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If I asked "When rounding to the tens place, how many different whole numbers will round to 50?" would the answer be "9" or "10". As in: does "50" "round to itself" (in which case, 10) or does it not count because it doesn't "go anywhere" (in which case, 9)?

I can formulate a common-sense rationale for either of these answers. Is there a more "mathematical" answer here?

Fly by Night
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bigfooot
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  • It depends on your definition of 'rounding'. But I see no problem with 50 rounding to itself. – Miguel Mars Sep 27 '16 at 18:30
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    If $50$ doesn't round to itself, what would it round to? – abiessu Sep 27 '16 at 18:31
  • Yeah, like I said, I can see an argument for defining 'rounding' either way. So, according to one hypothetical definition 45-54 would round to 50. In the other 50 would be excluded because it doesn't move at all. I was just wondering if there's any more "official" perspective on this.

    In the second definition 50 wouldn't round to anything (when rounding to 10's) because it's already "there".

     – bigfooot Sep 27 '16 at 18:38
  • You are defining a map from $\mathbb Z$ to $10\mathbb Z$ in the way that ${-15,-6}$ maps to $-10$, ${-5,\dots,4}$ maps to $0$, ${5,\dots,14}$ maps to $10$, ${15,\dots,24}$ maps to $20$, etc. But you can define it such that the multiples of 10 don't have an image. – Miguel Mars Sep 27 '16 at 18:59
  • Thanks for your comment, can you explain that in more layperson's terms maybe? – bigfooot Sep 27 '16 at 19:19

2 Answers2

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Wikipedia discusses rounding to integers in terms of the floor and ceiling functions. Rounding up $x$ is $\lceil x \rceil$, so if $x$ is already an integer, its rounded value doesn't change.

John
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  • Thanks for your answer and the link. Are you saying that the fact that its value doesn't change means that it doesn't round to itself? – bigfooot Sep 27 '16 at 19:18
  • I would say an integer does round to itself. It rounds up (toward $+\infty$) to itself, and it rounds down (towards $-\infty$) to itself. I've never thought of there bring a "rounding singularity" at the integers. The rounding function by most counts is idempotent. – John Sep 27 '16 at 19:27
  • Thanks, I see what you're saying! – bigfooot Sep 27 '16 at 19:45
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Being a programmer, I think of this in terms of "How would I implement the desired 'round-to-tens()' function? What would it return given 50 as its argument?" The only sensible answer seems to me to be 'round-to-tens(50) == 50'; otherwise one has to throw an exception, return some bogus value, set a global error code, or some such utterly gratuitous complication.

For a purely mathematical justification: The computation in question is really "Find the nearest exact multiple of 10 to 50" Obviously the answer is 50, since it is itself a multiple of 10.

Incidentally, notice that the answer to the original question (given this handling of 50) is ten, regardless of whether we mean 'round down', 'round up', or 'round off'. Justification: Two numbers whose difference is ten or more logically shouldn't round to the same value, so no more than ten consecutive values can round to the same number.

PMar
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  • Thanks, yeah I was trying to think it through computationally. I will mark this one as solving it, but John's answer is equally helpful! – bigfooot Sep 27 '16 at 19:45