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How can I refer to a number written out in its decimal expansion (e.g., 1.25) or binary expansion (e.g., 1.01) to distinguish it from a number expressed as a fraction? I am teaching students to use different bases so do not want a term referencing a specific base, such as "decimal expansion".

In computer architecture, the term "floating point" (or "fixed point", depending on the implementation) is used, but I don't think this is a mathematical term.

In case, my question is unclear, I want to complete this analogy: $\frac{1}{2}$ is to fraction as .5 is to ?

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    $\begingroup$ Maybe 'digits expansion' could work. I think it is used occasionaly, but I doubt it is a common term. $\endgroup$
    – quid ♦
    Aug 2, 2017 at 22:34
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    $\begingroup$ I think this is called "positional notation" with a radix point. $\endgroup$
    – user52817
    Aug 3, 2017 at 3:04
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    $\begingroup$ I would just call it "floating point number". Terminology diffuses between mathematics and computer science. It is a natural term to use and just takes a few seconds to explain. $\endgroup$
    – John Coleman
    Aug 3, 2017 at 15:44
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    $\begingroup$ @JohnColeman: But the point doesn't float. $\endgroup$
    – Daniel R. Collins
    Aug 3, 2017 at 21:27
  • $\begingroup$ @DanielR.Collins Depends on how you think about it. I remember being taught to multiply and divide by powers of 10 by shifting the decimal point. When I learned how to program much later in life, I thought of that the first time that I encountered the term "floating point" and it struck me as being incredibly natural. $\endgroup$
    – John Coleman
    Aug 3, 2017 at 22:30

2 Answers 2

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The point is definitely called, in general, the radix point (as stated in a comment by @user52817).

I'm not familiar with, nor succeeding at a search for, a general name for the representation method. I would be comfortable calling one an "$n$-ary representation", following the term decimal representation (similar to a comment by @mweiss).

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    $\begingroup$ Linguistically, then, based on the adjectival form of radix, you would think the term would be radical or radical representation, but sadly, those terms are already used for a different concept. $\endgroup$
    – shoover
    Aug 3, 2017 at 16:34
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    $\begingroup$ It is odd, though, that language seems to have settled on "$n$-ary" rather than "$n$-imal". I guess the latter sounds too much like "animal"? $\endgroup$
    – mweiss
    Aug 3, 2017 at 21:20
  • $\begingroup$ @mweiss Presumably "n-ary" is thought of as a generalization of "binary" and "trinary". $\endgroup$
    – Jim Belk
    Aug 5, 2017 at 17:46
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    $\begingroup$ Yes, obviously (although "ternary" is more common than "trinary") - but why those rather than a generalization of "decimal", "hexadecimal" and "sexagesimal"? $\endgroup$
    – mweiss
    Aug 6, 2017 at 2:00
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I have been searching the corect wording for a while in the same context, that is, implementing a function that represents the decimal expansion of $\frac{a}{b}$. But then, I landed in a word desert when I needed that function to handle base 2 to 10.

I naturally came to google "n-imal" fractions and found that the term "n-imal" is used here and there (@see ref. below) to designate a fraction expressed in base n, or the fractional part of a rational expressed in base n, but it seems no strict convention exists.

By fractional part I mean the non-integer part represented by the digits after the radix point (I'm french to complicate the thing so I'm not sure if it's clear).

Using "n-ary" is ambiguous, for example we already use "binary" to mean that something is "binary", but how do we name the fractional part of a binary number ? We're left with "binary" or "bit" (binary digit), something is definitely missing.

On the other hand, "n-imal" allows for words like "binimal" to be used, which can't be confused with "binary" (ie. for a binary system, or the binary representation of a number).

The problem is also that most of the time, we wrongly refer to base-10 numbers as "decimal" numbers, but we should call them "decanary" (or "denary" ?) numbers so that "decimals" can only refer to the fractional part of a decanary number, or by extension, to decanary numbers having a fractional part. The same logic should apply for "hexadecanary" vs "hexadecimal".

It seems most people don't care about this, but one of the most important thing in mathematics and computer science is to avoid ambiguity.

For consistency, I would recommend using something like :

base n-imal
2 binimal
3 trinimal
4 quadrimal
5 pentimal
6 heximal
7 septimal
8 octimal
9 9-imal (?)
10 decimal
... ???

So to answer the question, I would say that :

  • $\frac{a}{b}$ is to fraction as $.c$ is to n-imal (with $a<b$, where $c$ represents the fractional part of $\frac{a}{b}$ in base-n).
  • $\frac{1}{2}$ is to fraction as $.5$ is to decimal
  • $\frac{01}{10}_{2}$ is to fraction as $.1_{2}$ is to binimal.
  • $\frac{1}{2}$ = $.5_{10}$ : one half equals zero point five decimals, or zero decimal five.
  • $\frac{1}{2}$ = $.1_{2}$ : one half equals zero point one binimal, or zero binimal one.

Here some references using n-imal/n-mal wording :

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    $\begingroup$ Is there any evidence for this usage beyond that Gilmore & Lefranc book? For "binimal"; for "zero point five decimals"? By the way, the book does look interesting! $\endgroup$
    – Gerald Edgar
    Oct 24, 2021 at 19:30
  • $\begingroup$ I added some references. The proposal in the latter is somehow what we should already have (mathematicians and academicians should review, make proposal like this, and settle on a strict convention). The wording "zero point five decimals" sounds natural to me, and is also used by the author of that proposal. "zero point one" means $.1$ and we assume base-10 by default, so if you wan to say $.1_{2}$, you would say "zero point one, in binary" or just "zero point one binimal", no ambiguity. $\endgroup$
    – EricLavault
    Oct 25, 2021 at 11:34

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