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In grade school, numbers that use a positional notation along with a decimal point (to delimit integer and fractional parts of a number are called "decimals". This "point" notation is easily generalized to any base, e.g., in base 2, we can write the number 101.11, and the point is then called a binary point. But what is the number itself called—a "binary"? (doesn't sound right, somehow, but it is analogous to the base 10 term "decimal"). Surely we can't continue to call it a "decimal" given that the base isn't 10; that definitely doesn't seem right.

Is there a base-independent term for a positional number containing a radix point?

Thank you

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I don't call numbers written in decimal form "decimals" either, I call them numbers written in decimal form, or decimal expansions. I would call a number expressed in base $p$ a number expressed in base $p$, or the base $p$ expansion of the number. In some cases there are special names, like binary expansions for $p=2$, ternary expansions for $p=3$, octal and hexadecimal (8 & 16), and I don't know what else. Another concern is the terminology analogous to "digit". For binary there are "bits", but I don't know about other bases, even $p=3$; I've jokingly called them "trigits" in that case. –  Jonas Meyer Dec 21 '11 at 22:05
Yes but "a number expressed in base p" doesn't have to contain a radix point (it could be an integer), so I don't see how that terminology distinguishes numbers that contain a radix point from numbers that don't. –  Jack Dec 21 '11 at 22:25
Jack: Doesn't "noninteger" suffice in that case, or "fraction" if you prefer? I.e., a base $p$ expansion of a noninteger is precisely the type of thing that requires a radix point to be expressed in base $p$. –  Jonas Meyer Dec 21 '11 at 22:27
@Jonas: Well, I guess what I'm really looking for is a terminology that distinguishes fractional numbers notated using the vinculum ("fraction bar") vs. the radix point. –  Jack Dec 21 '11 at 22:43
@MichaelHardy: Is that supposed to be funny? If you find my question flawed in some way, explaining why you think so would be much more useful to me than sarcasm. –  Jack Dec 22 '11 at 3:36
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4 Answers

Recall that "decimal" is an abbreviation for the correct term, which is "decimal fraction." The base $2$ analogue is called a dyadic rational, but could be called a dyadic fraction. I have seen the name "ternary fraction" for the base $3$ analogue. We could (and someone undoubtedly does) also speak of octal fractions, or hexadecimal fractions.

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But, assuming base 10, "1/10" is also a "decimal fraction" isn't it, yet no one calls that a "decimal". They call it a "fraction," in fact. There is a difference in notation between 1/10 and 0.1, and I'm wondering how we can distinguish these notations terminology-wise. –  Jack Dec 21 '11 at 22:19
@Jack: The notation $1.234$ is explicitly an abbreviation for $\frac{1234}{1000}$. That is, the fact that they are the same is not a mathematical fact, but a linguistic one. But I can see the point of trying to find a word for the notation itself rather than the object. –  André Nicolas Dec 21 '11 at 22:30
Right. So you're saying that there is no terminology (that you know of, at least) that distinguishes the two notations (radix point vs. fraction bar), correct? –  Jack Dec 21 '11 at 22:36
The "baseline dot" notation may be a minority one. To someone like me, $1$,$23$ looks right. To others, a centered dot is correct. Over the years, in English-speaking lands, the name for the object has become conflated with the locally standard representation. –  André Nicolas Dec 21 '11 at 23:35
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The term decimal can be used used to denote a number system or a notation. Since it is a notation commonly used in everyday language it has acquired a non-rigorous naming pattern i.e. we call something in a "Decimal Fraction" notation a decimal. Since other notations/bases are not commonly used, numbers represented using such notation may not have names or common names. In such cases, i think you have leave-way to name notations as you please, the idea here is to communicate an idea as clearly as possible.

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You seem to be implying that a "decimal fraction" is a base 10 number that contains a radix point, and not a number written with a vinculum ("fraction bar") with numerator and denominator. And yet the latter are called "fractions", and if they are base 10 fractions it should be proper to refer to them as "decimal fractions". Thus, it doesn't seem appropriate to specifically call base 10 numbers that use the radix point a "decimal fraction", because that term doesn't distinguish them from base 10 numbers that use the fraction bar. –  Jack Dec 21 '11 at 22:31
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I would call numbers that are not integers non-integers, or possibly non-integral or fractional numbers if I really wanted an adjective. This is a representation-independent concept — it just happens that it coincides with the set of numbers that cannot be written in positional notation, for any given base, without a radix point.

For the digits after the radix point themselves, that's exactly what I'd call them. It's a bit verbose, but gets the point across unambiguously. Should I find myself wanting a shorter name for them, I might try fractional digits (and substitute digits with the appropriate base-specific name, such as bits, if there is one) or perhaps $n$-ary places by analogy with "decimal places" (again substituting the appropriate name for $n$-ary).

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I've always called it a "binary decimal" (etc. for other bases), as opposed to the regular "binary number." These are obvious contradictions of terms, but it's always been clear (or at least, it's always been clear with its context). One might also use "binary fraction," an allusion to the phrase "decimal fraction."

In another question, a poster claimed that the term 'binary decimal' is vastly outdated. He noted that binary numbers, which I suppose should just be called binaries (like decimals), are better known as binary numerals. If you pay a lot of attention, you'll notice that my use of binary decimal is a bit different than his.

Really, though, fraction is the best word. Recall that all digits after the decimal point are called the "fractional part" of the number, even if they don't terminate and are not periodic.

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Binary decimal... LOL; As if kids aren't confused enough in math! I have no problem with the term "fraction," except that it doesn't distinguish between the fraction bar vs. the radix point notations. –  Jack Dec 21 '11 at 22:40
Thanks for that link. That was very interesting. –  Jack Dec 21 '11 at 22:53
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