Talk:Positional notation

next guy's comment
I think it's odd that the issue of fractional numbers is addressed in the section on base-60 but not in base-10. MFH 13:56, 8 Apr 2005 (UTC)== Fractional numbers ==

Reorganize numeral system stuff
Also, there is really an important job to do consisting in clearly reorganizing all about base-p, decimal, p-adic, notation vs numbering vs numeral system: so many things are said about the same thing more or less correctly and more or less contradictionally in so many different places. MFH 13:56, 8 Apr 2005 (UTC)

Synonym
I added "place-value notation", a term commonly used in U.S. schools, as a synonym for this type of notation. Based on the description I believe this is accurate, but please someone double-check me. Thanks. Deco 01:58, 6 November 2005 (UTC)

Non-standard positional numeral systems
I have addressed certain issues by creating the article Non-standard positional numeral systems, and making related changes to Unary numeral system, Golden ratio base, Quater-imaginary base, Positional notation, Base (mathematics), and Category:Positional numeral systems. Perhaps the reference to the new article in the present article should be in the introduction, as some sort of disambiguation, rather than in the See also section. Apart from that, I suggest further discussion of these issues takes place at talk:Non-standard positional numeral systems.--Niels Ø 14:35, 26 February 2006 (UTC)

There's a simpler notation for balanced ternary. Because the only non-zero magnitude is 1, it's simpler, typographically, to simply use + and -, i.e., just the signs, for the non-zero digits. Nikevich 01:19, 17 January 2012 (UTC)

Mathematical implications
I removed the following paragraph:

''The real value of positional notation turned out to be its ability to invite the further study of numbers. Integers, rational numbers, and place-holders (e.g. zero) were long known about, but irrational numbers, infinity, transfinite numbers, and imaginary numbers were all concepts that could only be discovered once the idea of a continuous number line was implied by positional notation. ''

The concept of infinity and the invention of transfinite numbers are not related to the representation of numbers as points on a line, but is a purely set theoretic idea. Irrational numbers were found as solutions of geometric problems that had no corresponding numeral (rational) representation, long before positional systems came into use (or at least independent thereof).

Hylas 08:51, 20 March 2006 (UTC)


 * I, too, found that paragraph somewhat misleading. On the other hand, the notion of the real number line is an important one, and I think it is far more easily grasped if you have a mental concept of number tied to positional notation. Can this be said in the article in a way that is not misleading?--Niels Ø 11:57, 20 March 2006 (UTC)


 * I added links to the corresponding mathematical ideas. Sadly there is little details provided about the construction. Both articles deal only with the decimal system. Hylas 17:21, 21 March 2006 (UTC)

How is additive system easier for arithmetic?
aside: romans did not commonly use the preceding lower order symbol for substraction. I reckon it would have been more like this: (plus signs aren't needed now that order means nothing) IIII XII = XIIIIII = X IIIII I = X V I = XVI combine, re order, re group sometimes the process would have to be repeated -- I retract the earlier complain about the second paragraph being confused, but I still find it confusing. The problem is that it says additive systems are better for arithmetic, but this doesn't seem right at all. Could this be explained? How does positional notation require memorization of tables? Does this mean multiplication tables? Perhaps this matter should be moved out of the article lead. —Preceding unsigned comment added by 24.55.70.103 (talk • contribs)


 * I have serious doubts about that statement too. Roman numerals required the memorization of doubling tables (and possibly other multiplication tables) by everyone taking mathematics in school during the first millennium. Without such memorization the student was not considered competent. Positional notation also requires memorization of multiplication tables. Only when a machine is used (like the abacus) are such tables not needed. — Joe Kress 17:17, 2 June 2006 (UTC)

For simple addition and subtraction, the Roman numeral system is basically abacus-like; so for instance

IV + XII = V + XI = VI + X  = XVI

This isn't the actual computation someone would perform, but rather an attempt to replicate the abstract process the user of Roman numerals might engage in to perform an addition. For basic monetary transactions, it is slightly faster.

Positional Notation for Base-1
The article linked in the upper right box convers bijective numeration only. There does not appear to be a entry for base-1 positional notation. —Preceding unsigned comment added by 68.46.86.34 (talk) 19:33, 29 May 2009 (UTC)
 * See also Non-standard positional numeral systems (which confusingly says that unary is such a system despite the fact—also stated—that it's obviously not a positional system) and Talk:Unary numeral system, especially the Non-standard positional numeral systems section. I had forgotten all about this mess, but my viewpoint is the same as yours: there is nothing positional about the unary system. (The reason that true base-1 positional notation doesn't have an article is of course that it would be completely useless, not being capable of denoting a single non-zero number). — JAO • T • C 19:50, 29 May 2009 (UTC)


 * I don't see how unary isn't a positional system. You could view it as a base 1 system (seems kind of obvious). Its drawback is that it can only represent positive integers. Standard rules for constructing any base_n number work for base 1. 1^0 + 1^1 + 1^2 + 1^3... The position is meaningless, but it's a generalization of positional systems. Aleph Infinity (talk) 00:08, 12 February 2010 (UTC)
 * It is, it's just not an example we need.
 * EDITORIALIZING
 * Arithmetic times table unnecessary here.
 * Add to the article on Integers, and see what they say. &mdash; Cp i r al Cpiral  03:04, 28 September 2012 (UTC)


 * It might be of modest interest that the Friden EC-130 and EC-132 electronic desktop calculators (among the earliest such) used unary "notation" in their internal storage, which was serial, and recirculated stored numbers in a delay line. Each digit had its designated time slot, located by a specific delay after the pulse marking the beginning of data. If there were no pulses in a given time slot, that represented a zero. One pulse represented a one, five pulses a five, nine a nine. All computation was done basically by counting pulses; counting up past nine created a carry pulse. Subtraction was by counting down. Nikevich 01:29, 17 January 2012 (UTC)
 * Not convinced. &mdash; Cp i r al Cpiral  03:04, 28 September 2012 (UTC)

I want to add my base converter
What the subject says. It can do fractions (most of the time) to a many places, you can make up a base with whatever characters you want as long as each digit is only a single character, it has a list of common bases to choose from when converting. Plus its GNU GPL. I think mine has more features that the other links in this article. The link to my converter is. I tried to add an old version of this long time ago and some admins got all freaky on me and said that it wasn't allowed b/c you had to pay for something... But I'm trying again. Anyway I made it alot faster than my old one b/c its all client side and only requires javascript. Other than that it had all the same features as before. If somebody hosts it on their site and puts a link in the article to their site, I would like it if you would say on the page that you got it from my website. --Deo Favente (talk) 02:43, 20 January 2010 (UTC)


 * HA I added it anyways. Now admins, don't be noobish and say you have to pay for it. I'm still confused about all that. In fact, I'm a huge fan of freedom (libre), and with simple web stuff like this free (gratis) generally comes with. --Deo Favente (talk) 05:37, 23 January 2010 (UTC)


 * I'm sure that nobody wants you to pay to post the website. However, it is generally inappropriate to link to your own website. I looked at the link and your site is rather confusing, you seem to use base not as the numerical base, but as the list of digits. I'm not sure that this site is actually that helpful in it's current form. It would at least help to have some explanation on that page. Cheers, — sligocki (talk) 21:29, 23 January 2010 (UTC)


 * I have not been able to get this link to work any time that I have checked it, even when modifying the link to be ".com" instead of ".com.". If it's not corrected, it should be removed, as it is completely useless if it cannot be accessed. Aleph Infinity (talk) 00:05, 12 February 2010 (UTC)

Major edit
I did some major rewriting of the article, which apparently wasn't appreciated. Here is the article after I rewrote the section which here is called "Mathematics". First of all, this section is not about mathematics, just some random notes about positional notation that babbles on in different directions and doesn't do a very good job of explaining (in my opinion). If you disagree with my rewrite please let me know what about it bothers you. I'd be happy to make this more of a collaborative work, but I think it needs some major help. Cheers, — sligocki (talk) 01:04, 23 January 2010 (UTC)


 * I recovered the Babylonian and Greek non-positional forms because both are valid positional systems even though neither is positional within each position. They show that positional systems do not need a large number of symbols equal to their base, as you once stated. The Babylonian base-60 system only used three symbols (ten, one and zero), while the Greek (Hellenistic) system used fifteen (nine letters for units, five letters for tens, and a symbol for zero). The symbols were additive within each position. Not even modern sexagesimal systems use 60 symbols, such as those used for time and angles—they only use ten (0–9) within each position. — Joe Kress (talk) 01:52, 24 January 2010 (UTC)


 * Time and angles are not positional notations. Time is broken up into some blocks of 60 (but also some of 24, 7, 356.24, ...). In the same way that English weights and measures are not positional systems, even though they are broken up into blocks of 12s, 3s, 4s, 2s, etc. Babylonian numbers do have 60 symbols, they just happen to follow a pattern which can be defined by only 3 symbols and placements. But that is really aside from the point, my understanding of this article is that it covers modern positional notation. To me that means the extension of the way we write decimal to any base.


 * To be honest I don't know what you mean by positional within each position. Can you explain that? Cheers, — sligocki (talk) 03:26, 24 January 2010 (UTC)


 * This article is entitled "positional notation", so all aspects of that subject are appropriate, especially its history. I categorically reject your idea that Babylonian sexagesimal positions had 60 symbols. That is your "modern sexagesimal notation" voice talking. By that reasoning, Ptolemy's sexagesimal notation used 60 symbols even though he used Greek numerals that were normally used for all numbers. Unlike Babylonian numbers, Ptolemy only used sexagesimal notation for the fractional portions of his numbers. Those Greek numerals were simply Greek letters with assigned numerical values (e.g., α=1, ε=5, ι=10, τ=300), where numerals above fifty would have been used in the whole number portion, so even they were not new symbols. Even more absurd would be to argue that medieval sexagesimal notation used 60 symbols — that system used decimal numerals that were used for the whole number portions of those numbers. An example is 365 dies 14$I$ 33$II$ 9$III$ 59$IV$ ... (365.242546219... days) used in the 14th century for the length of the tropical year in the Alfonsine tables. Astronomical numbers usually had a whole decimal number followed by six decimal fractional positions identifed with superscript Roman numerals. The fractional positions were called minutes, seconds, thirds, fourths, fifths and sixths (in Latin). The first two superscript Roman numerals have now degraded to simple marks, &prime; and &Prime;. If hours or degrees were the whole number, its fractional sexagesimal positions were still called minutes, seconds, thirds, fourths, etc. By the 19th century, thirds, fourths, etc. were replaced by decimal fractions of the seconds sexagesimal position.


 * I said Babylonian and Greek sexagesimal numbers were not positional within each position, because both were additive within each position. Thus a Babylonian sexagesimal position like <<<|| meant 10+10+10+1+1=32. The corresponding Greek sexagesimal position $\overline{λβ}$ meant 30+2=32. But the corresponding medieval sexagesmal position was positional because it was the ordinary decimal number 32. Elsewhere, medieval numbers still used Roman numerals. — Joe Kress (talk) 01:43, 26 January 2010 (UTC)

Non-standard positional numeral systems (2)
I think the section Positional notation should be boiled down to a minimum, with a link to the main article Non-standard positional numeral systems, and with relevant content being merged into that article. Comments?--Nø (talk) 16:59, 13 June 2010 (UTC)

List of positional systems by base
Please see Category talk:Positional numeral systems.--Nø (talk) 10:55, 24 October 2010 (UTC)

Complex radixes
I carried this paragraph in the article "Complex Base Systems"--Solikkh (talk) 13:27, 26 April 2011 (UTC)

Digits and numerals
The section Positional notation says In order to discuss bases other than the decimal system (base ten), a distinction needs to be made between a number and the digit representing that number. Each digit may be represented by a unique symbol or by a limited set of symbols.

I think we can agree that a digit is a grapheme, not a "set of symbols" (as that passage could be interpreted). Also concerning "a distinction needs to be made between a number and the [representation of] that number", this seems to me not worth mentioning, because I take it to refer to the ordinary need to distinguish between an abstraction (a math number), and a symbol (a math numeral).

Does it want to say "In order to present numerals in another number bases, a new distinction needs to made for the symbolic notation of the digit in any one position"? But this is just saying "The numerals of other number bases have different meanings than the numerals of base-10 number bases." But that evades the rubric "digit". So how about "In order to present numerals in another number bases, a new set of digits is used that is not the set 0-9." &mdash; Cp i r al Cpiral  00:46, 22 June 2011 (UTC)


 * In Babylonian base-60 notation, a digit is composed of several graphemes - if I've gotten the terminology right.
 * From my childhood when I first came across notations other than decimal, I recall a period of confusion untill I had gotten used to the distinction between a number as an abstract entity and its familiar decimal representation - a distinction that is emphatically NOT made when we first learn numbers. Explicitly mentioning (and even repeating) this distinction in an article on postitional notation in a general encyclopedia seems highly relevant to me.--Nø (talk) 08:23, 22 June 2011 (UTC)
 * But how can such a multi-grapheme, Babylonian, ""digit take-up one positional notation? Making the distinction between a numeral and a number is explicitly stated in the article Number.  When I say "not worth mentioning" again here, my concern is mainly about stressing the meaning of  "digit" to do it.  Let's discuss the meaning of "digit".  When that is clear, then the entry can be written however you choose. In any case, I'm sure we can agree that the distinction is made overwhelmingly clear by describing the different number bases of a particular number, even without entering the statement I question.  &mdash;  Cp i r al Cpiral  17:50, 22 June 2011 (UTC)

I made the appropriate copyedits to the first two paragraphs. Also, BTW, I thought it not important to say "real numbers" can be represented in any number base. Firstly, integers can to. Secondly it is not on topic. Thirdly, I suspect any finite number can be represented by any base, and this suspicion extends into transfinites. I made it say "any number" instead of "any real number". &mdash; Cp i r al Cpiral  20:54, 25 June 2011 (UTC)


 * In my edit I initially wanted to fix the – usage (see Template talk:ndash), but then I kept going so maybe I got carried away :). I’d like to know if there are any specific problems with my version; the revert message didn’t explain much. From memory these are some of the problems I had with the earlier version:
 * The dashes shouldn’t be spaced.
 * The paragraph about the meaning of numeral and digit is confusing. I think numeral can sometimes mean an entire number written with of multiple digits (as used in Numerical digit).
 * One of the sentences really needs commas or brackets before I can make sense of it: “The decimal system is widespread, and, being the numeral system we learned with, its numerals are commonly assumed.” compared with “. . . we learned with its numerals—are commonly assumed?!”.
 * The use of quotes, italics and spelling for digits, digit values and other numbers seems arbitrary and potentially confusing.
 * Confusion of letters and numbers proved a little too well for me: “Except one in some fonts? Which one? Oh, except el in some fonts!”
 * If the first paragraph and start of the second paragraph is meant to introduce Nø’s distinction of decimal numbers, how about saying something like “A decimal number represents a particular numerical value. But these two concepts are distinct because a particular numerical value has different representations in different bases.”? But perhaps this point should be made further up in the article. Vadmium (talk) 03:12, 14 July 2011 (UTC).
 * Any numeral represents the numerical value of a number. The distinction between a numeral and its various numbers is made most clear when doing base conversion. &mdash;  Cp i r al Cpiral  05:54, 14 July 2011 (UTC)


 * A timid copyedit compressed my point to an "unseeable", two-sentence paragraph last month. This month, with your prompting, Vandium, I familiarized myself better with the article, edited more boldly, and emmulated some of the good math writers I've been reading by making my point clearly, and made the entire section about what digits and numerals are in the context of positional notation.  Your version did not contain as much about 'numeral' as what I think it could have, and what you said about 'digit' was fine, just not as strongly related to 'numeral' as I hope I have now done, thoroughly.  As far as using quotes or not and choosing between spelling-out or not, I think numerals and digits need quotes to turn them into "signs" to distinguish them from numbers, which are, in this simple case, pure, familiar, decimal magnitudes. I had to move the entire section further down the article to make my points since, sectionally speaking, Numerals use the Base's concepts, and Digits use Notation's concepts and Exponentiation's concepts. &mdash;  Cp i r al Cpiral  05:54, 14 July 2011 (UTC)

It was likely motivated by counting with the ten fingers.
"It was likely motivated by counting with the ten fingers. " Lol. Maybe, but you'd need to chop off a finger or count one of them as zero. Is there much research into the history of the base ten number system? — Preceding unsigned comment added by 96.33.158.121 (talk) 04:53, 12 September 2011 (UTC)

a note for Exponentiation
A section titled "Exponentiation" where the images display how the digits are laid out, is there a way I can have consistent font so I can edit the image to display parenthesis like this (or simply have them in text):

2506 = (2 * b3) + (5 * b2) + (0 * b1) + (6 * b0)

Because some users might do the formula in the incorrect order.✅

Also for mentioning different radixes/bases such as where &ldquo;In base-16 (hexadecimal), there are 16 hexadecimal digits (0–9 and A–F) and the number&rdquo; is mentioned, it can confused some users because using symbols 0-9 looks like decimal, when they could be hex numbers. I think it would be better to use subscript, for example: 1010 = A16❌

Please let me know what you think. 97.83.63.211 (talk) 23:07, 2 March 2018 (UTC)


 * I've implemented your first suggestion. Mathematically there is no need, but using the parenthesis does enforce what is happening to a student.  In the second case though I can't see where the ambiguity arises, unless you are suggesting not using the universal convention that the digits 0-9 retain their meaning in all systems in which they are present.  Subscripts work here where the text is processed, but are useless when computer sources are seen.  Consider: 1234567 = 4553207 = 12d687.  The last form is unambiguous, the first two can be confused.  Martin of Sheffield (talk) 11:27, 3 March 2018 (UTC)

Issues: Susceptibility to fraud in Chinese
From § Issues:

In the example of Chinese usage provided, 壹佰 (100) is contained within 伍仟壹佰 (5100). Given that the characters "伍仟" can be inserted at the beginning to change the value, how does writing out the number on a check in Chinese prevent fraud (in this instance, at least)?

Technotom2001 (talk) 19:56, 13 December 2018 (UTC)

Definition
Is ″Positional notation″ defined anywhere?2A02:A445:EB91:1:A5B6:B3DD:7FE5:9B1 (talk) 19:22, 19 February 2019 (UTC)
 * I'd say that's a good question - that is, I guess there is not one clear definition anywhere in the article. We might include
 * a quite technical definition,
 * or a more leisurely one, like the one I've included in the lead of Non-standard positional numeral systems,
 * or both.
 * It's not easy to put a definition into writing that actually is helpful - that is, it's easy enough to write a definition that is correct and makes sense to anyone who already "gets it"; it's really just one formula (like $$(a_n a_{n-1} ... a_2 a_1 a_0)_b = \Sigma_{i=0}^n a_i\times b^i = a_n\times b^n + a_{n-1}\times b^{n-1} + ... + a_2\times b^2 + a_1\times b + a_0$$), but writing a definition that makes sense to someone who does not already understand the concept is harder. I remember I had trouble when I first came across this kind of thing 40+ years ago -- like: If numbers are used in the definition, isn't it circular? Is there such a thing as a number, separate from it's representation? (Yes, of course, but I had to get used to that idea.) If we use subscript to indicate base for a digit string, how is that base encoded? (Basically, if it was coded the same way as the digit string, it should always read "10"!) Etc.
 * So... what can we do about it?--Nø (talk) 09:04, 20 February 2019 (UTC)
 * Possible to get inspiration in, with more details on the role played by the position. However, there is another issue in the article that requires clarification, the ambiguity in the meaning of the title: "Positional notation" may refer to any numeral system for which the value of glyphs may depend on their position. In this sense, Roman numerals is a positional system (VI ≠ IV). Or "Positional notation" refers to modern numeral systems in base $b$. Or something in-between as the Babylonian system. It seems that the lack of a definition may result of this ambiguity. D.Lazard (talk) 09:44, 20 February 2019 (UTC)

Please do not over-complicate the lead. The lead should answer the question in a simple summary, formal definitions belong down in the body. In this article the section "Mathematics" gives quite an involved description, whether it is a formal definition correct I leave to others, but this is the place to put it. I would suggest slightly rejigging the lead to simplify the explanation as follows:

Comments below. Martin of Sheffield (talk) 09:37, 20 February 2019 (UTC)
 * I like it, but I'd suggest changing this:
 * 1 means one but 10 means ten and 100 a hundred. The "1" symbol has moved position.
 * into this:
 * In 111, the three identical symbols represent one hundred, ten, and one, respectively, due to their different positions in the digit string.
 * --Nø (talk) 12:31, 20 February 2019 (UTC)


 * "In the Roman system, a digit has only one value: I means one, X means ten and C a hundred". This is wrong as I means one or minus one, depending on the position (XI ≠ IX), and similarly for X and C. I do not know how fixing this by keeping the lead short and clear, but we must not introduce assertions that are blatantly wrong. D.Lazard (talk) 14:17, 20 February 2019 (UTC)
 * Would you consider the addition of "earliest" as above? The original form seems to have used IIII for IV and VIIII for IX. Martin of Sheffield (talk) 14:50, 20 February 2019 (UTC)
 * Possibly, but this would need a footnote for explaining "earliest". Also a search on the sources is needed for clarifying which numeral systems have been qualified as "positional" in the literature: Should the Babylonian system system be qualified as "positional" or simply as a "precursor of the modern positional system"? WP must reflect the common use, not the opinion of its editors. A reliable reference for these questions seems The Art of Computer Programming, which has many pages devoted to numeral systems.
 * In fact, except for the history section, this article is about the modern positional systems, that is the extension of any base of the Hindu–Arabic numeral system. Therefore, I suggest the following beginning for the lead:


 * --D.Lazard (talk) 15:49, 20 February 2019 (UTC)
 * I'm happy with that. I always worry when academic experts get too rigorous in a lead, I tend to assume the lead readers may be 14yo boys or non-technical older folk and phrase accordingly.  The body is the place for rigour.  I think you've got the balance pretty well here. Martin of Sheffield (talk) 17:10, 20 February 2019 (UTC)

Base conversion
When converting to decimal from hex or octal I use a multiplicative method. I was taught it at school in the '60s, I can't say I've seen it written down, but it seems much simpler than either the division or Horner's method. The method is simple to code even in shell scripts or AWK.

Using the cited example: converting A10BHex to decimal (41227): Start with first digit:    A                  x 16 =>   160 add next digit =>  161 x 16 => 2576 add next digit => 2576 x 16 => 41216 add last digit => 41227

The system works just as well for smaller bases: 0b11111001 (binary) to 249 (decimal), just a bit more long-winded: 1         1      x2     2 1         3      x2     6 1         7      x2    14 1        15      x2    30 1        31      x2    62 0        62      x2   124 0       124      x2   248 1       249

In practice it's usually simpler to trivially convert binary to octal or hexadecimal before performing the conversion if doing it by hand. Martin of Sheffield (talk) 08:31, 6 August 2020 (UTC)


 * To me both examples look like Horner's method. – Nomen4Omen (talk) 08:57, 6 August 2020 (UTC)
 * Quite possibly. Like most maths articles Horner's method is written for specialist theoreticians, not for mere system programmers like me.  I scanned it but couldn't see anything on radix conversion.  Martin of Sheffield (talk) 09:27, 6 August 2020 (UTC)


 * OK, maybe I can help you.
 * $$A = \{a_n a_{n-1} \dots a_1 a_0\}_b$$ with $$b = \mathrm{Hex} = 16$$ or $$b = \mathrm{bin} = 2$$ is a shorthand for $$A = a_n b^n + a_{n-1} b^{n-1} + \dots + a_1 b + a_0$$. Thus, the number $$A$$ is a «polynomial» in the variable $$b$$ with the coefficients $$\{a_i\}_{i=0,\dots,n}$$. So far, we have the definition of positional notation.
 * Thereafter in the cited section, Horner's method is explained.
 * It is exactly what you are doing: You start with the most significant digit (= coefficient) multiply by the base (= variable) and add the next (less significant) digit (= coefficient). – Nomen4Omen (talk) 10:08, 6 August 2020 (UTC)

Positional factors alternating in sign: Original research?
A citation needs to be given to a reliable source proposing this. Peter Brown (talk) 01:57, 24 September 2020 (UTC)


 * If you mean Positional digits alternating in sign, then there is now given: E.g. Knuth. –Nomen4Omen (talk) 09:12, 24 September 2020 (UTC)

Issues Section
We should remove the "Issues" section from this page or we should put it under "Applications" within a subsection called "Application Issues: Legal and Economics". I believe this problem should be addressed in context and not part of the overall description of the notational system. Also, if we want to talk about 'fraud', we should provide better sources and explain why 'check fraud' is problematic in society. Audiacloud (talk) 19:32, 1 October 2022 (UTC)


 * I completely agree about removing this section of kind of «issues».
 * It is well known that the easiness of cheque fraud has been made as difficult as it was before by marking the specified digits on their left and right sides by surrounding lines, so that an addition is easily detectable. So if somebody fears the fraud of his digits he adds these surroundings. –Nomen4Omen (talk) 08:42, 3 October 2022 (UTC)