OK guys, I'll give it one more try. I think we're at the point now where egos have gotten in the way of objectivity so if everyone could just take a deep breath and look at the math for a minute, you'll see the point.

But first:

That's not proof of his statements.
you are quite correct. The fact that Microsoft and I agree on something does not make either on of us right, and God knows I don't like being in agreement with Microsoft on anything. Still, in this particular case, what make BOTH of us right is the underlying math. You can expostulate all you want but at the end of the day math is math and there's nothing you can do to change the fundamental correctness of my explanation.

Well I mean, for nearly whole numbers, its going to be nearly perfectly accurate
again, you are correct, but as leather pointed out, my entire point was that close is not exact. If errors of the type being discussed in this thread were very large, then computers would be worthless for numeric computations, but as you have correctly stated and as Microsoft has also point out, the errors are quite small. Again, I'm talking about exactitude and you keep sliding off into correct statements about the smallness of the error and making incorrect statement about the fundamental issue concerning exactitude, which you clearly refuse to understand.

Yeah but in this case we're not even rounding.
Its just 1.4

There is no rounding to be done.
brings us back to the point that I have been, apparently unsuccessfully, trying to make. You are absolutely incorrect in this statement. The decimal fraction 1.4, when turned into binary, is an irrational number. The computer representation is 2^1 * .10110110110110110 ... (1100 forever) and when you turn that back into decimal, you do NOT get 1.4 although you get something very close, call it 1.399999999. Again, I would agree with you that this is quite a small error, but that's not the point of this thread. The point is that 1.4 is NOT the same as 1.39999999 and that is the fundamental reason for this thread having been started in the first place. You postulate that there is some other reason. I assure you that you will search in vain for any other reason. I have explained the reason.

For further help in understand this, I point back to my earlier statement about a decimal computer getting the wrong answer if you were to add 1/9 and 8/9. The point there is that 1/9 is irrational in decimal, so it will have rounding errors. It is .1111111111 ... forever and at some point you have to truncate it and that makes it incorrect.

kayjay, the answer to your question is very simple in concept and not a lot of use in practice. The answer is this: all numbers that are rational IN BINARY avoid rounding errors. All numbers that are irrational IN BINARY will always have rounding errors. The big problem we face is that it is very tedious to determine whether or not a number is rational in binary. Plenderj, for example, automatically assumed that since 1.4 is rational in decimal it is also rational in binary, a "fact" which I have gone to the trouble to show is not the case. I HATE doing long division and doing it in binary is a REAL pain but anyone who cares to do the math can see conclusively that 1.4 does NOT get represented in the computer as 1.4 but as 1.399999... which is why this whole discussion started in the first place.

If anyone has any further quesitons on this subject ... fugeddaboudit !!! I'm sick of the whole thing. I first encounted this problem in about 1963 and I've explained the whole thing so many times in my career that once more made little difference, but enough is enough.