Talk:Significant figures

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)

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

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)

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)

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

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)

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)

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)

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)

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)

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)

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)

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)

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

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)

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)

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)