OK. I’ve sweated blood over this and I think I’ve got it.
This link and this one state the rule “if rounding 5, round to the nearest even number”. But they don’t explain why.
Which way to round 5 only becomes significant if you are rounding and subsequently processing a lot of numbers, as DMC illustrates.
Imagine we’re rounding one significant digit off of 100 randomly selected numbers and then summing them. 8.4 becomes 8, 13.9 becomes 14, etc. We would expect 10 of the randomly selected numbers to be X.5. Of those, five would be preceeded by an even number and five would be preceeded by an odd number. Rounding all X.5’s up and then summing would result in a sum that was 10 x 0.5 = 5 units more than rounding all X.5’s down. Rounding to the nearest even number would result in a sum halfway between these two results.
I don’t know if that’s clear or not, but I tried it and it works. I am such a pedantic prick. Who answers his own posted questions? Why did I even bother to post the question? I couldn’t figure it out yesterday, today it’s clear. I am an ass.
Your illustration serves only to underscore my original point which was that 5 is more than halfway between 0 and 9. Please note that between 0 and 5 there are 4 digits and between 5 and 9 there are 3. Ergo 5 is closer to 9 than it is to 0.
I would further like to go on record as saying that I do not dispute the issue of whether or not there are times when people choose to “round” 5 down or even always “round” 5 towards an even number and call it something like “banker’s rounding.” What I dispute is whether that is indeed “rounding.” I maintain that it is not. It is, instead, an aribtrary “rule” which has no basis in number theory and exists only to pacify the queries of people who ask questions like the OP (no offense intended) because they are suffering under the misimpression that 5 is halfway.
C’mon folks, we’re trying to fight ignorance here, aren’t we?
3.5 is exactly in the middle between 3 and 4. The absolute value of rounding up or down is exactly the same. You are not rounding up to 3.9, you are rounding up to 4. I can’t believe you can get confused by something this simple.
Your argument that 5 is closer to 9 than 0 is immaterial. The issue of the OP is whether 5 is closer to 10 than to 0, since the numbering system of interest is base 10, not base 9.
Imagine you cut an apple exactly in half. Which half is heavier? By definition, each half would weigh exactly the same. Since we can write “one half” alternately as 50%, 50% of the apple would be in your left hand and 50% would be in your right hand. Since we started with a whole apple (or 100%), 50% is exactly the same distace from 0% as it is from 100%. By the same argument, 5 is exactly the same distace from 0 as it is from 10. Not 9.
