09/21/2006

 
Upon contemplating the task of constructing a 64 row truth table, 
a student writes:


        Dr. Patt,

        Would you like me to devote an extra sheet of paper to contain all 64
        rows of the truth table for number 11, or make a simplified truth table
        clearly showing that I understand the concept (with 8 rows, omitting the
        redundant values of the 3 non-selected bit values that have no effect on
        the output, and also having a note that these have been omitted)? Do you
        prefer detail, completion, and tedium,  or  clarity, simplicity, and
        understandability?

We should probably send this question to the English department so they can 
include it in their examples of "rhetorical questions."

Of course, I do not like tedium.  In fact, I work very hard at not 
requiring you to do tedium.  That is why one of my TAs took the time
to tediously prepare the 64 row truth table.  ...so, you could print it,
fill it in (less than one minute if you "clearly" "understand the concept")
and move on.  ...and not suffer the tedium of constructing the 64 row table
yourself.

I could have added a part b, where I then asked you to write down the
8 relevant rows and an explanation of why only 8 of the 64 rows are 
relevant.  But I thought of saving that for the midterm.

        Also, on an unrelated note, is there a purpose for two 
        representations of the number 0 (+0 , -0) in the float 
        notation aside from convention?

When Professor Kahan (author of the IEEE Floating Point standard) proposes 
something, there is almost always a purpose.

        Are these equivalent values or not?

For all the applications you will encounter for awhile, yes.  And, for
the subtleties beyond that, I think it best to defer to some later 
graduate course in numerical computation.

        Can we imply that extremely small values
               ~~~~~
"infer," actually.		

        of floats after doing calculations are equivalent to zero?

The short answer: yes.  But that is another whole topic in itself.  
When is the result of a calculation sufficiently small that we represent 
it as zero?  We should defer this as well.  ...or google "subnormal numbers" 
and start reading.  And, if you get stuck, come and see me.  Again, this is
really well beyond the scope of 306, so I think I will save the class from
dealing with it in this email message.

        Or, is it recommended to avoid the value of zero in the float 
        notation completely?

No!  Not at all.  Zero is very, very important.  In fact, most believe
the definition of "zero" was one of the most important advances in the
history of mathematics.

        Thanks, (and more thanks for the great lectures)

        <<name withheld to protect the ...>>

Thank you.  I hope you continue to enjoy them.  Good luck with the course.

Yale Patt