Re: #48 Subtraction in +P-adics versus -P-adics; new textbook; "Mathematical-Physics (p-adic primer) for students of age 6 onwards"

Am 04.10.2007 08:38 schrieb Proginoskes:
On Oct 3, 11:23 pm, a_plutonium <a_pluton...@xxxxxxxxxxx> wrote:
A few articles back, I said that the roots of -P-adics was like
finding the roots of positive
P-adics and then attaching a negative sign to the final answer. So
that in the case of
the square root of (-)....000001 in Negative P-adics is the square
root of ....00001
and then attaching a negative sign.

Looks like AP is saying that if A*B = C, then (-A)*(-B) = -C. This is
almost as bad of his mistake as 0*0 = -1, since, in order for it to be
true, we lose the distributive property.



For example

.......000000002 - .......9999999 = (-) ......9999997

Another example

5....00000000 - 49.....999999 = .....000001

This is not a well-defined operation, since you could also have

50....0000000 500....00000
- 4....9999999 OR - 4....99999
_______________ ____________
46....0000001 496....00001

As by all operations in P-adics are exactly similar to Reals and where
the final answer
is what digits remain the same after successive place value.
[...]
So we have
50 - 49 = 01
then
500 - 499 = 001
then
5000 - 4999 = 0001

This is true only in base 10, so this operation really is base-
dependent. If we work in base 11, we'd find that

50 - 49 = 02
500 - 499 = 012
5000 - 4999 = 0112

and so on, which would give us an answer of ....1112, if we follow
AP's lead. My other objection also applies in base 11.

The issue of what ....9999 + ....0001 is will also show that AP is
really thinking of base 10.

--- Christopher Heckman

Some years ago I played around with these ideas and came to the
conclusion, that finally there is no use in it for me.
One may do a bit of arithmetic within this concept, and may base
all considerations primarily on the sum of a periodic part of in-
finite length) and an aperidoc part, let's first think of finite length.

Considering the periodic type only one can easily see, that an
arithmetic of this is a map of the same arithmetic of rational
numbers between -1..0, where ...9999 =: -1 , and for instance
.....3333 := -1/3 and so on.
One may apply addition, subtraction, multiplication and division
based on this mapping and the mapping will always be consistent.

Strings, which are sums of periodic strings and finite aperiodic
substrings can still be handled in the same way as the rational
numbers between 0..-1 summed with integers (the aperiodic summand
is just the mapping of a natural number).

Products of periodic strings get aperiodic, since they cannot
be mapped to a rational number from(0..-1)
Let x element( 0..-1) y element(0..-1) then
x*y not element(0..-1)
and must be expressed by an aperiodic expression.

That was nearly all what I was distilling out from this and was
sort of interesting. One could proceed and discuss roots, exponentials
and complex functions - but, well... it may be useful as a
compact notation for modular valuation, why not.

Gottfried Helms


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Gottfried Helms, Kassel
.