Math Is Fun Forum

Hi:

I've entered this debate a bit late, but I have a few things to add.

In your last comment Johnny, you said:

>If you care about consistency (like a programmer would), I guess I'd round up.

Well... I happen to be a programmer and I do care about consistency, but I don't round up. I'll explain how I round in a moment, but first, you said:

>Instruments of measurement don't truncate the last value they read, it's rounding it. Not >by design, but by the nature of the inaccuracy.

That's not correct. Numbers are man-made -- they don't exist in nature.  There is not an instrument (man-made or living) made that deals with absolute numbers, but rather differences in thresholds. If a voltage/resistance/amperage/whatever-physical-property has a certain value, it simply means that the signal measurement has obtained a sufficient threshold to register a number to whatever precision the instrument is capable. One can say that the number provided is always a result of a round-down method, but that's misleading. For example, the way your eye/brain determines colors is not due to a round-down method, but is rather through a step-like method of meeting certain thresholds -- one threshold for red, another for blue and so on. The computer is no different and there are many situations where a specific number is simplified via meeting a predetermined threshold. The concept of numbers, and the resulting algorithms in all forms, are man-made explanations to describe nature, not the other way around.

Barring real-world examples, you may want to consider that there is more than one way to round a number. For example, there is no completely accurate solution for rounding any real number. If you want absolute precision, then you must use absolute numbers to whatever precision they are. However, this may not be obtainable because of continuos numbers like pi. As such, you may be left with no choice other to round.

There are several different algorithms for rounding, such as truncation, floor (rounding down), ceiling (round up), common, and statistical. None are perfect, but some are more prone to errors than others. The least prone to error is the statistical, please review:

"Statisticians method of rounding":

http://en.wikipedia.org/wiki/Rounding

In previous post on this topic, people have mentioned that rounding when the last digit is 0-4 down and 5-9 up is a 50/50 arrangement, but that's not true either. In fact, you don't round at all when the number ends in a zero, you don't do anything to the number -- this leaves 1-4 rounding-down and 5-9 rounding-up -- a 4:5 statistical bias.

So to correct this bias, one should; a) do nothing is zero; b) round down for 1-4; c) round up for 6-9; d) and in the event of 5 round up or down depending upon the preceding number. If the preceding number is even, then round one-way and if it is odd, then round the other. It makes no difference which way (up/down) you round, just as long as you are consistent.

And, you may enjoy the products of round-off errors:

http://www.ima.umn.edu/~arnold/disasters/disasters.html

http://catless.ncl.ac.uk/php/risks/search.php?query=rounding

tedd
http://sperling.com