In article <see-047512.12463626042008@[EMAIL PROTECTED]
>,
Barb Knox wrote:
> In article <slrng14ggs.813.dbast0s@[EMAIL PROTECTED]
>,
> "Daniel C. Bastos" <dbast0s@[EMAIL PROTECTED]
> wrote:
>
>> The rationals form a ring with addition defined as
>>
>> p/q + r/s = (ps + qr)/qs.
>>
>> I'm used to the fact that this works for the addition of fractions as
we
>> want to; for example, a basket with half of 10 oranges if dumped on a
>> basket with a third of 12 gives 9 oranges. So I know that
>>
>> 10/2 + 12/3 = (30+24)/6 = 54/6 = 9.
>>
>> So if we think of adding oranges in baskets, we want the addition of
the
>> rationals not to give us any surprise. It seems to me then that what I
>> want is to prove that the abstract system of the rationals (with
respect
>> to addition) reflects what we want them to reflect.
>>
>> So if I ask for a proof that all fractions of oranges in one basket, if
>> added to another basket with another fraction of oranges, will give me
>> what my adding in the real world would give, how would I go about
>> proving this?
>>
>> I have a feeling that this is not possible because there is no
guarantee
>> that the real world will ever work as some abstract system does.
>>
>> On the other hand, I have mathematical induction in mind which would be
>> perhaps the strongest argument for proving that the definition of
>> addition will work as we expect.
>
> You don't need to use mathematical induction. Consider your 10/2 + 12/3
> example: Let your unit of accounting be 1/(2*3) of an orange. In terms
> of these units, you're adding 10*(2*3)/2 + 12*(2*3)/3. The denominators
> cancel in both, resulting in an integer addition (which I assume you
> find non-problematic in its correspondence with the "real world").
> Multiply the integer result by the unit's value of 1/(2*3), giving your
> result in terms of oranges.
That's a nice approach.
> Note this generalises to the p/q + r/s = (ps + qr)/qs formula.
I see. We multiply each term by qs so we get ps + qr and then we convert
it back to the previous unit by multiplying the integer amount by
1/qs. So the approach is really a conversion of units. It's a nice way
to think about it.
Also I realized that asking for a proof of that isn't very meaningful;
the most I can ask for is just a clear way to link the operation to
intuition, which the above seems to do well enough.
Thanks for your thoughts.
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