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Re: Non-Euclidean Arithmetic
Posted:
Sep 12, 2012 10:50 AM
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On Wed, Sep 12, 2012 at 9:37 AM, Joe Niederberger
<niederberger@comcast.net> wrote:
> Paul Tanner III says:
>>Yes, but you would not accept e(pi) as the answer to multiplying e and pi
>
> Paul - I'm letting you provide your definition of "answer". But juxtaposing the names of two real numbers only works for about 0% of all those numbers. What about the other 100%?
>
I gave the scaling model here that holds for every two real numbers:
"Re: Non-Euclidean Arithmetic"
http://mathforum.org/kb/message.jspa?messageID=7888565
(The model does in one sense what the repeated addition model does in
that without loss of generality, the two factors a and b are clearly
positive and thus in the cases when at least one factor is 0 or
negative, appropriate modification is taken to obtain two positives.)
Here is that model again:
Here is a real number based geometric construction using similar
triangles that gives point (product) ab on the real number line given
point 1 and arbitrary points (arbitrary real numbers) a and b on the
real number line:
Take a standard x-axis and y-axis in the Cartesian plane, and for sake
of simplicity, name the points on these lines according to the fact
they are each a real number line. Then:
Connect 1 on the x-axis to b on the y-axis, and, parallel to that
drawn line segment, connect a on the x-axis to the y-axis, and this
point on the y-axis is ab. That is, in terms of distance from 0 or
magnitude or absolute value: ab is to b as a is to 1 - that is,
written in terms of ratios or proportions: ab:b as a:1.
And via commutativity we have the other way as an alternative:
Connect point 1 on the x-axis to point a on the y-axis, and, parallel
to that drawn line segment, connect b on the x-axis to the y-axis, and
this point on the y-axis is ab. That is, in terms of distance from 0
or magnitude or absolute value: ab is to a as b is to 1 - that is,
written in terms of ratios or proportions: ab:a as b:1.
>
> But lets' move on - what is Devlin talking about when he speaks of this "process"?
>
> Is he confusing an algebraic property (that you call scaling) of multiplication with a process? Could that lead to some confusion? Will you help clear it up? Is scaling a property (as you yourself have said) or is it a process?
>
He said that repeated addition or scaling or any other model is an
algebraic properties. Multiplication is just one of many names we can
give to a binary function on a set, S x S -> S.
In my post
""Re: Non-Euclidean Arithmetic""
http://mathforum.org/kb/message.jspa?messageID=7883395
I showed how we can derive repeated addition using just some of the
algebraic properties in any field (with characteristic 0 and that
contains the natural numbers). Here again is that model, using
different variables:
For all field elements a,b and for every natural number n (excluding
0, for those who define them to include 0) there exist field elements
x,y such that ax = b = ny, so that from b/n = y we have ax = b =
n(b/n), then from n being equal to n instances of 1 added together we
have ax = b = (1_1 + ... + 1_n)(b/n), and then ax = b = [(b/n)_1 + ...
+ (b/n)_n].
I also said that this does not show that repeated addition is what
multiplication *is*, and that it actually shows the opposite, which is
that it is simply a property or model of multiplication.
Note that this model or property works for any such field as defined
above including those on which an order is not defined, and that
scaling works on any set (that can be more general than a field, such
as an ordered integral domain or OID [I did not say "well-ordered"
since that OID is isomorphic to the integers and we do not want to
limit the OID here]) on which there is an order giving magnitude or
absolute value.
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