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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?

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  • $\begingroup$ It is arbitrary, but if you have anything to right of that 5 then it is indeed closer to the higher number. So as a rule, look to that digit and if it is a $5$ round up, works out. But if you have multiple numbers and you are going to round before summing or multiplying you should consider rounding one up and one down. Or, sum first and then round. $\endgroup$
    – Doug M
    Apr 25, 2018 at 20:06
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    $\begingroup$ As you say, there is no compelling reason. In such a situation, humans tend to invent conventions. $\endgroup$
    – Hagen von Eitzen
    Apr 25, 2018 at 20:06
  • $\begingroup$ You can read more here: en.wikipedia.org/wiki/Rounding#Rounding_to_the_nearest_integer $\endgroup$
    – Naweed G. Seldon
    Apr 25, 2018 at 20:08
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    $\begingroup$ As junkquill's link notes, that is one of the more common conventions, but not the only one. If you are rounding to integers, you have to move .5 down or up, and the choice is really a matter of avoiding biasing your results. Always rounding .5 up isn't great for this, so another option is round to even/odd, where you always round .5 to the nearest even/odd integer. Assuming the numbers you are looking at aren't biased in some way with respect to proximity to even and odd integers, you will round it up half the time and down the other half. $\endgroup$
    – Tyberius
    Apr 25, 2018 at 20:17
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    $\begingroup$ "Why is 0.5 rounded up?" Well, it's not always. I bought of a piece of candy for \$0.50 and they didn't charge me a dollar. And I filled my tank half full and it didn't magically fill up. And another time my sales tax was calculated to be $11\frac 12$ cents and they rounded it down to 10 cents! So .5 is only rounded up... when it is rounded up. So the real question is "When they teach us to round to the nearest integer and .5 isn't closer to either, teachers tell us to round up, why?" And the answer to that is because it's easier for them to just make up a rule and have you do it. $\endgroup$
    – fleablood
    Apr 25, 2018 at 21:13

3 Answers 3

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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.

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  • $\begingroup$ Interesting, but what's so hard about scanning the other decimal places? In practice, I don't think I've seen real-world measures going beyond 6 decimal places. Very easy to check if any of them are not zero. I think I heard somewhere that even pi, to make a very accurate circle with galactic diameter, needs only 19 decimal places. (And btw, any irrational number will never be ambiguous, cuz it can't be exactly 0.5 + any integer.). This is why I voiced suspicion in the OP that it's more like a rule without any ambiguous cases, even if it has to be an arbitrary rule. $\endgroup$
    – DrZ214
    Apr 26, 2018 at 2:48
  • $\begingroup$ @DrZ214 In fact you can do a lot of rounding without ever invoking any of the special rules for rounding 0.5 up or down. As for why round at all, when your bank calculates interest, they compute it as a very small fraction but then have to round it to the nearest penny (or whatever the smallest unit of currency is). There are many other calculations that would give absurd numbers of decimal places (most of which are completely inaccurate anyway because of the inaccuracy of the input) if we did not round them. $\endgroup$
    – David K
    Apr 26, 2018 at 14:02
  • $\begingroup$ @DrZ214 you're correct that it's not very difficult to scan for further digits. However, we're still left with the fact that $.5$ needs to round to something. So we choose the rounding that is easier, even if the improvement is small. $\endgroup$
    – Dando18
    Apr 26, 2018 at 15:22
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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.

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  • $\begingroup$ Yeah I've always wondered how banks do this to avoid creeping losses, and figured they keep a more accurate track of numbers somewhere. I've also noticed that gas stations (in USA) always list their prices to three decimal places, and the last one is always a 9 lol. So apparently some businesses are allowed to be more accurate than others. $\endgroup$
    – DrZ214
    Apr 26, 2018 at 17:55
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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.

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