First, let me say I like these proposals. One comment below.

On Feb 28, 2007, at 11:13 AM, William Stein wrote:

>> 2) Automatically switch into scientific notation if str() would
>> otherwise have to add many many zeroes (my proposal would be to  
>> switch
>> if the number of added zeroes exceeds the number of original digits).
>> This also changes the result of hash(), but makes it constant time.
>> (Try hash(0.5^1000000000) -- but be sure you do it on a machine with
>> lots of memory.)
>
> Good idea.

Doing hash = hash(str(x)) can be expensive (we used to do this on  
matrices...) and I would propose changing the hash to something like  
hash(mantissa) xor hash(valuation) (+ extras? I'd have to look into  
the exact equality stuff for comparing real numbers from several  
fields to make a serious proposal). Also, even in scientific notation  
it would display the fixed number of digits, right?

- Robert

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