Why is $0.5$ rounded up to $1.0$? It's not closer to $1$ than $0$?

Question

Why is 0.5 rounded up? It is exactly halfway between 0 and 1, so it is not closer to 1...nor is it closer to 0. It is exactly halfway between them.

The same principle applies to other things like 11.5 or 23.5. The value is exactly halfway between integers. I don't see any objective round-direction in these cases.

You can say the same thing for something like 6.45 if rounding to the nearest 10th. 0.45 is no closer to 0.50 than it is to 0.40. It's exactly halfway between them.

Why are they rounded up? Is it just because we need an arbitrary rule for all cases?

Edit: I read everything on the duplicate question, but still feel left hanging. If the answer is, it's arbitrary, it just seems so unsatisfying. Is there a way to prove if something in math is arbitrary or not?

Answer 1

As far as I know, it's pretty much just an arbitrary convention. If you think about it, it really shouldn't matter too much one way or the other because, for example, while 3.5 rounds up to 4, 2.5 rounds up to 3 and so on. So, really it evens out more or less.

Answer 2

It is by convention that we define rounding as

$$ \text{round}(x) = \lfloor x+0.5 \rfloor = -\lceil-x-0.5\rceil $$

The wikipedia page on rounding explains how the process is done. The only incentive for this convention is we only have to look at one decimal place in order to determine whether to round up or not (i.e. if we see $4.500\cdots 001$, we only need to look at the $5$ and no further in the decimal expansion).

Also note that there are other conventions used, for example some programming languages round $.5$ away from $0$. Another custom is to round $.5$ to the nearest odd/even number.

Answer 3

There's actually quite a bit of background on this topic. Rounding should be considered in the context of working with large sets of numbers, either adding them all up or multiplying them together. Think of financial ledgers, for instance, where a long list of incomes and outlays must be summed to find a balance at the end. In this situation, accuracy and consistency are very important; if the method of rounding one uses introduces a systemic bias then the balance will be systematically low or high.

Rounding to the nearest integer is just a sub-problem of rounding to the nearest $n$th decimal; in many (but not all) financial transactions, amounts are rounded to the nearest cent. If one always rounds up, then extra money will be introduced into the system over time. This is why it's probably better to round away from zero, that is, $-3.5$ rounds to $-4$.

In contexts where only positive numbers are expected, rounding away from zero is the same as rounding up, so a different convention must be used to avoid bias. Thus, as mentioned in other answers, one can always round to the nearest even or odd number. Then you'll be rounding up half the time and rounding down half the time, so on average there is no bias.

When you're rounding a single number, then yes it's just by convention. But in practice single numbers aren't common, so how we round is actually quite important.