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If we cut 10 in half, then we get 5. This means two 5s makes 10, meaning, 6, 7, 8, 9, and 10, are the remaining 5 numbers that make it up. So why do we round up 6 numbers? It's such a small thing but it annoys me so much.

Luca Ghidelli
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Dillon B
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    Does this answer your question? Why do we generally round 5's up instead of down? – Samuel M. A. Luque Jun 20 '21 at 21:31
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    I think it just feels natural. 5 feels closer to 10 than 0, right? 150 feels closer to 200 than 100. If you want a more satisfactory explanation check the linked question by @SamuelM.A.Luque and the Wikipedia page: https://en.wikipedia.org/wiki/Rounding – KingLogic Jun 20 '21 at 21:39
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    I think the convention comes from the fact that we intuitively consider "numbers between zero and ten" to be $x$'s such that $0\leq x<10$ (note that the second inequality is strict, but not the first). That is, we see 10 as the beginning of the next set of ten numbers, not the end of the last one (which feels natural, since when you reach 10 you add one digit). Then, 5 really is in "the upper half", while 4.999... is in "the lower half". (This is only based on my own intuition, I've got no sources to back me up.) – Léo S. Jun 20 '21 at 21:40
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    another reason could be that if all you know are the first $k$ digits after the decimal and it ends in $5$ then it's more likely the other number aren't all zeros. – Asinomás Jun 20 '21 at 21:45
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    I removed the tag number theory because it doesn't fit the description of the tag. Here is the description of the tag "number-theory": Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebra geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc...... – Luca Ghidelli Jun 20 '21 at 22:11

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$5$ is as close to $10$ as it is to $0$. So that doesn't decide the issue. But if you always round these borderline cases up, or you always round them down, then you increase the rate that rounding errors accumulate over a sequence of calculations. And if you alternate between rounding up and rounding down, as some stone-age computers did, you lose reproducibility of results. So computers these days generally use round to even: if a number is equally close to two rounding candidates, choose the even one. Loosely speaking, this strategy makes it more likely that rounding errors will cancel out in the long run.

Here you are rounding to a multiple of $10$, so the strategy is to round to the nearest even multiple of $10$. So $5$ will round down to $0$. I hope this makes you feel better :-)

TonyK
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