# Can we have a tag for integer base $b$ representation?

I see a lot of interesting questions, broadly in the category of contest math and recreational math, about the base $b$ expansion of an integer, often $b = 10$. For example: the sum of the digits, when this expansion has repeated digits, how many $0$s at the end, and so on.

I would like a tag to use for these questions other than just . Although these questions fall into a clear common category, I have found tagging them to be an awkward task -- no existing tag seems to fit the bill for a question about the sum of the base $b$ digits of an integer, for example.

In the answer to this old question it is suggested to use . However the tag wiki seems to have changed since then, to make it even more broad and less about integers than it already was. The tag wiki discusses not just base $b$ representation of integers, but also roman numerals, floating point numbers, factorial base, Fibonacci base, and so on.

Other related tags include

• , helpful and appropriate for base $2$ but not for other bases;

• , possibly helpful for base $10$, but more appropriate for real number representation, and certainly not helpful for other bases;

• , which is unfortunately just a synonym of .

None of these tags are about base $b$ representation for a general $b$, which is a very common and important topic in elementary number theory. It also strikes me as odd that we have decimal expansion and binary tags, but no tags for base $b$, of which these tags are just special cases. Therefore, I am suggesting we have a new tag for base $b$ representation of numbers. As a rough suggestion:

: Questions about the base $b$ representation of an integer, where $b$ is an integer, including: arithmetic in base $b$, converting between bases, the number of digits, the sum of digits, and other features of such representations. Often appropriate with , but may also be used for base $b$ representation of rational or real numbers. For questions about base $2$ or base $10$ in particular, consider using or . For questions about some other representation, use .

Does the community agree this would be a useful tag, or have any suggestions for improving the scope and the above tag excerpt?

• Upvoting not because I necessarily support the proposed tag but because it seems worth discussing. There may be other actions that could be taken. – David K Sep 15 '16 at 1:55

Looking at the questions that are actually tagged number-systems, setting aside a few that clearly do not fit the wiki description at all (I have just untagged four of these), it seems to me the vast majority of them are about base-$b$ representation where $b$ is an integer.

There are a handful of questions regarding negative bases (which would be covered by the proposed tag, though that could be fixed by inserting the words "greater than $1$") or fractional bases.

Out of 473 questions tagged [number-systems],

• a search for [number-systems] float retrieved 2 questions;
• a search for [number-systems] roman retrieved 11 questions, but only 6 of them were actually about Roman numerals.

It would appear that only a tiny fraction of the questions properly tagged [number-systems] are really about anything that isn't base-$b$ representations of numbers, where $b$ is an integer. There are quite a few questions about representation of rational numbers, which is not about "about the base $b$ expansion of an integer," but would still be allowed under the proposed new tag.

I just don't see how a new "base $b$" tag would narrow down the search results much better (other than not returning many questions that fit the description but were not tagged).

I would rather discuss changing the tag description to emphasize the focus on base-$b$ numbers (for integer $b$) more and to encourage questions about such things as sums of digits and divisibility tests. (The current excerpt mentions algorithms only near the end, and neither the excerpt nor the full descriptions mentions the very interesting topic of digit patterns.) Frankly, I'm the one responsible for the current state of the tag description; it was (I think) my first attempt at a major edit of a tag, there was no existing full description to go by, and I may have gotten a little over-enthusiastic in some parts (especially in listing obscure subtopics in the excerpt).

In my own defense, the other things covered under the current tag description are mainly there because they are tangent to the main topic (floating-point number representations are strongly based on base-$b$, and fractional-radix or mixed-radix systems are extensions of the base-$b$ idea). Roman numerals are allowed because there are few such questions and they have no other obvious home (that I know of).

If there must be a new tag, I think the most reasonable thing is to convert the existing tag so it more specifically covers base-$b$ representation, create the new tag instead to cover the more "unusual" numeric systems, and move things that aren't really base-$b$ to the new tag. I'm just not sure there would be enough questions under the new tag to justify doing this.