Math Is Fun Forum

I don't see what your asking.  Something seems jumbled.
All I see is that your example keeps increasing by an amount that goes up by one each time.

I don't know how to do proofs, but aren't these just logical truths based on the
definition of Least Upper Bound I just found on the internet as:

The least upper bound, called the supremum, of a set S, is defined as a quantity M such that no member of the set exceeds M, but if ε is any positive quantity, however small, there is a member that exceeds M - ε.

The least upper bound of a function, f, is defined as a quantity M such that f(x) ≤ M for all x in its domain, but if ε is any positive quantity, however small, there is an x in the domain such that f(x) exceeds M - ε.

The -2 stays on the top (numerator).
The -2^-3 goes to the top as -2^3.
So now you have -2 times -2^3.
Then -2^3 is -8 because -2 times -2 times -2 is -8.
So that times -2 is 16.  So add 16 in the middle term of your
entire problem.

line2: y=2x

line1: y = -x/2  + a

line3: y=2x - a

intersection lines 1 and 2 is (0.4a,0.8a)

intersection lines 1 and 3 is (0.8a,0.6a)

small square side length is square root of ((.4² + .2²)a²), which is the square root of 0.2a².

If you square the side length for the area, you get 0.2a²,
which is five times smaller than the big square's area: a².

I think you missed my previous post that noted that 4.45 can round up to 5 when 5's round upward and the rounding
occurs more than once in a sequence on a particular data point.
This makes the exact opposite argument you made prior about rounding down being biased. 
So rounding up is biased because 56% of data rounds upward and 44% of data rounds down.

But rounding down is biased because 55% of data rounds downward and 45% rounds upward.
Did you catch the joke here with the numbers I used for percentages.
For 56-44, I rounded up, but for 55-45, I rounded down!

The tables turn again!
Your dumb method this time (5's round up):
4.445
4.45
4.5
5.
My good method this time (5's round down):
4.445
4.44
4.4
4.
So now we have seen that both methods, going up or down are flawed!
So now we can argue there is no good rounding method known to man yet.

I believe 9.50000000000000000001 should be rounded to 10.
I believe 9.49999999999999999999 should be rounded to 9.
I hope we can agree on this at least.

I have a hunch that your example of what statiticians do is incorrect.
If the input # is even in the place value we are keeping, then we go one way, and
if odd, then the other way, but that's just my guess.

Also 0 + 1 + 2 + 3 + 4 is not equal to (10-5) + (10-6) + (10-7) + (10-8) + (10-9).  (Your method)
However 1 + 2 + 3 + 4 is equal to (10-6) + (10-7) + (10-8) + (10-9).  (My method)

Thanks for explaining in more detail, however, I must admit that my knowledge
of matrices is far too limited at this point to even get a basic understanding of
what is going on.   I am pretty good with trig, however, and wondered if the
...97... number could be computed with trigonometry.

I'm very interested in knowing what c stands for?

In your 50 possibilities, the problem is that the sum of the distances you are rounding up is
greater than the sum of distances you are rounding down. 
How it really goes is:   .00 doesn't round anywhere; the value stays the same.
                                 .50 rounds up or down, so it's a don't care.
                               49 cases of .01 to .49 round down the same amount as 49 cases of .51 to .99 round up.