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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)[reply]

Also see Wikipedia:MOSNUM § Delimiting (grouping of digits). —[AlanM1(talk)]— 20:51, 21 July 2013 (UTC)[reply]

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)[reply]

The article uses the terms "significant figures" and "significant digits" interchangeably. Is there a mathematical reasoning to the usage of one word in some places and the other in other places, or is it simply a matter of preference? If it is a matter of preference then i suggest all instances of the term "significant digits" be replaced with "significant figures" in order to make the article clearer and because the article is about significant figures not significant digits. —Preceding unsigned comment added by 71.236.29.119 (talk) 03:47, 10 December 2010 (UTC)[reply]

I think that's just American vs. Commonwealth English. -- Beland (talk) 04:35, 3 March 2013 (UTC)[reply]

This article should mention the alternative term “significant digits” or “sig digs” for short; that is the term that I first learned when I learned about the subject in 8th grade. (I am from the U.S., not Great Britain).--Solomonfromfinland (talk) 19:18, 23 May 2016 (UTC)[reply]

I think the arithmetic section could use two quick examples, one for each rule.

For example, 1300 x 0.5 = 700. There are two significant figures (1 and 3) in the number 1300, and there is one significant figure (5) in the number 0.5. Therefore, the product will have only one significant figure. When 650 is rounded to one significant figure the result is 700.

For example, 1300 + 0.5 = 1301. There are zero decimal places in the number 1300, and there are is one decimal place in the number 0.5. Therefore, the sum will have zero decimal places. When 1300.5 is rounded to the ones decimal place (zero decimal places) the result is 1301. —Preceding unsigned comment added by 99.32.166.179 (talk) 01:30, 9 January 2011 (UTC)[reply]

This is covered under Significance arithmetic, and I think the second example is wrong. It should be 1300 + 0.5 = 1300, since addition uses the position of most significance in the least significant number, which would be the hundreds place of 1300. -- Beland (talk) 04:43, 3 March 2013 (UTC)[reply]

I think the example for logarithms in the Arithmetic section is wrong: 3.000 has 4 significant figures, and if the number of digits in the mantissa should be equal to the number of significant figures, then log(3.000×10^4)= 4.4771 (4 decimals), rather than 4.48 (2 decimals). I'm not 100% sure about this, so I would like to hear someone else's opinion before correcting... Oghin (talk) 21:35, 18 March 2013 (UTC)[reply]

I checked some additional sources, which confirm this, so I decided to go ahead and correct the example. Oghin (talk) 00:25, 19 March 2013 (UTC)[reply]

The important idea behind significant figures is uncertainty. For multiply and divide, the relative uncertainty is close to the worst (largest) relative uncertainty of the operands, so simple rules work. For log10(3.000e4), we mean log10(30000 +/- 5), log(30000)=4.47712..., log(300005)=4.47719, and log(29995)=1.47704, so we want 4.7712 +/- 0.00007, so 4.4771 is about right, though one would keep 4.47712 in an intermediate calculation. The precision is a little better than four digits after the decimal point, but not enough for five. Gah4 (talk) 19:14, 3 July 2017 (UTC)[reply]

The history of significant figures - who came up with it? When were they first used? When did they come into general use and start being widely taught? - would make a nice addition to the article. I couldn't find anything on this topic when I looked today, but someone must know. 99.65.213.158 (talk) 03:47, 19 March 2011 (UTC)[reply]

It appears to have been in the 1970s and 1980s, right when the government was starting to push for the removal of chemistry from chemistry, due to the protest movements. It was likely implemented as a filler in chemistry so that a full semester or year could be taught in bullshit. Sig figs serve absolutely no purpose and are just a distraction.--Metallurgist (talk) 00:24, 12 September 2012 (UTC)[reply]

Uh, no. Google ngrams and Google Books search 1800–1854. —[AlanM1(talk)]— 20:44, 21 July 2013 (UTC)[reply]

I suspect that the idea goes back pretty far, but it became much more important as pocket calculators became popular. Such calculators usually display 10 digits, no matter what you put in, and beginning students tend to write them all down. TAs would catch them, and students would eventually learn. With a slide rule, you got usually three digits, which was likely about right. Gah4 (talk) 18:48, 3 July 2017 (UTC)[reply]

The early introductory material says: ...

The significant figures (also called significant digits) of a number are those digits that carry meaning contributing to its precision. This includes all digits except:

leading and trailing zeros which are merely placeholders to indicate the scale of the number.

I've wracked my brain to realize, finally, that the context here is that the number 32,000,000 might have precisely two significant digits. But, the number 0.032000 probably has five. Thus, I think the introductory material should be clearer and is, as it stands, quite misleading. I'd fix it myself, but fear incurring the wrath of Those Who Care. — Preceding unsigned comment added by Rkolstad (talk • contribs) 17:17, 6 September 2012 (UTC)[reply]

Thanks for that comment; I've tried to clarify the intro. -- Beland (talk) 04:45, 3 March 2013 (UTC)[reply]

I went a little further, separating out leading zeroes are always insignificant (as stated correctly further down). —[AlanM1(talk)]— 20:35, 21 July 2013 (UTC)[reply]

I implemented the requested merge from Arithmetic precision; I left the comments at Talk:Arithmetic precision in place. -- Beland (talk) 03:41, 3 March 2013 (UTC)[reply]

Requested by whom? I do not see a discussion. Did at least two users express their support for this dubious deal? Incnis Mrsi (talk) 18:29, 20 March 2014 (UTC)[reply]

This article seems to have been eroded by a number of edits that sound like WP:OR at best, structure that's gotten kind of scattered, etc. One editor added a "cite" (Serway) that I fixed, but the referenced 1990 edition is way out of print and I couldn't even find an ISBN for it. I added some maintenance tags and cleaned up what I could, but it could really use a re-work against a good modern source or two if someone happens to be studying such a text and would like to do so. —[AlanM1(talk)]— 20:33, 21 July 2013 (UTC)[reply]

Mathematics software goes further. ex. in Mathematica one can specify a number with: [base^^][Real.m]([``]|[`])[*^n] where `` is acc, ` is prec, n means *10^n, and Real is not in rational form. Calcualtions take sig, acc and prec into account. Infact all non-rational numbers do, though new users are typically un-aware because they are displayed plainly. Competing software like Matlab have alternate ways of achieving the same.

How to sqeeze such numbers in a table is another topic because what to show (so the reader is not lied to) uses all the rules for sig fig, acc, prec mentioned above. Usually: the job is never done, excepting in formal science publications like CRC's handbook of chem & phy, and one should be aware it hasn't been.

fNBookForm2 displays scientific numbers in Mathematica tables in texbook style and does all mention rules (sig, acc, prec) and width, and has 4 rounding modes (off, normal: ignore fractional sig, round in fs, round off fs). It reads shorthand. It can display power letters as well. — Preceding unsigned comment added by Sven nestle2 (talk • contribs) 21:56, 6 October 2013 (UTC)[reply]

If I have a ruler marked with millimetres as the smallest division then how can I report a measurement as 2.54 cm. I can say it is 2.5 cm or 2.6 cm. But how can someone know that whether it is 2.54 or 2.55 or 2.52. If this scale has a least count of 1 mm then how can it shows measurements with 0.1 mm accuracy. Naveeagrawal (talk) 05:42, 24 January 2014 (UTC)[reply]

For most scales, one can estimate reasonably well in to one tenth of the marked divisions. This is more obvious reading analog meter movements, but should be true for rulers, too. Consider a ruler with divisions at 1cm, and that you can easily estimate where you are in between the divisions. Maybe one can only estimate to 1/5th of the spacing, but that is still a lot better than rounding to a whole division. Also, and I am not sure that the article explains this, when doing computations, one should keep one additional digit on intermediate values. Gah4 (talk) 01:59, 2 April 2017 (UTC)[reply]

I've found 3 articles on the web explaining the meaning of significant digits. All give many examples but none give an example with zeros in the middle. They do not address the fact that 20.002 has five significant digits. All the talk of leading zeros after the decimal and trailing zeros confuses this issue. foobar (talk) 01:16, 8 February 2014 (UTC)[reply]

Since the pages for most and least significant digit redirect here, there should be some explanation of what makes a digit more or less significant than another. At the very least, an explanation of the most significant and least significant concepts should be included. — Preceding unsigned comment added by Dstarfire (talk • contribs) 20:09, 18 October 2015 (UTC)[reply]

I suppose so, but it is really a different subject, and they probably shouldn't redirect here. Gah4 (talk) 02:01, 2 April 2017 (UTC)[reply]

One thing that appears to be missing is the infinite precision of constants. For example, in a calculation for the area of a circle A = ( PI * Diameter ^ 2 ) / 4, the number 4 has infinite precision. One digit, infinite precision. That it is a single digit should not detract from the overall significant digits. Similarly, in a calculation for the volume of a sphere V = (3 * PI * Radius ^ 3) / 4, the constants 3 and 4 are infinitely precise, as would the alternative representations 3/4 or 0.75 be. — Preceding unsigned comment added by 192.104.67.122 (talk) 16:20, 12 June 2018 (UTC)[reply]

At the beginning of the page, there is a broken link that contains the text "Numerical digit|precision|measurement resolution". RichMorin (talk) 20:17, 15 October 2019 (UTC)[reply]

Thanks for pointing that out! I've changed the lead back to the last good version. —The Editor's Apprentice (talk) 18:24, 19 October 2019 (UTC)[reply]

This is from the intro:

"For example, if a length measurement gives 114.8 mm while the smallest interval between marks on the ruler used in the measurement is 1 mm, then the first three digits (1, 1, and 4, and these show 11.4 mm)..."

How does 114.8 mm become 11.4 mm or "show" 11.4 mm?

Then (with overlap from the above-quoted line):

"(1, 1, and 4, and these show 11.4 mm) are only reliable so can be significant figures."

Are "only reliable"? That's not correct English if we mean to say something like "these are the only reliable digits" or some such.

The immediate dive in measurement issues and precision is odd. Is this how the topic of significant figures is normally introduced to students?

"Among these digits, there is uncertainty in the last digit (4, to add 0.4 mm) but it is also considered as a significant figure[1] since digits that are uncertain but reliable are considered significant figures."

Again, 114.8 mm is converted to 11.4 mm, with the last digit as .4. There is no actual uncertainty in the 4 if we are holding to the number that was initially presented: 114.8 mm on a mm-incremented ruler, so this is a mess.

It's not at all clear what it means for a digit to be "uncertain but reliable" in this context, and it's a big mistake to make these sorts of confusing proclamations in the introductory paragraph for this topic.

More broadly, the intro paragraph is diving into thorny measurement issues in ways that will likely confuse both students and experienced scientists. The marked increments on a device like a ruler do not carry any inherent standing as the limits of significance. Or reliability. Or certainty. It's not a given that the markings on a device are the governing unit limits for measurement. Depending on the device, we might decide that some fraction of the marking interval are the true limits on any of these dimensions (significance, reliability, certainty), for example a five-foot ruler with one-foot increments, no inches or lesser intervals – humans would be able to infer smaller than foot units, even without markings. The particular example in the intro is actually about humans reading the markings on a device like a ruler. That's a whole different epistemic phenomenon than a readout-based measurement that sprouts from various sensor and electronic operations, with whatever realities of precision and accuracy the readout represents. It's also possible that for marked devices meant to be read by humans, the true significance, reliability, or certainty limits are larger than the marking intervals – just imagine a ruler that marked microns, or even tenths of a millimeter, such that most humans could not reliably discern which tenth of a millimeter aligned with the beginning or end of the measured object. These are ultimately epistemological issues, and the mere presence of markings meant for two-eyed bipeds cannot be the governing unit limits for significance.

BlueSingularity (talk) 17:58, 28 April 2021 (UTC)[reply]

The figure illustrating significant digits is not consistent with the text describing significant digits. In general, the representation of numbers in Europe differs from representations of those same numbers in the USA. I work with the internals of floating point arithmetic and the important distinction between "significant digits (decimal)" and "significant bits (binary)." I recommend removal of the illustrative figure or replacement with a figure consistent with the textual definitions. Softtest123 (talk) 16:40, 20 May 2021 (UTC)[reply]

Significant figures is a rough approximation to uncertainty, and commonly rounded to integers. One decimal digit is worth 3.32 bits. Note that the actual precision can vary, such that 100 has two significant digits and 999 has 2.9996 digits. Sometimes there is need to be more accurate than the rounded value, other times not. Tradition is to do calculations with one extra digit, which is usually enough, and then decide the best value in the end. Also, with most analog measuring devices, it is possible to estimate to 1/10th of a scale division. In any case, if you try to be too accurate, you are wasting energy. Might as well do a full uncertainty calculation. Gah4 (talk) 00:21, 21 May 2021 (UTC)[reply]

I am surprised that this article makes no mention of the fact that significant figures are a poor attempt to explain error propagation to young students. Significant figures and their rules are not used in metrology. The idea that the number of digits represents something about precision is flat out wrong, and following sig fig rules always results in poorer estimates. In the "Relationship to accuracy and precision in measurement" section it states that significant figures are more related to precision than accuracy, which glosses over the fact that it *incorrectly* relates to precision. The number of significant digits does not tell you about precision. A separate statement of precision (such as the standard deviation of the measurement) tells you this. 140.177.239.221 (talk) 00:55, 6 October 2021 (UTC)[reply]

I wouldn't say that it is terrible, and it does mention a ways down that it is an approximation to actual error propagation. It is often enough used then the more accurate error propagation isn't needed. But yes, it could be better explained. Among others, one has to be careful not to use too few digits in intermediates, and rounding at the end. Also, most analog measurements can be made to about 1/10 of the finest divisions. Gah4 (talk) 05:14, 6 October 2021 (UTC)[reply]

It is terrible, because it implies that significant figures are in any way an appropriate way to handle error propagation. When you say "It is often enough used then the more accurate error propagation isn't needed." the problem is the assumption that significant figures are in any way accurate. They are not. They do not approximate the idea of precision properly. One is better off ignoring the concept of precision and keeping a bunch of digits than using significant figures. Describing significant figures is fine. They're commonly taught. But this is an encyclopedia, and it should be clear here that in metrology precision is handled entirely differently. 140.177.239.221 (talk) 15:46, 6 October 2021 (UTC)[reply]

I was in college not so long after scientific calculators were affordable for college students. I, and many others, would put answers to homework problems to 10 digits. We would get marked off by our TAs with notes like "SIGFIGS". (That was supposed to be enough, they didn't write a whole story about them.) When you do homework problems, and the study is not in proper error analysis (or metrology) significant figures is fine. When you want to figure the gas mileage for your car after a fill-up, you don't need an error analysis. There are plenty of everyday problems that don't need full analysis. One of the more common cases in Wikipedia is unit conversion. There is no need for excessive digits, but one also doesn't want to round off too much. The number of people doing actual metrology is fairly small, and hopefully they know how to do it right. Gah4 (talk) 19:15, 6 October 2021 (UTC)[reply]

All of that is fine, and is besides the point. Wikipedia is not here to help college students do homework or calculate mileage on their cars. It is here to convey information, and no discussion of significant figures is complete without describing their flaws, which are serious conceptionally and practically. It is a disservice to not describe how significant figures stem from error propagation in an incorrect manner. Students are perpetually confused by significant figures, because they are confusing. They are confusing because they are seriously flawed. By explaining the original purpose and how significant figures are a flawed solution to that that purpose, people will gain a better understanding of significant figures themselves. As is, the article is misleading, because it presents significant figures as a proper solution, not the gross simplification that they are 140.177.239.221 (talk) 13:50, 7 October 2021 (UTC)[reply]

I suspect that very often, Wikipedia is here to help college students with homework. Maybe we should go back to first grade and teach no 2+2=4, but instead 2+2=4±0.5. Oops, haven't learned decimal fractions yet: 2+2=4±1/2. For a very large fraction of every day math problems, it is fine, though, yes, the article should explain, and maybe closer to the top, more of uncertainties and error propagation. But you will have to simplify that, as most people don't have the background of statistics to understand standard error, or of calculus to understand error propagation. The first class I had was second year of college, designed as a physics lab, but everything taught was error analysis. (The physics is pretty dumb, like measuring the voltage of a battery with an oscilloscope, but all was meant to teach data analysis.) The book was Young,^[1] which is pretty good for an introduction, and only 172 pages. Also, it comes in paperback for a very reasonable price. I suspect most, though, don't get this until third year of college for physics and engineering students, and never for others. Gah4 (talk) 18:29, 7 October 2021 (UTC)[reply]

The article states that "Zeros to the left of the first non-zero digit (leading zeros) are not significant". However two senteces later, it states "0.00034 has 4 significant zeros if the resolution is 0.001". That is a contradiction, isn't it? 5.55.134.21 (talk) 06:08, 29 July 2022 (UTC)[reply]

I agree that it is potentially confusing. It isn’t contradictory because it is making use of the concept of “resolution” – see the second paragraph in the lead: “If a number expressing the result of a measurement...” Dolphin (t) 13:03, 30 July 2022 (UTC)[reply]

So the concept of resolution is ignored in the rule, but arbitrarily introduced in an example of application of that rule? 5.55.134.21 (talk) 10:11, 31 July 2022 (UTC)[reply]

Yes! It looks like it is intended to confuse readers who come to this site to gather introductory information. Dolphin (t) 12:20, 31 July 2022 (UTC)[reply]

The definition should also be clarified. — Vincent Lefèvre (talk) 12:37, 31 July 2022 (UTC)[reply]

Yep. Resolution should probably be broken out as a separate concept later in the article, with its own examples. The example chosen here is particularly and deliberately confusing, and arguably misstated. TenOfAllTrades(talk) 13:08, 31 July 2022 (UTC)[reply]

The article states that a measurement of 1500m has two significant figures because the measurement resolution is 100m! But a measurement of 1500m accurate to 1m is also: 1500m and in this instance the trailing zeros must surely be significant. Hence the rationale given here for trailing placeholder zeros being non-significant is surely ambiguous at best? Knolan7799 (talk) 23:18, 5 October 2022 (UTC)[reply]

Tradition says that is is two significant figures. If you want more, you are supposed to write it in scientific notation, where you can put in the extra sigfigs. Gah4 (talk) 06:42, 6 October 2022 (UTC)[reply]

I could argue the exact opposite. That 1500 has 4 significant figures, and to consider it as having 2, write it as 1.5 * 10^3. The problem is that a number such as 1500 is ambiguous in term of significant figure. To specify a number of figure, a scientific notation is necessary. So, as mentioned in the article, without further precision on the measurement resolution, 1500 can have 2, 3 or 4 significant figures. Dhrm77 (talk) 11:25, 6 October 2022 (UTC)[reply]

You could do that, but tradition says not. Consider the 4 significant digit values starting with 15, from 1500 to 1599. Only 1 out of 100 is 1500. But of the 2 significant digit values near 1500, starting with 15, all of them are 1500. Odds get worse with more digits. For similar reasons, there are a lot more values with 2 significant digits. (Not so many measuring devices have that much precision.) Gah4 (talk) 18:55, 6 October 2022 (UTC)[reply]

The introduction to this article uses the word “reliable” but it doesn’t give any indication of what it means. It would be great for it to link to the explanation in the article, if there is any, or for it to concisely give some idea of what it means. 191.156.227.97 (talk) 17:02, 11 February 2023 (UTC)[reply]

I find much of this article confusing to an unusual degree. Assuming it's not just me, this is unfortunate in an ironic kind of way, considering that the whole point of the topic itself is to reduce uncertainty and increase precision. Perhaps the article could be rewritten or, at very least, significantly revised. 72.182.56.43 (talk) 12:00, 4 June 2023 (UTC)[reply]

Thanks for taking time to give feedback. Unfortunately, your feedback is too vague to be actionable. Can you pinpoint a particular concept or section that is especially bad? For example, how is the introduction at the top? Are any of the examples good? Would you like more examples? Help us help you. Mgnbar (talk) 13:42, 4 June 2023 (UTC)[reply]

The previous article Significance arithmetic had years of talk page complaints that seemed to never be addressed, and it was an almost entirely unsourced mess with poor organization, style, and content, so I WP:BOLDly converted the title to a redirect to Significant figures § Arithmetic, which already contains the relevant material that was at that article.

I agree with posters above (e.g. the IP editor at §  This article is terribly misleading) that this article could use help. It would be nice if we had a much more accessible explanation at Propagation of uncertainty (the current one is totally useless for e.g. high school students or early undergraduates), and if we did a better job at this article explaining the problems with using (only) significant figures to represent uncertainty and propagate errors and comparing it as a method with better ones. –jacobolus (t) 02:28, 7 March 2024 (UTC)[reply]

I can't understand the first paragraph under this title. Can someone clear this one up? 2409:408D:3D06:B852:B4F6:1302:6FBF:B1F (talk) 08:48, 2 May 2024 (UTC)[reply]