Talk:Significand

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)

IBM consistently uses fraction in its description of S/360 and successors floating point formats. Characteristic is used to describe the biased exponent for hex floating point, just biased exponent for binary floating point. For binary (IEEE) floating point, the hidden one comes before the implied binary point such that the stored bits do represent a fraction. (At least it looks like that is the way IBM is describing it.) A large fraction of computer literature over the years has been written by IBM or about IBM systems. A google search for "fraction" probably won't be useful. Gah4 (talk) 22:32, 15 October 2009 (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.

If I remember my logs right, the mantissa of log(0.025) would be .398 and the characteristic would be -2, written as a bar over the two. Gah4 (talk) 12:46, 21 November 2008 (UTC)

'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)

I have recently noticed that the numbers that Google gives at the beginning of a search can be very wrong. I had a search that came up with a large number, but then clicking to the second page of hits, found that there were only two pages! It didn't do the full calculation until then. I believe it is well known not to trust them, but this is the first time I found out how far off they can be. Gah4 (talk) 14:36, 7 July 2020 (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.

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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).

'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

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Latest edits: nicely done; thanks. Mike mfc

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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

$1.0010011001\times 2^{21}$

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

Sorry to burst your bubble, but IEEE is not a universally accepted authority. If you are going to cite a publication along those lines as a reference, be sure to include the date. Dexter Nextnumber (talk) 02:55, 27 December 2009 (UTC)

"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))

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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.

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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)

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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)

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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)

Since IEEE-754 binary uses a hidden '1', the part actually stored is a fraction. The .2345 in your example. (Now that IEEE-754 includes decimal floating point, that doesn't apply.) As I noted above, IBM has consistently used "fraction" in describing S/360 style hexadecimal floating point where it is a fraction: 1/16 ≤ c < 1. I don't mind coefficient or fraction, but prefer significand. Gah4 (talk) 09:32, 4 March 2013 (UTC)

Some recent usage numbers

Since Gene requested, I did an updated literature search on the Compendex + INSPEC databases (which indexes most major computer-science and engineering journals, although it only searches title/abstract and not full text), and since 2000 there were 70 hits for "floating point AND mantissa" including three in 2005 (in IEEE Signal Proc. Lett., IEEE Trans. on Computers, and Computer Aided Design). If you do the same search for "floating point AND significand", I get 15 hits since 2000.

If you search on Google scholar for articles since 2000, I get 585 hits for "mantissa floating point" and 119 hits for "significand floating point".

If you read above, you'll also see a number of other search results that I gave a while ago, which all show basically the same trend.

I don't understand why anyone thinks this is controversial — it seems obvious that "mantissa" is still commonly used in its floating-point meaning in professional computing contexts. (If anything, the numbers suggest that "mantissa" remains more common than "significand" for describing the floating-point concept.) Yes, some prominent entities deprecate it, and our article states this, but Wikipedia should be descriptive, not prescriptive.

—Steven G. Johnson 03:54, 8 October 2005 (UTC)

Continuing the mantissa vs significand debate

I think there is absolutely no conflict between the definition of mantissa in terms of logarithems or in terms of floating points. A mantissa is originally "a very small addition", and in that way works for both cases (since a decimal can be considered small).

Not only that, in a logarithem, the mantissa is the part past the decimal - ie. it is NOT the significand. Note also that the mantissa is NOT the same as significand for floating points either. See mathworld's opinion.

In the case of binary numbers, the mantissa is the ONLY thing that is represented in the actual data, there is only one implicit digit to the left of the fixed point - and since that digit is always one it is "hidden" so that the number-representation can store more numbers. See floating point and read its section on the "hidden bit". In contrast, the significand includes this hidden 1, and thus is not a mantissa.

So to sum up:

• Mantissa is not the same as significand.
• Mantissa has the same definition for logs and floating points.

Fresheneesz 19:58, 25 May 2006 (UTC)

This is not how it is used in practice, as can be seen if you do a literature search (as I have done, repeatedly...see above). When practitioners use "mantissa" in a floating-point context, they mean the same thing as "significand", whereas the older version of mantissa is essentially the log of the signficand. Mathworld is giving the older definition of mantissa, not the recent floating-point usage. This argument was settled a long time ago to everyone's satisfaction; please, let's not restart it. —Steven G. Johnson 20:14, 25 May 2006 (UTC)

Alright, but let me get this straight. Significand has a different meanting in terms of floating point numbers, than it does with scientific notation? For example, in the number $4.598*10^{34}$, the significand is 4.598 correct? Same in binary, if you have a number $1.010111*2^{18}$ then the significand is 1.010111, correct? However, in floating point representation, the most significant bit of the significand it not used, and therefore the "fraction part" of the significand is the only thing written into the binary representation of the floating point number.

So is the mantissa 1.010111, or .010111 ? Fresheneesz 21:34, 25 May 2006 (UTC)

I also see that the word mantissa is used inconsistantly. For example, in this page, they describe "The combination of the integer bit and the fractional bits is called the mantissa (or significand).", but go on to say "The IEEE single-precision format has 24 bits of mantissa, 8 bits of exponent, and a sign bit.". Which is it? The IEEE format obviously doesn't contain the "hidden bit", but does, or does the mantissa not have that hidden bit in its definition. I think this term has be misused to the point of ridiculousness. Fresheneesz 21:40, 25 May 2006 (UTC)

Usage varies. Both "24-bit significand" (including the hidden bit, not stored) and "23-bit significand" (not including the hidden bit) seem to be widespread. Ditto for "mantissa". I'm sorry that you're unhappy with the way the English language has turned out, but there's not much we can do about it on Wikipedia. The best we can do (according to policy) is to describe the different meanings and the contexts of their use, and the recommendations of the standards committees, which is what we are doing now. —Steven G. Johnson 23:45, 25 May 2006 (UTC)

Alright, that seems reasonable. However, I think that the articles on floating point, significand, and IEEE floating-point standard do a poor job at explaining the different meanings and being clear as to what they mean. What was the consensus as to what we should call the 23-bit "fraction part" vs the 24-bit significand? Fresheneesz 02:31, 26 May 2006 (UTC)

I suspect that the use of log tables is a lost art by now. I barely remember learning them from my father, in about 8th grade, not so long before calculators became affordable, so I didn't need to use them. Maybe a little reminder will help. (I went to Common logarithm to be sure I remembered it right.) If you write a value in scientific notation with one digit before the decimal point, the log of the coefficient will be between zero and slightly less than one. This log value is then the mantissa. The power of 10, which might be negative, is the characteristic. (For a negative characteristic, a bar is drawn over it, as a minus sign would apply to the whole value.) The sum of the (possibly negative) characteristic and the positive mantissa, is then the logarithm of the original value. The mantissa isn't the fractional part of the logarithm in the case of a negative characteristic. (It is the 10's complement if anyone is counting.) The two parts are added, unlike the significand and characteristic (IBM name) of a floating point value. The mantissa is a small addition in the case of a large characteristic. That obviously doesn't apply for the significand of a floating point value, as it is multiplied not added. Gah4 (talk) 16:02, 7 July 2020 (UTC)

Recent edits by fresheneesz

Fresheneesz, please discuss proposed major changes to the article in Talk, especially as you seem to be new to this topic. I reverted your edit because it made the major change of suggesting that there are "two" definitions for the significand, depending upon whether you think of it as an integer or a fraction. I don't think that most practitioners would consider these to be separate "definitions" of the significand. The placement of the decimal point is arbitrary and could be chosen in many ways depending upon the desired range for a floating-point type (and, in fact, there are infinitely many possible choices and not just two). A subsection on notational conventions, such as whether one counts the hidden bit when talking about the significand or where the decimal point is placed, might be useful if it is properly sourced to reliable citations such as standards documents. —Steven G. Johnson 17:37, 26 May 2006 (UTC)

My edits weren't meant to in any way emphasize or change what this article says about treating the significand as an integer or fraction. As you can see from the history, I simply tried to make what the article already says about it, consistant with my changes.

My changes were meant to specify that there *are* multiple definitions of both "significand" and "mantissa". Something like my edits must be here, because the mantissa vs significand thing has been the subject of long debates, with no clear consensus as to what to do about it - except to note the different meanings. I attempted to do this.

This paragraph was in my edit:

"Depending on the interpretation of the exponent, the significand may be considered to be an integer or a fraction. In the case where the significand is considered to be an integer and definition 2 is used, the "integer" spoken of is made up of the digits that appear after the decimal point."

The bolded part is what I added. It was meant to make that "integer" thing consistant with definition 2 in my edit. However, it was badly explained. I have no problem removing that if thats the only thing you find wrong with my edit. Fresheneesz 18:17, 26 May 2006 (UTC)

There was a clear consensus, and it is reflected in the current form of the article. —Steven G. Johnson 18:36, 26 May 2006 (UTC)

(please don't interrupt peoples comments, its very hard to read) Fresheneesz 20:31, 27 May 2006 (UTC)

Also, theres a lot of discussion up there, and I can't seem to find a place where there was a clear consensus. Could you point me to the right spot to look? Also, past consensus, there *ARE* multiple definitions of "mantissa" *and* "significand". If you don't think those definitions belong on this page, where should they be? Fresheneesz 20:33, 27 May 2006 (UTC)

I put the definitions I had for mantissa on the page mantissa. But I will put a small note on the two uses of significand on this page. Fresheneesz 22:11, 27 May 2006 (UTC)

Fresheneeze, the page did already describe the different meanings for those words, so I'm not exactly sure what your objection was. As for the consensus, it was reflected in the article history as we eventually converged on a wording that everyone could agree was accurate, at which point the article stabilized.

I'm still not clear on the "two uses" of significand that you're referring to. The article already mentioned that it could be used for an integer or a fraction, but that this was just an arbitrary choice of exponent. You edited it to add the comment that the significand could refer to the "fractional part" of the significand. Was this in reference to including the hidden bit? If so, it was confusingly written, as no one would refer to e.g. 0.1234 as the "significand" of 56.1234 (unless you can provide a cite for this?). I've replaced your paragraph with a short section talking specifically about the hidden bit. I'm unclear as to whether that addresses your concern, however.

—Steven G. Johnson 23:12, 27 May 2006 (UTC)

It does address my concern thanks. I was refering to the hidden bit. My concern is that sometimes people say "significand" and it includes the hidden bit, and sometimes its meaning does not include the hidden bit. I would prefer having both definitions noted at the top, so that people don't get confused if, for example, they don't happen to read the section you just made. Fresheneesz 01:50, 28 May 2006 (UTC)

We probably shouldn't describe this as a difference in "meaning". It's simply a difference in how the width of the significand is described. —Steven G. Johnson 02:00, 28 May 2006 (UTC)

No, that is not the only difference. 1.2523 is fundamentally a different thing than .2523. Obviously their "widths" are different - but thats not the only difference. This is like taking a 5 meter cube, and saying its the same thing as a 2 inch triangle, except that their widths are different. Fresheneesz 10:41, 28 May 2006 (UTC)

Note that, to interpret your example in the sense of a hidden bit, the significand in both cases is 1.2523; whether the 1 is physically stored or not (as with a hidden bit) doesn't change the semantic/logical interpretation of what the significand "is", it's just a difference in storage formats. —Steven G. Johnson (talk) 22:05, 8 February 2008 (UTC)

significand: unfortunate name

In the last few sentences of the article it says: it is not etymologically correct since it means "that which is to be signified" rather than that which is significant.

What would right/better word be instead of significand? —Preceding unsigned comment added by 212.235.186.231 (talk) 10:55, 9 October 2009 (UTC)

Significand is not the same thing as significant. One is derived from the Latin signum and the Latin suffix for the gerundive while the other, significant, is derived from English.

This actually relates to the way you build your floating point number. Let's ignore IBM and IEEE for a moment, even though some people wish to cite them as an authority. If you go back 30 years, there was considerable debate as to how a computer should manage its floating point numbers. RAM was severely limited back then. People argued over 3 byte floating point, 4 byte floating point, and 5 byte floating point. Commodore Business Machines decided on a 5 byte floating point. Some people advocated 8, 10, or 12 byte floating point. Everybody had their own idea how many bytes to dedicate to floating point. But which of those bytes would be used for signifying the 'sign'? And should it be the top of the byte or the bottom of the byte? Bit 7 or Bit 0? Where would you put your sign bit? The 'sign' for positive or negative numbers is usually represented by a single bit, and it is often convenient to isolate the exponent in a single byte, and put the remainder (and the sign bit) in the mantissa. Other computer manufacturers decided to do it differently. I am not surprising any of you by noting how the Latin maneo ("stay/remain") gives rise to mantissa as the bits that stay there with the least perturbation, as you perform operation upon operation. I do digress much by noting how the English word for "mansion" came from the Latin verb maneo. A mansion was not a particularly resplendent place to sojourn, it was just a place where you stayed for a while, as wayfarers usually did. Significand is the part of the floating point number where you find the sign bit. The first part is the mantissa, the other part is the exponent. One or the other should have room for you to store your sign bit. Dexter Nextnumber (talk) 22:56, 26 December 2009 (UTC) Dexter Nextnumber (talk) 23:02, 26 December 2009 (UTC)

By the way, I decided to be bold and delete the part that says it's "unfortunate." Dexter Nextnumber (talk) 06:32, 27 December 2009 (UTC)

Sign is included in the significand/mantissa or a separate bit?

Discussions of "significand", which belong more to the literature on floating point, tend to use positive m and a sign bit as two separate data, so m is always positive. I'm not sure if this is as much the case in the non-floating point or STEM literature that primarily uses "mantissa" for m, but in both settings there is a tendency to treat m as implicitly or explicitly positive. But the Wikipedia articles on significand, scientific notation and related topics incorporate the sign into m, treating it as a signed number. What are the predominant conventions on this? (Here m denotes the multiplier of a power of 10 in some representation of numbers as m x 10^n)73.89.25.252 (talk) 05:51, 18 December 2020 (UTC)

To me, this article is more for the computer representation, and scientific notation more for printed or hand calculator usage. I don't see in scientific notation mention that the sign should or should not be part of it. Computer storage representations commonly put the sign first, then exponent, then significand, which tends to keep them separate. Some DSP chips use a format with a two's complement significand, in which case it is harder to separate the sign. The PDP-10 uses a format where the whole word is two's complemented, so it is less obvious that the sign applies just to the significand. Personally, except for the DSP formats, I would consider them separate. Gah4 (talk) 09:09, 18 December 2020 (UTC)

In the IEEE 754 standard (1985, 2008, 2019), the significand is always unsigned (nonnegative). In our Handbook of Floating-Point Arithmetic, this is not consistent (definitions would have to be fixed / clarified), and I believe that in the literature on floating-point arithmetic, it depends. There are contexts where it is more practical to include the sign in the significand, e.g. in proofs. But I think that it should be clear that the default definition, at least for floating point, should follow the IEEE 754 standard; perhaps say signed significand (at least once) when the sign is included. — Vincent Lefèvre (talk) 09:17, 18 December 2020 (UTC)

Definition and proofs related to rounding schemes, or any operation that should commute with change of sign, can be simpler with the sign kept separate. i.e. the operation is defined on positive numbers and extended to the general case by applying the sign to the result, so that it commutes by construction. Otherwise there is a non-uniform definition with one method for positive numbers and another for negative, and the result requiring a proof of compatibility with sign.

After the functions and conventions are developed to the point that they smoothly accomodate both signs of m, the non-uniformity is hidden in the notation and proofs can ignore it.

For scientific notation (I came here as a result of editing the article) the exposition is easier if one does not have to talk in every second sentence about "the absolute value of" the mantissa. 73.89.25.252 (talk) 16:10, 18 December 2020 (UTC)

In the case of two's complement significand, rare but they do exist, it isn't so easy to separate them. For the PDP-10 whole word two's complement, where there could be carry into the exponent, most obvious is that the system uncomplements before operating on it. Note that the latter has the convenience that integer compare instructions also work on floating point. Gah4 (talk) 00:17, 20 December 2020 (UTC)