Comment by h0mEDw

Yeah I meant that there's another layer to the symbolic representation of "sqrt(2)" than just the tree of operation, so that you can ask sensible questions about the value of the number. For instance a generator of decimal representation or a function like "given N return a fraction q/p that's within 1/2^N of the value".

And floats are really a whole other beast than fractions or naturals or real numbers. As I wrote earlier - the "+" operator isn't associative. Python snippet:

    def before():
        x = 0.0
        x += 1.0
        for i in xrange(1000):
            x += 1e-17
        return x

    def after():
        x = 0.0
        for i in xrange(1000):
            x += 1e-17
        x += 1.0
        return x


    print before() == 1.0
    print after() == 1.0

Will print:

   True
   False