on floating-point numbers

[Hope Rouselle]

Just sharing a case of floating-point numbers.  Nothing needed to be
solved or to be figured out.  Just bringing up conversation.

(*) An introduction to me

I don't understand floating-point numbers from the inside out, but I do
know how to work with base 2 and scientific notation.  So the idea of
expressing a number as 

  mantissa * base^{power}

is not foreign to me. (If that helps you to perhaps instruct me on
what's going on here.)

(*) A presentation of the behavior

>>> import sys
>>> sys.version
'3.8.10 (tags/v3.8.10:3d8993a, May  3 2021, 11:48:03) [MSC v.1928 64 bit (AMD64)]'

>>> ls = [7.23, 8.41, 6.15, 2.31, 7.73, 7.77]
>>> sum(ls)
39.599999999999994

>>> ls = [8.41, 6.15, 2.31, 7.73, 7.77, 7.23]
>>> sum(ls)
39.60000000000001

All I did was to take the first number, 7.23, and move it to the last
position in the list.  (So we have a violation of the commutativity of
addition.)

Let me try to reduce the example.  It's not so easy.  Although I could
display the violation of commutativity by moving just a single number in
the list, I also see that 7.23 commutes with every other number in the
list.

(*) My request

I would like to just get some clarity.  I guess I need to translate all
these numbers into base 2 and perform the addition myself to see the
situation coming up?