "sixty" etc., the "base"). The natural numbers are then
symbolized by strings of these numerals. The value assigned to any particular
numeral in a particular position within a string is that numeral times the base
raised to the power of that numeral's position in the string (positions in the
string are indexed right to left by the natural numbers starting at 0 at the
rightmost position). The value of the string as whole is the sum of the all
values assigned to each particular numeral at its particular position.
1. The simplest notation: binary - base 2, numerals {0,1}.
2. Common early computer world: octal - base 8, numerals {0 ... 7}.
3. Most common modern: decimal - base 10, numerals {0 ... 9}
4. Modern computer world: hexadecimal (senidenary) - base 16, numerals {0 ...
9 a b c d e f}
5. Most incredible: sexagesimal (Babylonian): - base 60, numerals {
ingenious! value of each numeral can be derived from its symbol}
All Babylonian children had to memorize the multiplication and addition tables
or Be Left Behind
Much follows from this incredible idea. We can create rational and even real
numbers out of the notation by adding to our set of numerals the point symbol (
usually ".") and assigning negative powers of the base to the positions right
of the point. All of our algorithms for adding, subtracting, multiplying and
dividing numbers are consequences of the positional notation.
Joshua Thorp is exactly correct. The formula he presents is the very
elementary formula for the sum of a finite geometric series.
Dean Gerber
________________________________
From: Robert J. Cordingley <rob...@cirrillian.com>
To: The Friday Morning Applied Complexity Coffee Group <friam@redfish.com>
Sent: Monday, October 8, 2012 3:15 PM
Subject: Re: [FRIAM] Nines: Trivia Question?
...and I guess (base) n can be rational, irrational or even imaginary.
Thanks
Robert
On 10/8/12 12:02 PM, Joshua Thorp wrote:
> I think you just replace '9' with 'n-1' in Dean or Frank's answer and you
> have a general proof, for n>=2.
>
> I suppose you may need to convince yourself that a number like n^k - 1 ==
> (n-1)*n^(k-1) + (n-1)*n^(k-2) + … + (n-1)*(k-k).
>
> --joshua
>
> On Oct 8, 2012, at 11:37 AM, Robert J. Cordingley wrote:
>
>> May be I should reframe the question.
>>
>> How do you prove there isn't a system of numbers to base N where it doesn't
>> work?
>>
>> Thanks,
>> Robert
>>
>> On 10/8/12 11:00 AM, Tom Carter wrote:
>>> Robert -
>>>
>>> There's a reasonably good discussion of this here:
>>>
>>> http://mathforum.org/library/drmath/view/58518.html
>>>
>>> Thanks . . .
>>>
>>> tom
>>>
>>> On Oct 8, 2012, at 9:20 AM, Robert J. Cordingley <rob...@cirrillian.com>
>>> wrote:
>>>
>>>> I probably should know this...
>>>>
>>>> So when you rearrange the digits of a number (>9) and take the difference,
>>>> it is divisible by nine. A result that sometimes points to accounting
>>>> errors. If the numbers are not base 10 the result is divisible by
>>>> (base-1).
>>>>
>>>> What is the associated theorem for this?
>>>>
>>>> Thanks
>>>> Robert
>>>>
>>>>
>>>>
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>>>
>>
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