Talk:Subtraction

Problems?
This article needs to be fixed. The way it's written a and c are points while b is a distance. This is not a proper geometric analogy to the way subtraction is carried out, and the original writer is subtracting two different classes of objects.

Perhaps you've heard it mentioned that you can't add or subtract apples and oranges 1) Start with 3 apples

2) Take away 2 oranges from those 3 apples

3) One shouldn't expect to have 1 apple left, as subtraction in this way just doesn't make sense.

Points are quite different than distances, and just as in the case with apples and oranges, one should not expect to be able to subtract distances from points. Instead a, b, and c should all be distances from the origin, a point who's numerical value is zero. In this way the distance to c minus the distance to a is the distance b.

Could somebody who posesses both the tools to generate decent images as well as a strong math background fix this article? -User:AlfredR


 * I'm responsible for the ambiguity in the article. I was adding and subtracting vectors but I did not make it clear. When I wrote the article, people were discussing the need for a simple explanation of subtraction for those with little or no math background. My best attempt was to use two position vectors (a,c) and a displacement vector (b) to illustrate addition and subtraction on a number line.  I hid the details since those who wanted an explanation of subtraction would not appreciate them.


 * For example, if vector a = (1, 1), b = (2, 0), c = (3, 1) then


 * a + b = c


 * c &minus; b = a


 * We need to determine a simple (but obviously rigorous) subtraction explanation or decide that a simple explanation is not needed. The Addition page does not offer a simple explanation so we could ignore simple explanations on the Subtraction page as well.


 * I removed the "Basic subtraction" section that I wrote since it is ambiguous.


 * jaredwf 02:59, 27 October 2005 (UTC)

AlfredR is misinformed, and I'm going to restore jaredwf's Basic subtraction section. I'm also going to give a lengthy argument here on the talk page, because the good name of mathematics itself is at stake, and becuase I have nothing better to do.

Geometric vectors (in our case, signed distances) are equivalence classes of "differences of points"; they are exactly the correct objects to add to, or subtract from, points. We need not summon the chimeras that are "position vectors". In some sense, it is more fundamental to subtract a vector from a point than from another vector.

On a more general note, the apples-and-oranges analogy is worse than wrong. Teenagers don't even blink if you subtract the number 2 from the letter x. No one complains if I multiply a matrix with a column vector or an algebraic number with a spinor field. Half the fun of geometric algebra is adding quantities with different dimensions, and hang the taboos. Some of the most interesting mathematics arises when fundamentally different objects interact and when familiar concepts are revealed to tolerate a whole lot more ambiguity than they let you know about in school. Moral: mathematicians are clever. If you think something is meaningless, they will give it meaning.

(By the way, if we make the usual assumption that apples and oranges are unrelated, then 3 apples minus 2 oranges is an element in the free abelian group over apples and oranges. If, for some application, we later find it useful to equate apples and oranges, we can pass to the quotient group by this equivalence through the canonical projection and, yes Virginia, we will have one apple left.)

Now that I'm done ranting, I should comment that the section in question is a little unencyclopedic in tone. It should be condensed or rewritten, but not deleted. Melchoir 10:34, 27 November 2005 (UTC)


 * Your ranting aside, I stand by my previous statement. Your notion of "differences of points" requires that "subtraction of points" is already defined in a manner consistent with the formation of a vector space in which each point labels a vector. With this in mind, it is an obvious tautology to say that these are "exactly" the correct objects to add to or subtract from "points" because you've already assumed that "points' behave like vectors under subtraction.


 * Now since we're doing vector like addition and subtraction the obvious geometric interpretation is to treat each point like a geometric vector. A mixed interpretation is unclear without further information on how to pass between or identify one space with the other. for instance, a point + a vector might be intuitive, but what's intuition tell you about a vector plus a point? (assuming you know nothing about commutativity - just looking at a picture).


 * There's no reason to, geometrically, expect it to be anything in particular. for instance you may think it places a point at the end of a vector, but where should that vector start? why should one starting point be better than any other?


 * Thus my original point, that the geometric interpretation of subtraction employed in this article is not proper, stands. I did not claim, however, that it was incorrect. Once we build the appropriate, trivial, isomorphism we can identify the spaces and go about abusing notation however you see fit.

AlfredR 21:37, 3 October 2006


 * I argue that points and distance are not apples and oranges but rather naval and Florida oranges. In vector math the difference between a point and a direction are an interpretation. In video games (2d or 3d) I subtract my velocity from my origin and come up with an origin which is my new position. I could then subtract my two points, original and new origins, to find a distance. I can now move another object by subtracting this distance from it's origin. In physics there would be a definite difference, in math they are just synonyms. Is there a semantics label that can be placed on articles? LOL Cabbruzz (talk) 15:34, 17 April 2009 (UTC)

P.S. If there are 3 apples and you take away 2, you have 2 because you took them. --116.14.26.124 (talk) 01:15, 23 June 2009 (UTC)

Algorithms for subtraction section
Please explain this. It starts out as 100-11, then jumps to 90-11=79, which in no way answers 100-11=(89). I may be completely confused. Oh yeah, and please don't go into vectors, I mean this is a Subtraction page, and while that is possible, the page links to Arithmetic and Elementary School! Chrishy man  02:59, 4 April 2007 (UTC)


 * Yes, this gives an algorithm by example, which is not the way to do it. The correct way would be to give the algorithm first, then an example of how it works.  The algorithm would involve m and n digit numbers c_m...c_1 and b_n...b_1 (where n<=m perhaps) and a look-up-table that gives the answer for c-b where c is a one or two digit number >= b and b is a one digit number. InformationSpace 06:43, 19 June 2007 (UTC)

I made a substantial edit. Apologies if I stepped on anyone's toes. My motivation is that a lot of parents need to know about the differences in American-style and European-style carries. This is a major problem for parents when their kids come home from school with a subtraction style different than the one the parents know. I also wanted to document the history of the subtraction methods. I am sure that the education literature is full of comments here. Thanks for everyone's patience. -- Ozga 16:21, 23 August 2007 (UTC)

more complex subtraction
Currently this article has nothing at all about subtraction of objects other than numbers. Perhaps links to other things (on wikipedia) that have subtraction operators defined could be added. InformationSpace 06:45, 19 June 2007 (UTC)

X out
Usually when teachers teach their class, there are several items. The student usually Xs out the apples. —Preceding unsigned comment added by Pkkao (talk • contribs) 01:26, 8 July 2007

Negative Difference
In this article, difference is described as "the restult of subtraction". Is it actually the result of subtraction, or the positive result? Take for example The difference of 2 and 3 = 3 - 2 = 1, then The diference of 3 and 2 = 2 - 3 = -1 or |2 - 3| = 1 ? —Preceding unsigned comment added by 137.222.234.90 (talk) 06:37, 12 October 2007 (UTC)
 * Interesting question. Does "difference between a and b" mean b - a, a - b, or |a - b|? -- memset (talk) 09:18, 20 June 2008 (UTC)

Multiple meanings of "difference" can be annoying or confusing and maybe ought to be pointed out by someone more mathematical than me however as I understand it this is not "negative difference" but instead the difference between the "difference as comparison" and the "difference as subtraction result" where the difference (comparison) between {5, 14} as well as {14, 5} are equivalent and 9 (the comparison in this case is in each set the gap on the number line and they give the same result in both directions from one to the other) and the difference (result of subtraction) where the difference of 5-14 = -9 (14 places less than 5 on the number line) in the first set or in the second set 14-5 = 9 (5 places less than 14) and where both sets give different results and one of them is the same result as the comparative difference. One might have to take extra care to avoid ambiguity when writing descriptive verbose math explanations such as "two even factors produce an even product, and subtracting two from the product leaves an even difference" (for those interested this is part of my (possibly wrong or poorly written) answer to question 7 in Exercises 1.1. in "An Introduction to Combinatorics and Graph Theory" which is a CC by-nc-sa licensed maths textbook).90.149.36.98 (talk) 23:29, 26 July 2018 (UTC)
 * Many mathematical concepts are named by a word that has a different meaning in common language. This may produce an ambiguity. Usually, the context makes clear whether it is the common-language or the mathematical meaning that is used. Here, "difference as comparison" is the common-language meaning; in mathematics, the meaning of "difference" is "result of a subtraction". This was not clearly said in the article, and I have fixed this. D.Lazard (talk) 10:01, 27 July 2018 (UTC)

Unambiguous definition for subtraction needed
Subtraction simply reverses movement on the line of real numbers, does it not? This is especially comprehendable when subtraction is viewed as addition (as in algebra.) Consider:

5-3=2 backward movement of |3| on the number line.

5-(-3)=8 forward movement of |3) on the number line; that is to say, the "-" immediately proceeding "5" is reversing the operation that would otherwise be done if it was only "-3": addition. Children in junior-high often say this as "minus a minus."

What I am saying is very obvious, but subtraction defined as "reversing movement on the line of real numbers" is an all-encompassing definition that would cover both the algebraic and arithmetic understanding of subtraction. This has probably been said and proven somewhere in academia, but perhaps we could place it on Wikipedia. —Preceding unsigned comment added by CPRdave (talk • contribs) 01:07, 26 October 2007 (UTC)

The article should state that subtraction is the inverse of addition. The subtraction of a number from itself gives 0, the identity element for addition. For a given element a of an additive group, the inverse -a is unique. This should also be in the article IMHO. John (talk) 16:34, 25 June 2014 (UTC)

Line Segments
I applaud the work of the person who took the time to carefully explain how subtraction works despite there being no one who can read this article and not subtract. :) -- MQ Duck 07:45, 4 March 2009 (UTC)

Citation Needed
Is that "citation needed" tag after the explanation of the terms minuend and subtrahend really necessary? It is surely common sense to assume that these particular words are seldom used in mathematical parlance. --T.M.M. Dowd (talk) 23:49, 1 January 2010 (UTC)

Subscript text 224,940-175,000= to —Preceding unsigned comment added by 121.1.53.54 (talk) 23:33, 19 February 2010 (UTC)

Subtraction and Precision
It would be nice if a section on the problem of precision loss in limited-precision subtraction (if numbers are subtracted that are much larger than their difference) and on common workarounds could be added. As this is probably the most important source of precision loss in real-world algorithms. -- 92.229.180.109 (talk) 13:58, 22 March 2010 (UTC)

Minuend and Subtrahend archaic? A single, opinionated, reference footnote on the article about subtraction?
I take some issue with this article. "Minuend" and "subtrahend" are commonly used in computer science as well as discussion of arithmetic. However, some editor of this article has made the effort to elevate the assertion:
 * "The words "minuend" and "subtrahend" are uncommon in modern usage."

While this point is arguable, I question that it should appear:
 * 1) In the first paragraph of the article or even the lede section.
 * 2) As the only referenced assertion of the entire article.
 * 3) As an assertion of objective fact based on a reference with an opinionated title.
 * 4) As an assertion made without qualification in a technical article, subject is a mathematical operation.  Consider this possibility:
 * "The words "minuend" and "subtrahend" are uncommon other than in technical writing."

Heathhunnicutt (talk) 17:41, 12 June 2010 (UTC)

 Subtrahend  is certainly not an archaic term. (I'm in agreement with Heathhunnicutt.) "Subtrahend" is a specific entity that refers to the value (or item that contains the value) by which another value will be decreased. It's taught in Computer Science, Engineering, and Mathematics courses. Even 2nd grade Math (where U.S. students are generally about 8 years old) are taught the meaning of "subtrahend". And to cite a satirical work as a reference suggests that the poster is having fun, or on a personal agenda. — Preceding unsigned comment added by KentOlsen (talk • contribs) 18:48, 8 December 2011 (UTC)


 * I teach math in New York. Kids at all grades (2-9) are unfamiliar with subtrahend and minuend, and they cannot reliably say which is which. Ditto for dividend and divisor. The know (or can guess that two addends form a sum) and that the answers to multiplication and division problems are products and quotients, but that's about it.


 * Whether this means the terms are archaic or simply that our schools aren't doing a good job on math vocabulary is an open question. --Uncle Ed (talk) 15:31, 8 December 2013 (UTC)


 * I am 61 years-old, a Berkeley graduate, a teacher since 1999, and today, while reading about the Mayan number system, is the first time I learned the words minuend and substrahend -and spellcheck thinks they are both misspellings. And so, even though I like them and think they are groovy and useful, I'd also be fine with assigning them the label 'archaic'. 2601:648:C180:E10:ADCD:434:7A09:C7B6 (talk) 15:45, 2 December 2023 (UTC)


 * Labeling terms as "archaic" would imply that they've been superseded by others, which isn't the case here. "Minuend" and "subtrahend" are the equivalents of similar latinate designations in more common use, such as "divisor", "multiplicand" (and "latinate" is the only word, of those I've just written, that's flagged by my spellchecker). Dhtwiki (talk) 06:42, 3 December 2023 (UTC)
 * I agree that "minuend" and "substrahend" could not be obsolete, if there was no terms to replace them. But we have "first term" and "second term" (of the substraction), which are unambiguous, very commonly used, and clearly less pedantic. The fact that many teachers do not even know these terms shows clearly that they are indeed obsolete and pedantic.
 * The situation is different with "dividend" and "divisor", because the various presentations of divisions (inline, as fractions, as ab^{-1}, etc.) make the use of "first" and "second" ambiguous. D.Lazard (talk) 14:08, 3 December 2023 (UTC)
 * May I recommend the excellent word obsolescent? It recognises the status of a word or technology in the real world without need for a mathematical proof. --𝕁𝕄𝔽 (talk) 15:31, 3 December 2023 (UTC)
 * That's a good description. Technically the words aren't quite disused, but I think the proportion of English speakers who know even their rough meanings is low, and most of us even in that group don't remember which is which. Certes (talk) 16:36, 3 December 2023 (UTC)
 * I myself can't find "minuend" and "substrahend" used in math books outside of my math dictionary, which is possibly due to the fact that advanced math treats subtraction as the addition of negative numbers. But I also don't see the terms marked as obsolete in any English dictionary that I've checked. Dhtwiki (talk) 10:47, 21 December 2023 (UTC)
 * And that is a great illustration of obsolescent. Not provably obsolete but not used in current literature. --𝕁𝕄𝔽 (talk) 10:52, 21 December 2023 (UTC)

borrowing
The article currently states that "some" American schools teach the borrowing method. As far as I know it is virtually universal in American schools. Can any-one find a citation so we can beef up the statement to reflect reality?Kdammers (talk) 03:10, 9 January 2013 (UTC)

counter-intuitive
"In mathematics, it is often useful to view or even define subtraction as a kind of addition, the addition of the additive inverse. We can view 7 − 3 = 4 as the sum of two terms: 7 and -3. This perspective allows us to apply to subtraction all of the familiar rules and nomenclature of addition. Subtraction is not associative or commutative—in fact, it is anticommutative and left-associative—but addition of signed numbers is both." the first part makes the "naive" reader think that subtraction should be associative ("allows us to apply to subtraction all of the familiar rules ... of addition"), but this is followed -- with-out a "however" and only a delayed "in fact"-- by a sentence that says "Subtraction is not associative...." This makes for unnecessarily tough reading. Kdammers (talk) 03:15, 9 January 2013 (UTC)

Brownell did not invent crutches
The article states that "… crutches are apparently the invention of William A. Brownell who used them in a study in November 1937." This is clearly false. See ref 1, which I just added for "crutches" and the Austrian method. You can read it on google books, click on the link. This is a book published in 1916 by an American, Paul Klapper. He talks about crutches and their usage in subtraction algorithms at least 21 years before Brownell's article. John (talk) 04:01, 24 June 2014 (UTC)

I went to the library and looked up the article by Brownell by looking in the comprehensive index of the journal, The Mathematics Teacher. The text is wrong, his essay appears in vol. 37, year 1945 (not 1937) of The Mathematics Teacher. There is no mention of crutches. So the statement about Brownell has two errors. I'm rewriting the paragraph. John (talk) 00:48, 25 June 2014 (UTC)

Correction: it was the wrong article. Brownell conducted his study in 1937. It was apparently published in 1949, according to Susan C. Ross and Mary Pratt-Cotter, Subtraction From a Historical Perspective, published in 2010. — Preceding unsigned comment added by John Palkovic (talk • contribs) 00:58, 25 June 2014 (UTC)

Partial method
+ 300 − 40 + 2 = 262. This begs the question. Using this method, how is one supposed to know that 300-40=260? It seems at least one step is missing. Kdammers (talk) 06:29, 27 November 2014 (UTC)

Anticommutativity
"Subtraction is anti-commutative, meaning that one can reverse the terms in a difference left-to-right, and the result is the negative of the original result. Symbolically, if a and b are any two numbers, then    a - b = b - a. " Have I forgotten all my math or is old age dementia clouding my understanding? Or maybe it's my vision. This looks patently wrong to me. The text says the value of the two differences is opposite in sign (i.e., |a-b| = |b-a|. Right, that's clear.  But  then it continues with an example that shows to my eyes some-thing quite different, .i.e., that the differences not only have the same absolute value but the same value, which is only true for a few special cases, i.e., where a=b.  Since this part of the article seems to have been around for some time, I am loathe to change it even though I just cannot at all see how it can be correct. E.g., let a=5 and b=2.  Then a-b=5-2=3 and b-a=2-5=-3, 3NOT=-3. Hence a-bNOT [necessarily]=b-a. Or am I blind or crazy????? Kdammers (talk) 07:49, 27 November 2014 (UTC)


 * It looks as though the example should be written "a-b = -(b-a)", to match the text. Dhtwiki (talk) 09:14, 27 November 2014 (UTC)

rows of zeros?
Try subtracting this:

1005

-928

Notice that you are left with only 7. You cannot barrow after you turn the 5 into a 15, as the "1" in 1005 is consumed. On a calculator, it would be 77. Is there a way around this stuck? Joeleoj123 (talk) 23:16, 26 February 2016 (UTC)


 * The phrasing in the article continuing the borrow leftwards until there is a non-zero digit from which to borrow should mean that you take from the tens, making -1, then from the hundreds, making the tens column 9, but the hundreds now -1, which is immediately made up by taking the 1 from the thousands. For the subtraction above, you should have something like this, based on the article explanation, if it isn't too confusing:


 * Dhtwiki (talk) 19:56, 27 February 2016 (UTC)

Citation needed for claim about "general" treatment in advanced algebra
Wcherowi has reverted the that I associated with the claim that "In advanced algebra ... an expression involving subtraction like A − B is generally treated as a shorthand notation for the addition A + (−B)." Minimally, a claim that a usage is "general" implies that it is used more than half the time. This is an empirical claim; why should it not need a citation? The edit summary says this usage is "standard", but this term is open to the same problem as "generally". Subtraction can be taken as primitive and −x defined as the result of subtracting x from the additive identity. What is the source for the implication that this is nonstandard? If there is one, it should be included in a citation. Peter Brown (talk) 19:33, 21 June 2019 (UTC)
 * What we have here is yet another example of the conflict between "school mathematics" and "advanced mathematics". Subtraction is considered a primitive operation in elementary treatments and it certainly can be made a primitive operation in advanced mathematics, but these days no one does this. The ruse of subtraction is dropped as soon one starts to consider abstract mathematical systems and the same is true of division. This is due to the importance of the concept of inverses in such systems, which overpowers the plethora of operations that are taught at the school level. Since the article is being written for a variety of readers and the passage is clearly marked as being for advanced mathematical areas, I felt that BLUESKY would come into play. --Bill Cherowitzo (talk) 21:51, 21 June 2019 (UTC)
 * Once again, the text under consideration is


 * In advanced algebra ... an expression involving subtraction like A − B is generally treated as a shorthand notation for the addition A + (−B).


 * No claim involving the term "generally", however, is a candidate for BLUESKY. As the Collins dictionary notes, "You use generally to say that something happens or is used on most occasions but not on every occasion." BLUESKY is inapplicable to claims that explicitly refrain from asserting universality.


 * The edit summary directs attention to Field (mathematics). That article presents an alternative definition in terms, inter alia, of subtraction rather than the additive inverse.  In admitting an alternative, does the article betray the tenets of advanced algebra?


 * Peter Brown (talk) 16:38, 22 June 2019 (UTC)
 * Nobody asserts that one cannot use subtraction as a primitive operation. The fact is that the use of subtraction as a primitive operation is possible, but rare; this is the exact meaning of "generally". However, for being more accurate, one would need a study of all textbooks for counting how many define subtraction as a primitive operation. As such a study clearly not exists, it is impossible to provide a citation, and making this study would be WP:Original synthesis. So the claim for which you want a citation cannot be sourced. As this claim can be clearly useful for many readers (giving them indications to what is the current trend in mathematics), it must be kept, although no source can be provided. IMO, this is a case of WP:IAR. D.Lazard (talk) 17:20, 22 June 2019 (UTC)


 * It is surely not "impossible to provide a citation". If the claim is indeed as useful as D. Lazard asserts, surely there are reliable sources somewhere that do make the claim, perhaps even using terms like "generally" or "rare".  Why not leave my  alone until someone acquainted with such a source provides it? Peter Brown (talk) 21:54, 22 June 2019 (UTC)

Should compound subtraction be on this page? Or in a separate page?
A century ago, it was common to teach compound subtraction as an explicit method during arithmetic. Examples of this include taking measurements in cups, tablespoons, and teaspoons and subtract, or to take an age at death and the date of death and work out the age it birth. Efficient methods for hand calculation were taught, and there are also some subtleties: for instance, the inferred age at birth from compound subtraction depends on the order in which the subtract is done, i.e., whether the days, months, and years are subtracted first, second, or third. There are numerous (old) books and articles on the subject, plenty enough that it should be covered somewhere on Wikipedia. My question: should it be a subtopic of this page, or should it have its own page? Barryriedsmith (talk) 20:57, 13 September 2021 (UTC)

Related to Subtrahend and Minuend usage
Subtrahend derived from 'subtrahendus' means 'to be taken away. Minuend derived from 'minuendus' means 'to be lessed/reduced'. Minus means 'less' These are given in wikipedia and other websites. For example 10 - 3 = 7, here - 3 in words minus 3 which means less 3. It means 3 is 'to be lessed'. It means 3 is the minuend by the meaning. Considering another example a boy is expected to be taken 10 marks. But he takes 7 marks, in other words the boy took '3 marks less'. This means '3 less' to 10. It means 3 is the number 'to be lessed'. So 3 is the 'minuend' by the meaning. Here expected 10 'to be taken'. So by the meaning 10 is the 'subtrahend'. Inkartumac (talk) 09:16, 13 February 2022 (UTC)


 * Where did you obtain such information? - S L A Y T H E - (talk) 13:10, 14 December 2022 (UTC)


 * You have stated the opposite of what the article says, what my mathematics dictionary (James & James) tells me, and even by your own linguistic analysis in the beginning of your post. 10 is the minuend, the number 'to be lessed/reduced', 3 is the subtrahend, the number 'to be taken away'. The result is the remainder or difference. Dhtwiki (talk) 23:31, 14 December 2022 (UTC)
 * As the article says, minuend − subtrahend = difference. That matches the language above: the minuend 10 is reduced as the subtrahend 3 is taken away from it. Certes (talk) 00:29, 15 December 2022 (UTC)
 * No I mean the etymology. Sorry if I didn't clarify. - S L A Y T H E - (talk) 16:14, 15 December 2022 (UTC)
 * "Minuendus" and "subtrahendus" are the future passive participles, or gerundive, of the Latin 3rd-conjugation verbs "minuere" and "subtrahere", respectively. Is that what you're asking? Dhtwiki (talk) 02:21, 16 December 2022 (UTC)
 * The sources. - S L A Y T H E - (talk) 09:58, 16 December 2022 (UTC)
 * That isn't saying much about what you're specifically looking for. According to the OED, "minuend" was first used by William Jones in his 1706 book Synopsis Palmariorum Matheseos: Or, A New Introduction to the Mathematics, which may well contain "subtrahend" for the first time (I didn't confirm that). The book also contains the first occurrance of 'π' in print, according to Jones's Wikipedia article and its references. — Preceding unsigned comment added by Dhtwiki (talk • contribs) 03:03, 17 December 2022 (UTC)  (edited 03:58, 17 December 2022 (UTC) and 04:01, 17 December 2022 (UTC))

"Restar" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Restar and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 February 15 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. User:1234qwer1234qwer4 (talk) 20:19, 15 February 2022 (UTC)

What needs citations?
Are there any specific claims that need citations? At least give me 3. - S L A Y T H E - (talk) 05:52, 27 December 2022 (UTC)


 * The template more citation needed that is used is not the right one, and should be replaced by templates at the beginning ot sections,  and . For the other sections, there are enough citations, even if many of them are not WP:reliable sources and could be tagged with better source needed. D.Lazard (talk) 10:48, 27 December 2022 (UTC)