# What is a base-neutral equivalent of the word “decimal”, when used to denote a “non-integer”

In common parlance, it's usual to refer to 2.346 as a "decimal", to mean that it's not a whole number. This might be a British English thing, so sorry if this isn't familiar to you, but I would likely call "4" a *whole number and 2.7 a decimal and 22/7 a fraction

But what should we use outside of base 10? You could call it a "fraction" however this has 2 pitfalls:

1) it might be an irrational number
2) even when talking about only rational numbers, to say "fraction" conjures up the image of a number written as a numerator and denominator. This is misleading at best, but also useless if you want to disambiguate between different ways of writing a rational number i.e. vulgar fraction, mixed number and a... decimal...?

You could call it a "non-integer", but as well as being unwieldy, it's not great to definite something by what it isn't rather than what it is. Non-integer can mean different things in different contexts (for example, in computer science where it might refer to datatypes)

3 is a whole number.

1/7 is a fraction

3.14159 is a decimal

But what is 3.243F6A?

A hexadecimal decimal? Surely not. A hexadecimal fraction, no, not really, that would imply something like B3/39.

• In Italy we say "numero con la virgola", which can be translated in "number with the point". This turns out to be independent on the basis ;) . – Crostul Apr 28 '17 at 16:08
• Your "decimals" are, in fact, fractions. $3.14159 = 3\frac{14159}{100000}$. Every "decimal" can be written as a fraction, and every fraction can be written as a (possibly repeating) "decimal". Because you can only write a finite number of digits, all of your decimals are rational numbers, so can be written as fractions. Some fractions may have nontrivial infinitely repeating tails (e.g., $\frac13 = 0.\overline{3}$). So in the end, I would stick with "nonintegral base 16 fraction". – MPW Apr 28 '17 at 16:11
• Would you call "1.0" a decimal? If so, perhaps "real number" neets your needs. If not, "non-integer" may in fact be the best option. – Mark S. Apr 28 '17 at 16:15
• Uh. Isn't it called position notation? – user251257 Apr 28 '17 at 16:40
• My gut said "radix notation", but that seems to be just like "base notation" in that "54" probably would count too, which you don't want. – Mark S. Apr 28 '17 at 17:11

## 2 Answers

It is a Laurent series, at some integer value of $x$.

For example, with $x=10$ $$\frac{4}{33} = \sum_{i>0} a_{i}x^{-i}$$ with $a_i = 1$ for odd $i$ and $2$ for even $i$; but with $x=11$, $$\frac{4}{33} = \sum_{i>0} a_{i}x^{-i}$$ with $a_1 = 1$, $a_i=7$ for odd $i>1$, and $a_i=3$ for even $i$. That is, in base ten, $$\frac{4}{33} = 0.12121212\ldots$$ while in base eleven, $$\frac{4}{33} = 0.137373737\ldots$$

If I may rant a bit.

It seems the lack of a general terms is due to the the happenstance that not a lot of mathematics concerns itself with the peculiarities of particular bases (but as I will show the problem has legs). Most of the time I've seen the word "digit" used with reckless abandon, e.g. "binary digit." Because of the basis representation theorem only artifacts of base representation find themselves in need of such terminology and, for good or for ill, a lot of this is recreational mathematics. Mostly "digit" and "decimal" have come to stand in for positional notation of arbitrary precision in arbitrary bases when the need arises.

You might note that even outside of this the word "fraction" is not particularly well-defined. For instance, consider the use of "fractional part" to mean "the difference between the number and the floor of the number." Fractional part of a fraction isn't the fraction—go figure. Unless the fraction is less than one and not an integer, but that's a coincidence. Don't even get me started on "improper" and "mixed" fractions. Even then the point is usually around its magnitude, not its representation. I'm not saying I've never heard something like this, but honestly the need to specify something to the level of "non-integer rational" is not so common.

"Rational" too has side-uses which are far and away from $\mathbb{Q}$, for instance "rational functions," "linear fractional transforms," and so on. I can already hear you objecting, these are named-by-analogy—but I am immune to your criticism. This named-by-analogy business gives the impression that the underlying term is clear and I suggest that it is not. What do you call $\frac{1.3}{2}$? Another-way-to-represent-division-in-base-ten-under-the-rationals? (We start chasing our own tail.) Then there's the disaster of pointing out that the constant of proportionality in a circle $\pi = \frac{C}{D}$. Much effort went into $\epsilon-\delta$ and virtually nothing here, which is just a swampy mess of notation and terminology. The linguistic bedrock is not secure, we just stand on whatever bit of detritus can bear our weight and proceed quickly, hoping no one notices us swaying. Convenience is worth a lot and it's only an abuse of notation if we had a well-defined notation in the first place. So let's skip that part and go wild, and try not to think too hard about all the numbers represented by $10_{10}$ (every base is base 10 when you sincerely work in another base and aren't just slumming).

Largely I think this is ends up a non-problem because the context of the discussion makes the words clear and there's no pressing need to have a well-defined special word for these. It can be clarified once at the start of a discussion and then move on. But devising some better jargon here would probably help novices a lot as they learn math in school and lead us away from some of the possible abuses I mentioned above. While I, of course, never experienced any confusion between a use like $\pi = \frac{C}{D}$ and $\frac{1}{3}$ (I was abusing notation before I could walk you've got nothing on me), I did know a few unfortunate souls in my time who did find it mysterious: "but aren't we writing pi as a fraction?" It seems we must introduce transcendence theory much earlier in primary education.