Joe Pfeiffer <pfeif...@cs.nmsu.edu> writes:
> Hope Rouselle <hrouse...@jevedi.com> writes:
>> Christian Gollwitzer <aurio...@gmx.de> writes:
>>>
>>> I believe it is not commutativity, but associativity, that is
>>> violated.
>>
>> Shall we take this seriously? (I will disagree, but that doesn't mean I
>> am not grateful for your post. Quite the contary.) It in general
>> violates associativity too, but the example above couldn't be referring
>> to associativity because the second sum above could not be obtained from
>> associativity alone. Commutativity is required, applied to five pairs
>> of numbers. How can I go from
>>
>> 7.23 + 8.41 + 6.15 + 2.31 + 7.73 + 7.77
>>
>> to
>>
>> 8.41 + 6.15 + 2.31 + 7.73 + 7.77 + 7.23?
>>
>> Perhaps only through various application of commutativity, namely the
>> ones below. (I omit the parentheses for less typing. I suppose that
>> does not create much trouble. There is no use of associativity below,
>> except for the intented omission of parentheses.)
>>
>> 7.23 + 8.41 + 6.15 + 2.31 + 7.73 + 7.77
>> = 8.41 + 7.23 + 6.15 + 2.31 + 7.73 + 7.77
>> = 8.41 + 6.15 + 7.23 + 2.31 + 7.73 + 7.77
>> = 8.41 + 6.15 + 2.31 + 7.23 + 7.73 + 7.77
>> = 8.41 + 6.15 + 2.31 + 7.73 + 7.23 + 7.77
>> = 8.41 + 6.15 + 2.31 + 7.73 + 7.77 + 7.23.
>
> But these transformations depend on both commutativity and
> associativity, precisely due to those omitted parentheses. When you
> transform
>
> 7.23 + 8.41 + 6.15 + 2.31 + 7.73 + 7.77
>
> into
>
> 8.41 + 6.15 + 2.31 + 7.73 + 7.77 + 7.23.
>
> it isn't just assuming commutativity, it's also assuming associativity
> since it is changing from
>
> (7.23 + 8.41 + 6.15 + 2.31 + 7.73) + 7.77
>
> to
>
> (8.41 + 6.15 + 2.31 + 7.73 + 7.77) + 7.23.
>
> If I use parentheses to modify the order of operations of the first line
> to match that of the last, I get
> 7.23 + (8.41 + 6.15 + 2.31 + 7.73 + 7.77)
>
> Now, I get 39.60000000000001 evaluating either of them.
I need to go slow. If I have just two numbers, then I don't need to
talk about associativity: I can send 7.23 to the rightmost place with a
single application of commutativity. In symbols,
7.23 + 8.41 = 8.41 + 7.23.
But if I have three numbers and I want to send the leftmost to the
rightmost place, I need to apply associativity
7.23 + 8.41 + 6.15
= (7.23 + 8.41) + 6.15 -- clarifying that I go left to right
= 7.23 + (8.41 + 6.15) -- associativity
= (8.41 + 6.15) + 7.23 -- commutativity
I see it. Cool. Thanks.
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