Gadgetopia

which way do you round 0?

or, to make it more obvious:

5.0, 5.1, 5.2, 5.3, 5.4 => 5

5.9, 5.8, 5.7, 5.8 => 6

which way should 5.5 go, if you want to keep things balanced?

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i think your greater question of looking for math rules that are arbitrary bypasses something about the nature of mathematics. it's all arbitrary rules, that happen to behave interestingly when you poke at them enough. math is a thought exercise - it is an abstraction. mathematicians make up rules that allow them to manipulate those abstractions in useful ways.

(err, that second 5.8 should have been 5.6)

Rounding up isn't a mathematical rule anyway, it's a convention - and not the only one. In commerce, sometimes folks round from everything above .4 up. In working with columns of figures, I was taught to round to the nearest even number, in order to minimize creep.

If you're looking for interesting arbitrary mathematical rules, try 0! = 1, "by definition".

I?m looking for a mathematical rule that was arbitrary.

hah!

i've got it!

it's a notational shorthand that if you push two things together without any operators in between them, you multiply them: 3x = 3*x.

completely arbitrary, and potentially confusing two people who are learning their way around operators and notation.

I think rounding up came from dealing with money, where it would mean more profitable for the person setting the price.. :-) On a scale of 0.1 to 1.0 there seems to be only 5 that rounds down and 6 that rounds up. I've heard of a rule - again, arbitrary

0.0 to 0.4 rounds down 0.6 to 1.0 rounds up 0.5 rounds up if the whole part is even, and rounds down otherwise.

[...] bypasses something about the nature of mathematics. it?s all arbitrary rules [...]

No, I don't agree with that. The laws of mathematics are logical and universal. If you live on the planet Warblon 7 and you have one moon rock, then you get another moon rock, you now have two moon rocks. The concept of 1 + 1 is 2, no matter where you are or what you call the numbers.

But on Warblon 7, they may round 2.5 down to 2. Why? Just because that's what the Elder Warblonion Chieftain decided way back in the day.

Or how about:

x^0 = 1

That never made sense to me until I realized that you could define exponentiation as:

x^y = 1 * (x multiplied by itself y times)

Then x^0 = 1 made sense.

Which explains why 0^0 is also equal to 1.

Some forms of rounding are not strictly round-up.

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

There is also banker's rounding (round to the nearest even number, so 0.5 -> 0 and 1.5 -> 2) and stochastic rounding (which rounds randomly).

So when I round -1.5, do I round to...-1, since that's the next larger number?

On some rounding schemes.

Generally the common rounding scheme is to round away from zero, so you'd round to -2.

Axioms in theories are arbitrary. You could as well select a negation of some axiom and you would get some other theory. Usually mathematicians tended to select those axioms that described their real-world experience the best, but sometimes our senses lie and sometimes non-real-world theories are very interesting.

You could round inward (towards zero), or round outwards (towards infinity - increase in magnitude) to the nearest integer. You can alternate these rules between odd and even integers. The possibilities are endless..... it just depends on what the data represents. Significant digits: to round or to truncate. That is the question!