Talk:Significant figures

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[edit] Major Edit Proposal

At present, Wikipedia has no article which defines significant figures as a way of representing numbers. Rounding doesn't do this - it links to here. It has two articles (this and Significance arithmetic which deal with the problems involved in using significant figures as a way of representing errors. These articles pretty much duplicate each other. So we have 2 articles for one topic, and none for another... I suggest that this article should define 'representation to n significant figures'. It should mention their use as a crude form of error representation and link to Significance arithmetic. Please respond with feedback on this idea. User:Winterstein 2006 Nov 10th

OK - no responses, so I'll go ahead and do the edit. User:Winterstein 2006 Nov 10th

[edit] Merging

The article Significant figure is being merged into this one. As the redirect command appears to have wiped the rest of the article from this page's current revision, use the second-latest revision (as of now Dominus's version, from 10:29 2004 Feb 04).

There was a discussion about the other article at Talk: Significant figure. While the comments are rather old, some of them may still be pertinent.

Jxg 20:35, 2004 Mar 21 (UTC)

I did the merging, but because the other version of the sig figs page was very different from the system I'm used to, I'm not sure the synthesis page is that great. Comments/ideas/discussion would be apprectiated. --Alex S 03:46, 23 Mar 2004 (UTC)

Must be merged with significance arithmetic.

[edit] Merging with Rounding

Bad idea .. Rounding is a well-defined concept. Significant Figures is full of poor science and rules-of-thumb which are very much in dispute. quota

Merging with rounding is indeed a bad idea. Rounding is used in many contexts in which conveying a number of "significant figures" of measurement error is not the intended purpose. For example, rounding is used in fixed-point arithmetic to avoid an excess number of bits when performing computer calculations. It is also used when printing the result of computations using floating-point arithmetic to fit the result into a reasonable amount of text with no bearing on how many digits should actually be considered "significant". The definition of rounding here could use some improvement, but it is a concept that applies outside of the concept of significant figures. --Cat5nap 04:24, 29 September 2005 (UTC)

No-merge:- I agree that merging is bad idea. The two articles treat different enough things that they deserve separate articles. Oleg Alexandrov 07:02, 29 September 2005 (UTC)
No-merge:- Indeed, merging with "rounding" is not the brightest idea. Significant figures are notably different from rounding. Kilo-Lima 12:03, 9 October 2005 (UTC)
No-merge:- I am very against merging with rounding. Rounding only applies to decimal places, significant figures applies to units and above as well. Ukdragon37 12:05, 9 October 2005 (UTC)
Merge: There is another article - Significance arithmetic which deals with performing arithmetic with significant figures. This article should be short and sharp - just a paragraph or

two. At that point it makes sense to merge it into Rounding. User:Winterstein

[edit] Leave Rounding as an Unmerged Article

Rounding is a concept in it's own right. Sure it results in less significant digits, but that is a side subject. To merge it would result in people wanting to learn one thing having to wade through other material. Just make proper references. MathsIsFun

Don't merge: The two concepts may be related but they are distinct. There are other reasons for counting sig. figs than for rounding purposes. There are other ways of rounding than considering the sig. figs. Jimp 13Oct05

[edit] Remove Merge Tag

The votes seem to be against any merge with Rounding. Is it time to remove the tags yet? Jimp 13Oct05

[edit] Yes, done

Open2universe 11:32, 14 October 2005 (UTC)


I'm going to try moving the stuff in this article about rounding into Rounding Thincat 16:13, 15 November 2005 (UTC)

[edit] Splitting

I think it would be good to split this article into 'significant figures' -- the concept and measurement rules, which are widely used, and 'significance arithmetic', which is worthy of note but needs much more careful treatment as in almost every case it is best avoided. Any objections? quota 11:13, 26 Nov 2004 (UTC)

I added a few caveats about the "rules of thumb." Pakaran (ark a pan) 05:09, 3 Dec 2004 (UTC)

[edit] Significant figures and exponentiation and roots

Taking roots can gain significant figures; raising to powers (if exponent is greater than 1) can lose significant figures.

Is it not that one maintains the least number of significant figures of an element used in a calculation. Such that in the cases you have described above, the sig figs would remain the same? Seeaxid

[edit] Showing significant zeroes to the left of the decimal point

I was taught that a decimal point could be used to notate significant zeroes to the left the decimal point, IE 2000 is one sig fig, but 2000. is four; 193,000 is three but 193,000. is six. The article almost seems to touch on this ("In order to express the degree of precision to which a value was measured, decimals are used."), but would more information on the subject be helpful? Or first, has anyone else heard of this notation? Thanks. - Masmith 8/24/04

I've never heard of that, but I suppose it deserves a mention. Seeaxid 6 July 2005 15:14 (UTC)
Neither have I, can it be that this is not an international practice?

Here's how I'd read and write 2000 correct to various numbers of significant figures (sf):

  • 1sf: 2.\times10^3 or 2000\pm500 (or 2000 according to Masmith, but I disagree)
  • 2sf: 2.0\times10^3 or 2000\pm50
  • 3sf: 2.00\times10^3 or 2000\pm5
  • 4sf: 2.000\times10^3 or 2000\pm0.5 or 2000.
  • 5sf: 2.0000\times10^3 or 2000\pm0.05 or 2000.0
  • 6sf: 2.00000\times10^3 or 2000\pm0.005 or 2000.00
  • etc.
  • \inftysf: 2000 (i.e. exact integer value)

So, I think it's natural to read "2000." as 4sf, but we still need scientific notation or \pm e.g. for 2sf. Unlike Masmith, I wouldn't read "2000" as 1sf, but as an exact integer. Is there a general consensus on this? --Niels Ø 07:42, 28 April 2006 (UTC)

I think there are different 'camps' of viewpoint, here (but I'm in yours) quota 11:23, 28 April 2006 (UTC)

[edit] Computation with significant figures

[I have moved the following text to here because it is already covered in a different article (significance arithmetic) .. however that one needs some work, so some of this text would be good over there! quota]


It should be noted that when adding two numbers with different numbers of significant figures, the sum is reported using the smallest number of decimal places. Multiplication and division are sublty different: the resulting answer contains the smallest number of significant figures between the operators, and not the number of decimal places. Adjust the answer based on the measurements' number of decimal places or significant figures, ignore any constants that are known exactly.

For example:

1.234 + 9.7 = 10.934 => 10.9 (round to one decimal place)
1000 * 5.0 = 5000 => 5.0e3 (two significant figures)
K = 2
1.001 * 5.0 * K = 10.010 => 1.0e1 (two figures, ignore one siginificant figure of K)

Note that the above rules don't cover trailing zeroes which are placeholders. They are often considered undefined, simply because we can't tell if they are placeholders unless the number is reported in scientific notation. Scientific notation provides a convenient way of communicating precision and helps avoid errors because zeroes are hard to count when there are many of them.

[edit] Rounding

This article is redirected from rounding, but has less on rounding than I really wanted to find. In scientific circles, 1.678 would always round to 1.68 (correct?)

Depends on the rounding mode. Yes, if 'round to nearest'. quota

but what should 1.675 be rounded to?

1.68 (default) -- rounds to nearest even digit. quota

Also, if it is really about rounding it should mention that accounting rules may be different (e.g. requiring consistent rounding up or down). Notinasnaid 08:58, 19 May 2005 (UTC)

Indeed, Rounding is a topic in itself -- should not really be redirected here. See also IEEE 754]. quota

I may a topic on rounding in a second (or so ...). Seeaxid 6 July 2005 15:14 (UTC)
actually, on further thought, rounding is a particularly small topic ... Seeaxid 6 July 2005 15:53 (UTC)
Okay, I've created a section for it, but I think it still need some examples, and perhaps langauge refinement. Seeaxid 6 July 2005 16:05 (UTC)

You may want to define or explain the term "reportable digit", which seems to not be referenced prior to its own "defining" use in this section. (Karl Kaiser)

[edit] Counted sig figs

I've reworked this section, had added some things, I may have lost some information in while.

Anyway, overall I think the may need to be reworked so as to be more understandable. Seeaxid 6 July 2005 15:14 (UTC)

[edit] Measuring with significant figures

"In order to express the degree of precision to which a value was measured, decimal numerals are used. When using significant figures rules, it should be assumed that the last significant digit of every measurement was estimated. Using the previous example, if the observer read the amount of liquid in the cylinder to be exactly at the 12 ml mark, the observer would write the value as 12.0 ml, which would indicate that the tenths place was the precision obtained, and the 0 was estimated. If the cylinder were marked off to every tenth of a ml, the observer would write the value as 12.00 ml."

Are we sure about this? What if, say, the uncertainty of said instrument was marked as ±1 cm^3. I would think the limit of the reading would be the degree of the significant figures, although I'm not too sure. Seeaxid 6 July 2005 15:39 (UTC)

No, if you specify the uncertaintly then you don't have to fall back on counting the significant figures as a means of guessing it.

I don't get the quoted passage above. Isn't it standard to judge a measurement to the closest marking on the scale? For instance, if I desire to measure as precisely as I can to the nearest millimetre, I rely upon the marks on the ruler's scale that show millimetres. I don't use a ruler that only shows centimetres and then try to guess the millimetre part. Also, when reporting the uncertainty range, using my ruler would give me for example a measurement of 14 ± 0.5 mm. (± 0.5 mm because a measurement simply stated as 14 mm can arise from an actual distance anywhere from 13.5 up to 14.5, but was rounded to the nearest millimetre mark on the ruler, 14.) How would I report the uncertainty if I am eyeballing 1/10 of the precision showing on the scale of the measuring device? Because I would have to rely upon my eyeball to be as precise as 1/10 of the distance between the closest markings on the scale, my personal uncertainty figure could be different from someone else's, depending on how relatively good at it we are.

[edit] One drawback of significant figures

Suppose I tell you that something were 936 metres long and you wanted that in yards. What would your conversion be? According to a strick application of the significant figure rule of thumb it would be 1020 yards, right? However, wouldn't 1024 yards be the better answer? This doesn't involve (much) false precision. "936 m" can be taken to mean "936 ± 0.5 m" whereas "1024 yards" can be taken to mean "1024 ± 0.5 yards". "1020 yards", on the other hand, would be taken to mean "1020 ± 5 yards" whereas the original measurement had been far more precise than this. Jimp 12Oct05

That's an issue that comes up sometimes. The reason is that a number with a low initial digit actually has less "accuracy" in its last digit. Consider that the difference between 10 and 11 is 9%, but the difference between 98 and 99 is only a tad over 1%. Pakaran 18:56, 12 October 2005 (UTC)
Which is why the whole concept of 'significant figures', as an arithmetic, is flawed. It is definitely valid and useful for a single measurement .. as it indicates the error bounds for that measurement. But it is not valid when arithmetic (such as conversions .. a multiplication!) is done on the grounds of 'significant arithmetic'. quota

Perhaps a short warning to this effect could be useful in the article. Oh, and things are worse going the other way of course. 1024 yards should be converted not to 936.3 metres, indicating accuracy to the nearest 100 mm, but to 936 metres. The former would introduce substantial false precision. Jimp 13Oct05

[edit] Sig figs with math?

I've noticed that computing sig figs with the basic math functions (+, -, *, /) is not mentioned. Is there a single set of rules that can be used and added to the article that would be useful? I'm familiar with the rules as I was taught them, but after reading the article, I'm not sure if everyone is even taught the same way. JumboBrian 3 February 2006

That's covered in significance arithmetic (see above in Talk). quota 17:02, 3 February 2006 (UTC)

[edit] Sig figs are informal and this article has no references

Sig figs, as a concept, was invented by textbook writers as a way of getting around writing about error analysis and error measurement. While the current article is correct in its introductory explication (sig figs are a "rule of thumb" of sorts), it should be much more clear that in the profession of measurement (at NIST, for example) sig figs are not referenced. Rather, what determines the precision and accuracy of the measurement is the associated error. This removes all ambiguity which is present in the arbitrary rules of sig figs.

Since this article has no references (though practically ever intro science text has a section on sig figs, much to my chagrin) I feel that its explication of sig figs may wander a bit toward original research. Unfortunately, many science texts themselves don't rely on standard practices from measurement when discussing this subject. It's really one that receives way too much attention.

--ScienceApologist 09:26, 4 February 2006 (UTC)

Indeed, yes. Some of us have been revisting this article over time and gradually cleaning it up. (Go back to some of the early versions in the history, and be horrified :-))

Next, please improve the article and add references! quota

[edit] Large additions by Patdw

I have reverted the additions by Patdw for two reasons:

  1. This looks like the same text that was claimed to be copyright and was inserted and removed a short while ago
  2. It duplicates the content that's already in the text of the article, and without pointing out which parts are hard-and-fast rules and which vary from person to person (e.g., whether 3210 is 3 or four significant digits)

A few more examples would be good, indeed, but why not put them at the bottom of the article? quota 15:32, 1 March 2006 (UTC)

[edit] Pitfalls of Floating point, etc.

I have removed this paragraph because it is deals with arithmetic on numbers (which may or may not be correct to however many significant figures are in the operands). This is quite orthogonal to the definition of significant figures which is what this article is about.

There is an article on significance arithmetic, so some of that material would be appropriate there, but I think it is mostly covered there already.

The two articles were split some time ago to make clear the distinction between significant figures (reasonably clear and uncontroversial) and significance arithmetic (a different kettle of fish).

Thanks. quota 10:44, 15 June 2006 (UTC)

I previously reinstated my submission about Pitfalls etc. because it is needed to redress several generations of neglect of the matter. If anything, because of that neglect, the question of its arithmetic is even more important to the student than the definition itself.

(Which is why it merits its own article, which already exists.) quota

In fact, I disagree fundamentally with the definition of significant figures proffered in the article, for the following reasons: The article states: "The concept of significant figures originated from measuring a value and then estimating one degree below the limit of the reading; for example, if an object, measured with a ruler marked in millimeters, is known to be between six and seven millimeters and can be seen to be approximately 2/3 of the way between them, an acceptable measurement for it could be 6.6 mm or 6.7 mm." I would maintain that there is a critical difference between measuring and estimating. A measurement is a figure that is consistently repeatable and agreed by all observers. An estimate is debatable by definition. If a student attempts to supplement his measurement with an estimate of an additional so-called "sig.fig." he/she is simply exceeding the accuracy of his/her measuring instrument. That instrument was calibrated to a certain accuracy from an entirely arbitrary base, which, suppose it had been 0.11111 longer, would have resulted in a valid exact 6mm with a single sig.fig. My point is that no figure can be regarded as significant if the measurement is not consistently repeatable by all observers. Thus the correct measurement of the object exemplified is 7mm, as that calibration mark lies closest to the edge measured, and all observers will agree that that is the case. (That implies that in reality it could lie anywhere between 6.5 and 7.5, although we openly admit we cannot be certain.)

That's all good material. There's a whole discussion, there, worth having. Many rulers (measuring bars) have a 'half-way' point marked between (say) a 0.1 and an 0.2 mark (i.e., 0.15 is marked in this case). Skilled users can usually, with that shown, estimate to give that extra significant figure. All this is definitely with the scope of the article, I would think. quota 14:35, 17 June 2006 (UTC)
Perhaps the midpoint graduations should be considered limits of confidence in the primary graduations. Determination of the least size significant figure should not depend upon the skill of the observer as practical measurements should be equally accessible to everyone using the same tools. Estimating an additional figure, and calling it significant, is no substitute for investment in better measuring equipment that can give generally repeatable results at a more refined level of significance.Geologician 16:24, 17 June 2006 (UTC)

Returning to the question of where the Pitfalls should be outlined, no measurement exists in isolation from other quantities. Thus its relationship to those other quantities is an important parameter in its definition and its proper place is as part of the definition.

Agreed, but that has little or nothing to do with some calculation that might be done on such numbers. The calculations that can be done are quite well defined (and unfortunately the 'rules of significance arithmetic' are very approximate for one calculation and badly wrong if the result is carried forward for another calculation). But I suggest the place to discuss/describe that is in the article that is specifically about those rules (significance arithmetic). quota

I shall await the reply of 'Quota' to this argument before reinstating Pitfalls as before in its rightful and helpful place. I should also be interested in the opinions of others who have experience in this field. Geologician 12:42, 17 June 2006 (UTC)

It's helpful, but not here. Inevitably if here the section would have to pull in all the material from significance arithmetic. That's not helpful to anyone who is trying to understand the concepts of measurement, etc., which you rightly point out are complex and subtle. Let's get the concept of significant digits well-defined, and then one can discuss arithmetic on numbers separately? quota 14:35, 17 June 2006 (UTC)
I'd like to hear other views on the advantages of separation of these concepts, that seem to me to be closely interlinked. There are many precedents in other parts of Wiki for lumping rather than splitting. (Even at the cost of paraphrasing or repetition). Geologician 16:24, 17 June 2006 (UTC)

Let me observe that uncertainties can be expressed in several ways, e.g.

  • "x=31.7 with 1 sf" (meaning perhaps 31.65 \le x \le 31.75),
  • double inequalities (like in the previous parenthesis),
  • "\overline{x}=31.7 with σx=0.05" (indicating mean and standard deviation - so in this case, the error could in fact be somewhat larger),
  • specification of a probability distribution for x,
  • "x=31.7±0.05" (sometimes meaning the same as the double inequality above, sometimes σ=0.05, sometimes 2σ=0.05).

Using any of these methods except sf, it is possible to indicate any degree of uncertainty; with sf, only \pm 5 \times 10^{n} where n is an integer can be specified. With some measuring equipment, the experimental uncertainty is a magnitude of this type; with other equipment, it is not. Discussing how to use significant digits to represent the uncertainty in such cases really is a waste of time.

Exactly what implications these observations should have for the articles is beyond me.--Niels Ø 08:50, 28 September 2006 (UTC)

[edit] Specific cases

How should the following special cases defined?

0.97 => taking 1 sf

0.97 to one significant figure is 1. Similarly to one decimal place it would be 1.0. Raoul 08:36, 1 August 2006 (UTC)
I find it more interesting (or at least confusing) to ask: How is 0.97120545613 written to 2 sf? Are 1.0 and 0.97 both OK? I think so, but in most contexts I guess 0.97 should be preferred as the most accurate.--Niels Ø 08:20, 28 September 2006 (UTC)
I would say that 0.9712... to 2 sf is 0.97. 1.0 is simply 1 given an extra significant digit, which has nothing to do with the representation of 0.9712..., it is just to do with the representation of 1.
I would also suggest that the example given "0.1 becomes 0.10 (the trailing zero indicates that we are rounding to 2 significant figures). " is also not very sensible because it implies that extra information is extracted that wasn't present in the original figure (for example if a length was recorded as 0.1m, regardless of the accuracy of the original measurement, from the figure given all that is known is that the measurement lay somewhere between 0.0500...m and 0.1500...m (to the significant figures of the measurement), however expanding 0.1m to 0.10m implies that the original measurement lay somewhere between 0.095...m and 0.105...m which is not the case unless some additional information is known).(131.111.200.200 12:57, 21 April 2007 (UTC))
The problem with significant figures is that we must write an integer number of decimal digits. If the goal is that the number of significant figures correspond to the desired relative accuracy (dx/x), then we are excessively precise around 9.9, and excessively sloppy around 1.0. In practice, I think that if it really mattered then the error range would be given explicitly (0.971 +/- whatever). —The preceding unsigned comment was added by 24.91.135.162 (talk) 11:00, 22 April 2007 (UTC).

[edit] Tone and Content

This article is really an overwound and off-topic dissertation on the disadvantages of significant figures. This really doesn't belong in the article, or should at least be put under its own heading. I think eliminated all of this criticism would require a rewrite. The article needs to be a brief and succint definition of significant figures, and some guidelines on how and when to use them. Cephyr 23:32, 31 October 2006 (UTC)

I agree completely. I suggest we give the author a couple of weeks to respond, then look at how to edit this page. Daniel 1st November 2006


Most definitely. It seems the original author may have had a bad falling-out with someone who really likes significance figures. Are significant figures really such a bane to the world of knowledge as made out in this "article?" User:Benjamin.mcclure 8th December 2006

[edit] Erratum

it is more sensible to report it to four significant figures (8.540 m/s), because the time is only measured to four significant figures.

It is even more sensible to report it to one significant digit (9 m/s), because the distance represented in this question has only one significant digit (100 m, it doesn't say 100. m). The principle of minimum applies to SD calculation. The SD count of the answer is the same as the lowest SD count of all the numbers used to arrive at it.


Assuming 100. m, it would still not be four SD. —The preceding unsigned comment was added by 76.209.50.222 (talk) 00:41, 27 February 2007 (UTC).

reply: The above observations are too pedentic. In the context (running a 100m race), we may assume that the 100m track was laid out to a very high degree of accuracy. I'd be very surprised to find a "100m" event at the Olympics being run on a 97m track!! It is not necessary to refer to a "1.000 x 10^2 meter race" in this context.142.205.213.42 13:43, 30 April 2007 (UTC)

It's a pretty poor example, then. — Omegatron 14:11, 30 April 2007 (UTC)

[edit] non-standard_decimals

Can we get a reference for the "non-standardness" of "1000." to indicate four significant figures? I seem to remember this being the recommended notation in school... — Omegatron 23:55, 17 March 2007 (UTC)

That's what I learned as well, from my Zumdahl Chemistry textbook. ➪HiDrNick! 23:57, 17 March 2007 (UTC)
I've seen this in both high school AP physics and college physics courses, as well. Exigence 00:52, 30 May 2007 (UTC)
Though I've taken both chemistry and physics in highschool and is currently an undergraduate physics student, I have never seen this notation. I don't know if it is a US convention (but I have had some english textbooks)? The scientific notation is IMO the safest way (and the method I've learned) to express significant figures. It's in the article, but I think it should recieve more information in the article. JKBlock 16:46, 18 June 2007 (UTC)
I learned that a decimal point after 1000 means it is precise up to that point. We would fail to receive extra credits and lose points on exams for failing to take that into account, such that when we wrote equations, we had to make sure that decimal point at the end was there, and had to take that into account when giving the answer to the required number of significant figures. We learned this from Dr. John Trafton, a man with a Ph.D. in Physical Organic Chemistry, and one of the smartest men whom I've ever met.
http://www.secinfo.com/dSUyh.837.htm
And his YouTube videos, just for fun (and to prove he exists and is a teacher).
http://www.youtube.com/watch?v=krFBLbQkIPc
http://www.youtube.com/watch?v=1D07_mVT2zE&mode=related&search=
I know this is probably not a reliable source, but at least I should like to lend credence to the idea that the decimal point is MOST DEFINITELY used in this manner. Perhaps not in other countries, but definitely in the US.
Here is a source.
http://student.ccbcmd.edu/~cyau1/WhichDigitsareSigFig.htm
Scroll down to question 1. "1. Give the number of sig. fig. and the number of decimal places in each of the number below." The answers are here.
http://student.ccbcmd.edu/~cyau1/WhichDigitsareSigFig.htm#Ans#1
Notice that 3000 says "assume 1" sig fig, but 3000. says 4 sig figs.
That is from The Community College of Baltimore County, http://www.ccbcmd.edu/. I would say they are a good source. XD Jaimeastorga2000 10:23, 14 July 2007 (UTC)

I decided to be bold and remove the "dubious" tag and also the dubious statement. I also added a reference. In our high school we teach students to use a decimal after zeros to indicate their significance. Redhookesb 07:10, 31 July 2007 (UTC)

[edit] Topical agreement with content, and simplicity

Since this article is supposed to be about significant figures, not rounding, I suggest restoring an earlier article on this topic, then clarify the introduction to say that significant figures is an indication of how precise a value is. I think starting with that sort of simple article, then adding a See also for more technical articles, would help the readers. --SueHay 19:54, 6 May 2007 (UTC)

[edit] Reminder on thousands and decimal separators

Note to editors from non-English-speaking countries: please be aware that this article follows the usage of most English-speaking countries in using a dot (.) to indicate decimals, and a comma (,) to separate thousands. Quite a few non-English-speaking countries do it the other way around, which has caused confusion on this and other articles. See Decimal separator for more information. (I wonder if it's worth creating a template for article space?) --Calair 12:28, 1 June 2007 (UTC)

[edit] Bar Notation

I was wondering why bar notation was removed; this is the sometimes-used convention that zeros left of the decimal point are only significant if there is a bar over them. I don't recall seeing it used since I was in high school (25 years ago!) but it got a mention in the Wikipedia vs EB debate here: [1] , where it is described as "...common in chemistry, but seen in other scientific fields as well." It was first added in [2] and was removed (along with all the rest of the content) in [3]; other bits of the article were restored but not that. Google has a bunch of references from .edu domains for its use, some of which describe the convention as 'common' and some as 'rare' - [4].

It does seem error prone compared to scientific notation (eg you can't tell whether '1000' was written using the convention or not), but from an NPOV it should be in here as 'sometimes used' Bazzargh 13:40, 24 July 2007 (UTC)

[edit] Zero

How many significant figures are there in '0.0000'? My instinct would be 'none' or 'undefined', but if there are any sources that establish a convention on this it would be nice to cite them. --Calair 03:31, 1 August 2007 (UTC)

No, zeroes after the decimal point are significant. Think about it this way, if you were measuring the exact temperature of the triple point of slightly impure water with a very accurate thermometer, and you wrote down the temperature as "0.0000 °C", you'd be indicating that you know the temperature to be between -0.00005 and 0.00005 °C. This is a very accurate measurement. Which brings us to another interesting point: What if the thermometer were calibrated in Kelvin? Then you'd have written down the temperature as "273.1500 K". I leave it as an exercise to the reader to figure out where the extra significant figures came from. --Slashme 15:39, 1 August 2007 (UTC)
It is indeed a very accurate measurement... but the number of significant figures indicates not absolute accuracy, but accuracy relative to the number's magnitude.
The temperature example is a red herring; while "0.0000 °C" and "273.1500K" represent the same temperature, and indicate the same absolute level of accuracy, "0.0000" and "273.1500" are not the same number. (Otherwise, I could create a relative scale of weight that takes -MEarth as its zero and declare that my weight is 5.9736E24kg - look, five significant figures!) So let's keep this simple and stick to absolute scales for now.
If my scale gives a reading of "111.11 grams", and is known to be accurate to the displayed precision, that's five significant figures. If it reads "11.11 grams", that's four SF. If it reads "1.11 grams", three SF. At "0.11 grams", we have two SF, and at "0.01 grams" we have just one; leading zeros are not significant. Continuing the progression, how many SF does "0.00 grams" have? The obvious answer is 'zero'; there's also a pretty good case for 'negative infinity' (to see why, note that the number of SF is approximately equal to log10(magnitude of measurement/magnitude of uncertainty), which tends to -Inf as the measurement goes to zero).
But it certainly doesn't go back to 'two'. --Calair 17:17, 1 August 2007 (UTC)

No, you're right, I was wrong. Zeroes after the decimal point are indeed not significant. Absolute and relative scales mess a bit with significant figures, though. If you have weighed out 0.0001 g of powder on a 0.1 mg balance, and you do a titration, your weighing is clearly done to one significant figure, and your rule of thumb that tells you that you can only expect one significant figure in your calculated concentration is working quite nicely. If you measure temperature in degrees Celcius, and you get the following results:

 1.0000
 0.5432
 0.0213
 0.0014
 0.0001
 0.0000
-0.0001

each of them is indicating the same amount of information.

All of this once again underlines the point that significant figures are a useful rule of thumb when you are multiplying numbers, but it's a system with limits. You have to be working in absolute scales to do sensible multiplication most of the time, anyway. Remember, if you add two numbers, it's not the significant digits you're worried about, but the tolerance of the least precise one. So for example, 0.00 degrees Celcius plus 0.1 degrees Celcius is 0.1 degrees Celcius, not 0.10 .

Coming back to your question of how many significant figures "0.0000" has, it might well be undefined. If you weigh out 0.0000 grams of some powder to do a titration, it means you don't know quite how much you have, but it's less than 50 micrograms. No significant figures there at all. If you push your car 0.0000 kilometers, it means you are not sure how far you have pushed it, but it's not more than 50 millimeters. Still no significant digits.

I might put a short note in the article to this effect, but it would be nice to have something to cite. (But I suspect anybody fussy enough to worry about this particular case is probably using uncertainties rather than SF anyway.)--Calair 03:02, 3 August 2007 (UTC)

I was taught about significant figures at my university. I should have sued the buggers. This has been slowly dawning on me for a while now, but reading John Denker's discussion of measurements and uncertainty I'm now absolutely convinced that it's generally a very bad system. He discusses the issue of zero quite nicely. --Slashme 08:35, 3 August 2007 (UTC)

[edit] Suggestions

Just a couple of suggestions if I may be so bold:

1) rules #2 and #4 in the "Identifying significant digits" section could be considered contradictory. E.g consider the number "130300". A strict reading of #4 would imply that the 1st "0" is not significant. However, rule #2 implies that it is significant (which I believe it is). This could be resolved, I believe, by changing rule #4 to say:

All zeros appearing in a number without a decimal point and to the right of the last non-zero digit are not significant unless indicated by a bar. Example: '1300' has two significant figures: 1 and 3. The zeros are not considered significant because they don't have a bar. However, 1300.0 has five significant figures.

2) in the "Rounding conventions" section I think it might be a bit clearer if each of these rules began with:

If the digit immediately to the right of the nth significant digit...

The first time I read it, I was confused because I wasn't sure if 'digit' included all the digits in the number or just the significant ones. On further thinking it seems obvious it must only be the significant digits. However, I think stating it explicitly would be clearer.

192.88.66.254 18:16, 1 August 2007 (UTC)Bill

Agreed on both counts. --Calair 18:34, 1 August 2007 (UTC)

[edit] Duplication of Significance arithmetic

Should we keep this page as it stands, or work towards merging with Significance arithmetic? --Slashme 07:45, 3 August 2007 (UTC)

[edit] Identifying significant digits - Example 4

In the fourth example of the "Identifying significant digits" section it says, "All zeros appearing in a number without a decimal point and to the right of the last non-zero digit are not significant unless indicated by a bar." The example shows that "1300" has only two significant digits. According to the Alberta Physics 20 course, this is wrong. One handout I have says, "The Learner Assessment Branch considers all trailing zerso to be significant. For example: 200 has three significant digits." I do not know if this only applies in Canada or maybe only in Alberta, but I thought I would point it out. --bse3 20:39, 8 September 2007 (UTC)

This is actually a common idea used in many educational facilities, I have found. In the courses I teach I say "for all numbers between 1 and 1000, all zeros shall be considered significant." It is also used by the British Columbia government on their provincial examinations for sciences. I do this because it is just easier to write "100" rather than "1.00x102" when I want to express three significant digits. That being said, the conventions in this article are correct. "The Learner Assessment Branch" are just simplifying things for the purposes of the assessment of the course(s); if simplifying is the correct choice of words here. Grishmak 22:19, 11 September 2007 (UTC)

Thanks. I just wanted to be sure. —Preceding unsigned comment added by Bse3 (talk • contribs) 00:38, 15 September 2007 (UTC)

[edit] Misunderstanding

Hi there. I was reading the article on sig. fig. and was trying to fathom the process. This paragraph leaves me thinking there must be a typo:

3. All zeros appearing to the right of an understood decimal point or zeros appearing to the right of non-zero digits after the decimal point are significant...

..The number '0.00122300' still only has six significant figures (the zeros before the '1' are not significant).

0.00122300 has 2 zeros appearing to the right of an understood decimal. These are significant. Then it has 1223 which is significant also. Then it has 00, which appear to the right of a non-zero digit after the decimal point. so we have 2+4+2=8 not the stated 6? so I am confused.

Someone please please help! Am i just reading it wrong!? —Preceding unsigned comment added by Dirtyraskal (talk • contribs) 01:10, 27 October 2007 (UTC)

I agree that it's confusing. I don't understand what is meant by an "understood" decimal point here. All leading zeroes are insignificant. The significant figures in 0.00122300 start with the 1. --Slashme 05:20, 8 November 2007 (UTC)

[edit] Thousands separator

I have now (for the second time) fixed someone's change from , to . in a number where the comma is used as a thousands separator. I have now also removed all the commas which are used as thousands separators and replaced them with spaces (I guess non-breaking spaces would be even better) because we have the least chance for misunderstanding if we use a point as a decimal and no commas at all. --Slashme 05:20, 8 November 2007 (UTC)

[edit] math

significant figures —Preceding unsigned comment added by 124.217.84.214 (talk) 09:45, 19 November 2007 (UTC)

[edit] misunderstanding in the article

i am asking for some clarifications on this article.In the first part of the "Rounding conventions" section of the article,6 examples of rounding numbers to 2 s.f are given.but i had it edited for there was the rounding of 19 800 to 20 000 missing as the last example. i quote to you what is written later in the same section: "This occurs when the last significant figure is a zero to the left of the decimal point.For example ,in the final example above, when 19 800 is rounded to 20 000, it is not clear from the rounded value what n was used - n could be anything from 1 to 5."What i wish to know is that if the last example that was missing is "19 800 becoming 20 000" or something involving decimals like 19.800? i feel that the last example shoud be involving decimals,as mentioned in the quotation.i wish if you could make appropriate editing about that as soon as possible. —Preceding unsigned comment added by 41.212.232.44 (talk) 12:55, 31 January 2008 (UTC)

[edit] misunderstanding in the article

i am asking for some clarifications on this article.In the first part of the "Rounding conventions" section of the article,6 examples of rounding numbers to 2 s.f are given.but i had it edited for there was the rounding of 19 800 to 20 000 missing as the last example. i quote to you what is written later in the same section: "This occurs when the last significant figure is a zero to the left of the decimal point.For example ,in the final example above, when 19 800 is rounded to 20 000, it is not clear from the rounded value what n was used - n could be anything from 1 to 5."What i wish to know is that if the last example that was missing is "19 800 becoming 20 000" or something involving decimals like 19.800? i feel that the last example shoud be involving decimals,as mentioned in the quotation.i wish if you could make appropriate editing about that as soon as possible. —Preceding unsigned comment added by 41.212.232.44 (talk) 12:58, 31 January 2008 (UTC)