On Wed, Sep 12, 2012 at 9:37 AM, Joe Niederberger <email@example.com> 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:
(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.
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.