Talk:Significand

Contents

1 Mantissa vs. Significand 2 OED entry on mantissa, for reference 1 "The" established mathematical meaning 2 Fractional coefficient

Mantissa vs. Significand

It seems like there is some disagreement about whether mantissa is "less correct" or "less formal" than significand in the context of floating-point numbers, and it seems better to discuss it here. To begin with, let me lay out some facts, or at least strongly-grounded points:

• The use of mantissa with floating-point dates back to at least 1946, and possibly earlier (the cited reference does not seem to be self-consciously coining a new term). It is widespread (probably in all fields where floating-point arithmetic is discussed).

• The use of mantissa in the floating-point context is, arguably, more widespread now than either significand or the older logarithmic usage.

• Searching on Google, "mantissa floating-point" gets 28100 hits, while "significand floating-point" gets 6830. "mantissa logarithm" gets 4180.

• Searching on the INSPEC literature database (1969-present) gets 183 hits for "mantissa AND floating-point" (from 1970 to 2003), while "significand AND floating-point" gets 11 hits (from 1991 to 2002). "mantissa AND logarithm" gets 5 hits, the most recent of which is from 1986.

• The Compendex database gives 113 hits for "mantissa AND floating-point" (1970 to 2003), while there were 16 hits for "significand" (starting in 1981, and that reference explicitly equated it to the mantissa). The most recent citation for mantissa in the logarithmic sense was from 1975.

(Is it possible that significand is a neologism, coined more recently by someone unsatisfied with the dual meaning of mantissa? From all the citations, it seems likely that mantissa is actually the original term used for that floating-point component.)

• The floating-point usage does not directly correspond to the older usage for the fractional part of a logarithm, but it is analogous: one is the logarithm of the other.

• There are several older meanings of mantissa; the original meaning is not even mathematical (see below for OED quote).

It seems to me that Wikipedia should be descriptive, not prescriptive; it is not our place to describe one widespread usage as more correct than another, or even more formal since both are used in formal professional contexts. Nor should we be saying that the logarithmic usage of mantissa is more commonly or generally used when, arguably, it is not. Now, if you want to quote some authority or standards body that says significand is better, go ahead...but still describe the common usage of mantissa. —Steven G. Johnson 17:58, 13 Jan 2004 (UTC)

I did find one place where Kahan parenthetically calls mantissa "wrong", so certainly some professionals, and perhaps the IEEE 754 standards body, share this opinion. However, English is not like a computer language—standards bodies cannot invalidate usages by fiat (although some have certainly tried). It looks like mantissa is the original term for this concept in floating-point, and retains enduring popularity, while significand is only slowly catching on. —Steven G. Johnson 19:37, 13 Jan 2004 (UTC)

I recently noticed that Knuth is adamant that mantissa is wrong, too. He writes (in The Art of Computer Programming, Vol 2, section 4.2.1):

“Other names are occasionally used for this purpose ... but it is an abuse of terminology to call the fraction part a mantissa, since that term has quite a different meaning in connection with logarithms. Furthermore, the English word mantissa means ‘a worthless addition’”

(Of course, fraction part is not always appropriate, either, if the significand is considered to be an integer and the exponent related appropriately. Overall, I prefer Kahan’s suggestion of coefficient.  :-))

mfc 17:07, 14 Jul 2004 (UTC)

This discussion strikes me as odd, as I have just queried 'mantissa' on dictionary.com and turned up "mantissa - n. - the decimal part of a logarithm". The only etymology given is "Latin makeweight, possible Etruscan origin". Also, the American Heritage Dictionary of the English Language gives "mantissa - n. - Mathematics. The decimal part of a logarithm when the logarithm is written as the sum of an integer and a decimal. [Latin mantissa, makeweight, probably from Etruscan]". Neither of these sources has significand as being an English word. It appears as if mantissa has been around far longer than significand, which is still seeking total acceptance from the rest of the English-speaking community, as opposed to mainly computer scientists. ~Dwee

That's exactly the point. 'Mantissa' is generally understood to mean the decimal/fractional part of a logarithm. The coefficient of a floating-point number is not the decimal/fractional part of a logarithm. It's a plain fraction, not a logarithm at all.

Example: using decimals, the value 0.025 might be written as the floating-point number 0.25 × 10⁻¹. However, as a logarithm, it would be (approximately) -1.602. The mantissa is 0.602 (the fractional part of the logarithm). The thing stored in a floating-point representation is the fraction to be multiplied by the power of ten (0.25). These are different.

'Significand' was coined or invented or hauled in by the people working on the IEEE 754 standard to make that distinction. It suffices. mfc 16:25, 2005 Jun 11 (UTC)

OED entry on mantissa, for reference

[< classical Latin mantissa, mant {imac} sa, of doubtful meaning and unknown origin; said by the Latin grammarian Festus (as preserved in a later epitome) to be an Etruscan word meaning `makeweight', elsewhere in classical Latin app. meaning `sauce', and `fuss or profit'. Found in two early 16th-cent. German glossaries, and thereafter as a common element in the titles of post-classical Latin books, esp. by German authors or printed in Germany, from 1608 to the 19th cent, with the sense `supplement'. Sense 2a has been attributed to H. Briggs 1624, but appears in fact to have originated with J. Wallis in his Opera Math. (1693) II. x. 41, where the Latin word is used for the digits of a decimal number following the decimal point (in the English edition of his Algebra (1685) Wallis used `appendage'); the Latin word was used in relation spec. to the decimal part of a logarithm by L. Euler in his Introductio in Analysin Infinitorum (1748) I. vi. 83.]

1. An addition of comparatively small importance, esp. to a text or discourse; a supplement. Obs.

1641 H. MAISTERSON Serm. 20 Trifles, which..should..as a mantissa or an overplus be cast in at their bargain. 1642 R. CUDWORTH Disc. Lords Supper i. 9 It will not be now amisse, if we adde as a Mantissa to that discourse, something of the Custom of the Heathens. 1671 R. MACWARD True Non-conformist 5 Spurning at the righteousnes of Jesus Christ, and aspiring to adde a Mantissa, an addition of your own, to his sole purchase. 1781 E. DARWIN Let. 29 Sept. (1981) 112 Mr. Lightfoot advises the botanical society to introduce the plants in the Mantissa in their proper places.

2. a. Math. The part of a logarithm after the decimal point. Cf. CHARACTERISTIC n. 3.

1846 J. W. KAVANAGH Arithm. (ed. 2) 217 The decimal part [of a logarithm] is sometimes, but more frequently in old treatises, termed the mantissa (over-weight) of the logarithm. c1865 J. WYLDE Circle Sci. I. 519/1 The decimal part of a logarithm is called the mantissa: the whole number is called the characteristic. 1917 Biometrika 11 May 376 Degen gives the logarithms of the factorials..to eighteen mantissa figures. 1941 C. D. HODGMAN Math. Tables from Handbk. Chem. & Physics (ed. 7) 1 A common logarithm, in general, consists of an integer which is called the characteristic and an endless decimal, the mantissa. 1961 L. J. COMRIE Chambers's Shorter Six-figure Math. Tables p. xii/1, When logarithms are taken to base 10..it is advantageous to keep the mantissa positive. 1989 W. GELLERT et al. VNR Conc. Encycl. Math. (ed. 2) II. ii. 58 Having found that lg 2.37 = 0.3747 one has at the same time the common logarithms of the numbers 23.7, 2370, 0.237, 0.00237, etc... The actual digits 3747 to be calculated for a logarithm are called its mantissa, and the integer before the decimal point its characteristic.

b. Computing. A number, usually of a fixed number of digits and between 0 and 1, by which a power of the base (e.g. 2 or 10) is multiplied to represent a number in floating-point representation.

1959 M. H. WRUBEL Primer of Programming for Digital Computers ii. 19 The second representation employs a mantissa, containing the significant digits, multiplied by 10 raised to a power. 1960 M. G. SAY et al. Analogue & Digital Computers v. 142 After multiplication has been completed the digit following the binary point must be examined and, if this digit is 0, a corrective shift must be applied to the mantissa together with an adjustment of the exponent. 1979 Sci. Amer. Dec. 87/3 In most computers the decimal point in the mantissa of a floating-point number is by convention placed at the far left, so that the number 3.24 ? 10^6 would be represented in floating-point form as .324 ? 10^7, or the pair of numbers 7, 324. 1985 Pract. Computing Aug. 102/4 Single-precision variables use a three-byte mantissa and a one-byte exponent.

---

OK, thanks. Yes, you are mostly right here -- though outside computing circles I have only heard mantissa being used to mean the fractional part of a logarithm. It is certainly in current use in that sense here in the UK. (Even the OED has no post-date since Darwin for the older meaning.)

I don't think the OED takes post-dating very seriously unless they think a word is obsolete (or changes in meaning/usage); otherwise they don't try to cite the very latest examples of usage, which would be a never-ending endeavor. I don't think the logarithmic usage is completely obsolete—you can still find examples on the web as well, and there are plenty of people with living memory of logarithmic tables. But the search results do reinforce the impression that it has greatly decreased in popular usage. Steven G. Johnson

But, in truth, I don't like the term significand; in my papers & documents I use coefficient. Another good, though less specific, term is fraction, though the latter implies something about the exponent which may or may not be true. (Nowadays it is realised that describing the coefficient as an integer is simpler and cleaner, and this is the way it is described in the current draft of the IEEE 754 revision (see: 754r.pdf (http://754r.ucbtest.org/)).

'Significand' is indeed probably a newer name for the term; it is the name the IEEE 754 committee chose (and perhaps invented), circa 1981 at a guess, though Kahan and Ris avoided the use of 'mantissa' in 1976). It is also the usual term used in the current IEEE 754 committee deliberations and draft. It is certainly true that the members of that committee and people working in this area consider the use of 'mantissa' to be "just plain wrong", and correct their students and others if they should use that term.

I've also done a quick scan of the papers at Arith16 (IEEE Conference on computer Arithmetic); there, 'significand' appears more often than 'mantissa'. I only remember one speaker at the conference saying 'mantissa', and she quickly corrected herself.

In short; floating-point in computing almost invariably means IEEE 754 floating-point -- so it makes sense to use the terminology in that standard in a primary reference.

So .. it is fine for Wikipedia to mention mantissa, and there should perhaps even be an entry for it, but I also think it should warn the reader that in some circles at least it is considered 'wrong'. Otherwise the reader will be misled into using the term in an inappropriate context.

See the current revision.

I'll leave it as-is for now; off on vacation in a few hours...

mfc

---

Latest edits: nicely done; thanks. Mike mfc

---

I have seen significand used as described here and mantissa used for the fractional part of the significand of a normalized binary floating point number. In other words, for the number

<math>1.0010011001 \times 2^{21}<math>

the significand is

1.0010011001

and the mantissa is

.0010011001

I don't recall where I read this, but it makes some sense. It is certainly useful to have a separate term for what I call a mantissa here, since that is how many floating point numbers are stored. CyborgTosser 06:18, 13 Jul 2004 (UTC)

In the IEEE 754 representation these are one and the same thing; the integer part may be either 0 or 1, but is in either case implied by some other part of the encoding. mfc

"The" established mathematical meaning

I don't see why Mfc feels the need to push the older meaning of mantissa as "the" established mathematical meaning of "mantissa", as opposed to simply calling it "a previous usage". This not just a stylistic question — the latter is clearly true, while the former is clearly false: as demonstrated by the citations above, the floating-point usage of "mantissa" has been well established in professional literature for 30 years, and this usage is clearly mathematical (unless you think numerical computing and numerical analysis are somehow not "math"). My version is factually correct, points out the common usages, and points out the preference of the official standard and some prominent professionals. What is your problem with it?

—Steven G. Johnson 18:33, Jan 28, 2005 (UTC)

By the way, your dismissive comment that the fp meaning of "mantissa" is only used in the "rather smaller field of computing" is not supported by the fact that a literature search above (including science, engineering, and math journals) turns up far far more fp usages of mantissa than logarithmic usages. Moreover, it's a pointless point since the logarithmic usage was arguably also used only in the "rather smaller field of computing", since its main use was for for computing things with logarithms. You're hardly likely to find the word used at all in more abstract mathematics from any time period. —Steven G. Johnson 18:44, Jan 28, 2005 (UTC)

I'm willing to compromise; how about calling it "the traditional mathematical meaning"? —Steven G. Johnson 18:51, Jan 28, 2005 (UTC)]

Calling it a "previous usage" is incorrect. It remains the normal usage in that context. Your proposed "traditional" has no advantage over "established", and some disadvantages. Gene Nygaard 18:54, 28 Jan 2005 (UTC)

That's not what "previous" means. If I say "I wrote an book on foo, and a previous book on the subject said bar," I'm not saying that the older book has ceased to exist. I just mean that it precedes mine (look up "previous" in the dictionary). However, I suppose it could have the connotation you don't like, so how about "a pre-existing usage" instead? —Steven G. Johnson 03:13, Jan 29, 2005 (UTC)

BTW, it looks to me that it is the other adjective over which you two are really at loggerheads: mathematical. Gene Nygaard 18:59, 28 Jan 2005 (UTC)

What I started to say yesterday, then deleted after I posted it because I had gotten myself all mixed up in trying to express it, was along these lines. What this article needs is a better explanation of the way in which the two meanings of mantissa are so very similar, before it tries to distinguish the two meanings. mfc mentioned this above on the talk, but not as clearly as it should be done in the article itself; you should get together on wording to make that clear in the article. Gene Nygaard 19:21, 28 Jan 2005 (UTC)

I agree. As I mentioned above, the older mantissa is essentially the logarithm of the significant. I actually wrote this in the article at some point, but it was deleted. —Steven G. Johnson 03:13, Jan 29, 2005 (UTC)

• (Have reverted the text to the original of a few days ago, not from spite but because it was starting to spill, and I have been hors de combat for the the last couple of (4) days with the 'flu, so haven't had a chance to follow the discussion here. Apologies! Will read this properly in the morning and try and answer/propose a compromise.

Mike (mfc 20:48, 2005 Jan 29 (UTC))

---

Well, I've re-read the above and the original article, and for reference, here's the text of the note in question (note that is it is just a note :-)):

The word initially used, in computing, for the significand was often mantissa (see Burks et al., below). This usage of mantissa, while still common, is discouraged by the IEEE floating-point standard committee and professionals such as William Kahan and Don Knuth. It conflicts with the established mathematical use of the term of mantissa for the fractional part of a number or logarithm (17thC, from the Latin for makeweight), although this older meaning has grown less common with the disappearance of logarithmic tables in favor of computers (see common logarithm for more on the older meaning).

I think the problem here is that there's a jargon meaning in computing and there's the usual meaning of the word, and the note doesn't quite make that distinction properly. First some comments on the two meanings:

1. The jargon meaning in computing, if one follows the literature, seems to have been an accidental misuse at first which then got picked up as a useful 'handle' to name the oject in question (the fractional part of a biary floating-point number). Knuth, Kahan, and others have pointed out that this is a confusing usage for anyone who has to use it in the other sense too, but more recently it has also become apparent that it is particularly inappropriate when a floating-point number is being described as a pair of integers (as is common for decimal floating-point, and is almost a necessity for any arbitrary-precision floating-point such as that in Java).

Lots of English usages are by extending older words via analogy (and in this case there is a clear analogy) or even (gasp) by misuse. Get over it...it's in the language now. —Steven G. Johnson

1. The mathematical meaning is still in everyday use; when logarithms are taught in school, 'mantissa' is the word used for the fractional part. A Search on Google for 'mantissa logarithm' finds thousands of current uses of the word mantissa in that sense/context (and Google Scholar similarly finds many recent examples in the more specialist literature).

I just tried searching Google scholar, and I'm having a hard time finding many uses of "mantissa" in the older sense. I found one from 1967, and one in a historical quotation from a 1992 article, and pages upon pages of results for the floating-point usage.

Did you search on 'mantissa logarithm'? 8 of the first 10 results are the 'older' use...

I just tried that exact search on Google Scholar and got the exact opposite results. 8 of the first 10 "mantissa logarithm" results are in the floating point sense (i.e. number = mantissa * base^exponent), while only 2 of the 10 were in the older sense. One of those two was from 1952, and the other was describing a 1973 paper (and both of those two papers were about electronic computing). —Steven G. Johnson 02:29, Feb 4, 2005 (UTC)

mfc

(The question of continuing current usage of the older sense is more one of curiosity for me; it's a side issue to my objections to the factual accuracy of your wording.) In any case, see below.

So we need to craft the note to reflect these uses (and warn the reader of the related concerns). How's this for a new version? (Links and emphasis to be added back in later.)

(If I personally were to warn someone, I would warn them against using the word "mantissa" at all in mathematics. If you use it in the fp sense, you are going against Kahan and the IEEE. If you use it in the log sense, you are going to confuse people because the fp sense now seems to be more widespread.

That's just not true. 'mantissa' is the word that (all?) math teachers use when describing logarithms at school. Anyone who learns about logarithms at school will know this usage of the word, and few, if any, will have heard the computing jargon use. Just to test this, I called the math teacher at a local school here. He used the term in a class last week (in the logarithm context), and was completely unaware of the computing usage. mfc

Every objective measure (e.g. literature searches) so far shows that the computing sense is far more common in new writings. As for your anecdotal claim that "all" students learn the older mantissa sense, I can just as easily (and truthfully) anecdotally say that I never heard it in school myself. (Which is not surprising, since was mainly useful to distinguish the mantissa in conjunction with logarithmic tables.) —Steven G. Johnson

But this kind of recommendation is POV and doesn't belong in Wikipedia (yes, I agree, I meant 'alert' not 'warn' mfc) ...we should just state the two usages, their historical order, and the viewpoints of the relevant authorities. —Steven G. Johnson 00:11, Feb 2, 2005 (UTC))

Isn't that exactly what this proposed wording does?:

No, because it states as fact the POV that the log usage is "the" (please note "the", which implies "the only") established usage. —Steven G. Johnson

That's the current text in the article. My proposed text is right here (and has been here for 3 days). The word 'established' does not appear in it! I have just added <hr>s to highlight it.

Can we please agree on it so we can do something productive...? :-)

Sorry, I forgot which proposed wording you were referring to. As I said below, the older meaning was also used only in computing, albeit the pre-electronic version, so I don't like the distinction you are making.

---

Note: The word used in computing for the significand is often 'mantissa' (see Burks et al., below). This usage of mantissa is discouraged by the IEEE floating-point standard committee and by professionals such as William Kahan and Don Knuth because it implies that the significand is a fraction (which it need not be) or a logarithm (which it never is). Outside computing, the term mantissa is used for the fractional part of a number or logarithm (17thC, from the Latin for makeweight) when logarithms are taught and in other mathematical uses (see common logarithm for more on this other meaning).

mfc 16:10, 2005 Jan 31 (UTC)

---

The problem I have is "outside computing"...you don't use the mantissa in the older sense except for computing too (albeit in the pre-electronic sense of computing). Your perspective (as you have expressed it in the change comments as well as above) seems to be that the older "mantissa" is somehow used in a larger subset of mathematics than the "narrow" realm of computing that uses the newer sense, and that just doesn't seem to be true.

Well, in my experience it most certainly is...

Give me an example of a log usage of mantissa where they are not involved in a numerical computation. —Steven G. Johnson

(Besides the fact that the number of modern citations that use the new sense seems to be vastly greater than the number of citations for the old sense on every literature database I've tried.) —Steven G. Johnson 00:02, Feb 2, 2005 (UTC)

Why can't we just take the current page and replace your "the established meaning" with the neutral "a pre-existing meaning"? What's not to like? —Steven G. Johnson 00:02, Feb 2, 2005 (UTC)

Just the fact that you won't admit that it is "established", I guess.

For the umpteenth time, I agree that the older meaning is established, but not that it is the established meaning. The newer meaning has been well-established in usage too. It's a little frustrating that you keep misrepresenting me. —Steven G. Johnson

1. There is no dispute that the meaning of mantissa as it is used with logarithms is correct.

Is a correct usage, yes.

1. There is no dispute that that is the older usage.

Agreed.

1. There is no dispute that that usage continues, something which you at least mislead people about in earlier edits, whether you intended to or not.

I said that it has become less common in the article. Which is true, and there seems to be no dispute about this. I never wrote in the article that no one uses the older meaning anymore (although I pointed out in my comments that it is virtually absent from recent literature searches...also true).

1. There is no dispute that the different meaning with respect to floating point is common.

Agreed.

1. This is dispute whether the usage in connection with floating point is correct. In other words, there are reliable authorities who say it should no longer be used this way. That's something we do not have with respect to the meaning used for logarithms.

We agree that that some professionals deprecate the usage, and the article reflects this. Where we apparently disagree is you think(?) that Wikipedia should also push the POV that the fp usage is "incorrect", beyond stating the opinions of Kahan et al.

1. So, in at least some definitions of the word "established", the meaning in connection with floating point is not "established". It is either considered incorrect, or it remains in dispute.

"Established" could also be understood as "established in common usage", not "accepted by an official standards body". Why push this when "pre-existing" is unambiguously correct and doesn't push a POV on the issue? —Steven G. Johnson

1. That's the very reason the word "significand" exists—to step around that problem. Gene Nygaard 05:15, 2 Feb 2005 (UTC)

As we said, there's no dispute that some prominent professionals don't like the fp usage of mantissa, and the article reflects this. However, standards bodies can't dictate the English language, and Wikipedia should not imply that they can. —Steven G. Johnson 21:34, Feb 2, 2005 (UTC)

---

Fractional coefficient

The article calls 1.2345 a "fractional coefficient". But a (proper) fractional coefficient would be 0.12345. I realize that normalized IEEE floats use coefficients in the range 1 ≤ c < 2, but not all floating-point systems do (many older ones have 1/2 ≤ c < 1), and it is confusing to call 1.2345 a "fraction". We could either we replace or add the representation 0.12345 × 10⁺³ to the list. Thoughts? --Macrakis 16:53, 17 Apr 2005 (UTC)