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Talk:Chinese numerals

Separate info on Unicode representation into separate section?
For someone like me who wants to know about how numbers are written in Mandarin Chinese, the information about how they are represented in Unicode is really irrelevant. Could the statements like "such-and-such was an error which was corrected in the next version of the standard" be separated out into a different section? How numbers are represented in a language and how that language then is represented in a computerized form are two different subjects - related enough to be in the same article, yes, but not mixed together, I think. Evan Donovan 03:11, 19 October 2006 (UTC)

The numeral 2 (liang and er) (两 and 二)
The translations for numbers such as 1,200 can be both yi qian liang bai (一千兩百) or yi qian er bai (一千二百). Although this is addressed in the article, too many veriations exist in various dialects as to render the simple rules stated rather inadaquate. Maybe the way to do this is to remove all instances of 2 in the examples of forming Chinese numbers except in the instances where we are illustrating the liang (兩) versus er (二) idea? --Sjschen 19:13, 9 December 2005 (UTC)

 * Alternatively, this point can be noted somewhere early on the page, and the entire page refers to Standard Mandarin (and maybe Standard Cantonese). After all, you wouldn't want to include every dialect here... Neonumbers 02:57, 10 December 2005 (UTC)

 * I cannot speak for other dialects. However, in Mandarin and Cantonese, the use of liang and er are very well defined and they are consistent across the two dialects.  Perhaps the same rule applies to all Chinese dialects too.  The description in the article is pretty accurate regarding its general use.  You may be able to find some exceptions, but you should keep those as what they are: exceptions, not the rule.  The word liang is used only when the number is used as a counter (e.g. 兩個人).  The word er is used as a number or an ordinal (e.g. 二號, 第二) and in certain situations, used in the sense of double, couple or duet etc.  (e.g  二人世界, 二重奏).  However, you seldom see these two words used interchangably because they are not the same.  For example, you'll never see 二個人, 兩號, 第兩.  The only situation these two are interchangable is when they are used in the sense of double, duet with the same meaning as 雙. e.g. 雙人世界, 兩人世界, 二人世界 etc.  Kowloonese 22:22, 12 December 2005 (UTC)

 * In Taiwanese, the rules specified don't work. >200 and 2 all use 兩 (Min: Neng) while 20 and 12 use 二 (Min:li). This makes me wonder if differing rules exist out there for different Chinese dialects, which is why I brought up the idea of eleminating 2s from examples in the article while creating a better section to deal with the issue. --Sjschen 00:41, 13 December 2005 (UTC)

 * I tend to think of liang as a word meaning "pair", as such it is not a number. The english equivalent would be discussion of the word "pair" or "dozen" in a wikipedia article about numbers, inappropriate in my opinion. Kotika98 (talk) 05:45, 21 March 2013 (UTC)

 * The rule is only clear when someone say single digit 2 as in two and nothing else, then 二 is the proper choice. Even then depend on dialect or local slang, 兩 can be perfectly understandable as "two" by a visitor. But when "2" is in combination meaning 2 of something, then its most probably interchangeable. As a native you may find it odd but at the same time understandable and acceptable when speak to certain origin of the speaker and preference. Taking example above 二號 (number two as in "room number two") and 第二(number two as in "room number two or second room from the right hand side". The other use case 兩人/二人 (two person), and moving two complex syntax 雙人世界, 兩人世界, 二人世界 (two person world, usually means couple) works perfectly find for every native speaker. Then you have specific noun that must use one form, example 二重奏 (duet). As a native Cantonese I also find Kowloonese's explanation not representative and contradicting. Example 二個人 does not work because it is a special case. Consider 二 pair with other collective counter: 二家, 兩家 both mean two families or two shops/businesses. Finally back to his final example 二號 vs 兩號. These two can and cannot be interchanged. One stand-alone used and not interchangeable: 二號 number two, 兩號 two numbers. Context dependent scenario 1: 跳了二號 （either means skipped number two or skipped two numbers)，跳了兩號 (mean skipped two numbers only). Context dependent scenario 2: 二號人物 （as in the number two of the organisation)， 兩號人物 (two kinds of people). This differential comes from the meaning 號 is use as either positioning or type as required by context. As demonstrated, the rule is definitely not clear cut and certainly double, duet or with the something in pairs are not the determining factor. — Preceding unsigned comment added by 14.192.209.176 (talk) 15:37, 13 October 2020 (UTC)

Hangzhou
Though the name HangZhou numerals implies they originate from HangZhou, China, these numerals are not used in HangZhou anymore. I asked a 30 year old lady who grew up in HangZhou, she had not seen these numerals in her life. Her mother recognizes what they are. However, she had not seen them in use during her 40 years in Hangzhou. She remembered seeing these numerals in Malaysia over 50 years ago.

Several of my friends from Taiwan, who are in the 40s, had not seen these numerals ever.

On the other hand, the Chinese herbal doctor I used in San Jose, CA wrote me a herbal prescription using these Hangzhou numerals. It is evident that these numerals are still in use in some Cantonese communities in the US. But with just a small sample, it is hard to determine how extinct it has become.

Kowloonese 11:05, 29 Jul 2003 (UTC)

 * The Unicode standard corrected the name to SuZhou numerals since the above question was posted. That explains a lot.  Kowloonese 03:23, 18 Mar 2004 (UTC)

I have removed the question that asks whether the system for writing Suzhou numerals in Hong Kong is the same. It is indeed the same. I've also added a vertical example.

I remember that the digits 6–8 also gets rotated when a string of such digits are written. But I don't have my math textbook from primary school any more :-( Can anyone check? - Gniw 05:09, 6 Feb 2005 (UTC)

Zhao

 * got a problem. In my dictionary, it says "Zhao4" is
 * 1 million
 * 1 trillion (ancient use)
 * maybe someone should make a note there. --FallingInLoveWithPitoc 06:36, 25 Oct 2003 (UTC)

 * Is a "trillion" according to the long scale or short scale? 81.232.72.53 12:58, 9 January 2006 (UTC)

According to Unicode standard zhao means trillion in Japan and Taiwan, it means a million in China. It also means trillion in Hong Kong. --Anon2

 * After some research, I have found this to be the current situation: all Chinese communities defines zhao4 兆 as 10^12 when it is used in writing numbers. In Mainland China, Hong Kong and Macau, it means 10^6 when used as an SI prefix, e.g. zhao4he 兆赫 = megahertz. That convention came from the fact that in ancient China there were three systems to define number words larger than 10^4: yi4 億, zhao4 兆, jing1 京, gai1 垓, zi3 秭...
 * 10^5 10^6 10^7 10^8 10^9 ...
 * 10^8 10^12 10^16 10^20 10^24 ...
 * 10^8 10^16 10^32 10^64 10^128 ...
 * The second system became the standard. The first system survived in the translation for mega because Chinese otherwise lack a monosyllabic word for one million.

 * In Taiwan, however, zhao4 still means 10^12 when used as an SI prefix. So zhao4he 兆赫 = terahertz. The megahertz is called bai3wan4he4 百萬赫.

 * That should explain all the confusions. Dictionaries and Unicode standard may show inconsistencies because they may not be looking at the whole picture when they seek definitions of individual characters. I have already made major changes to the main text to resolve the values of all numerals > 10^4. I will continue to make some more minor changes to make it clearer.
 * Felix Wan 2004-07-08 02:30 (UTC)

About those big numbers, I found something a little bit different. The second one among those three ancient systems should be close, but different from, the modern way.

《數術記遺》(東漢徐岳) ：「黃帝為法，數有十等，及其用也，乃有三焉，十等者， 億、兆、京、垓、秭、穰、溝、澗、正、載。三等者，謂上中下也。其下數者，十十變 之，若言十萬為億，十億為兆，十兆為京。中數者、萬萬變之，若言萬萬為億、萬萬億 為兆、萬萬兆為京。上數者數窮則變，若言萬萬曰億，億億曰兆，兆兆曰京也。下數淺 短，計事則不盡，上數宏廓，世不可用，故其傳業惟以中數耳。」

So technically, it goes up like 10^8, 10^16, 10^24, 10^32, 10^40, 10^48 etc. --Liuyao 01:52, 21 Nov 2004 (UTC)

You are right. Recently, I have also noticed that there exist two conflicting sources, one saying that "萬萬為億、萬萬億為兆、萬萬兆為京", the other saying that "萬萬為億、萬億為兆、萬兆為京", which is the modern way. I am going to include the other ancient way in the main text. By the way, welcome to Wikipedia. You may be interested in the China-related topics notice board. -- Felix Wan 10:08, 2004 Nov 21 (UTC)

Hao/li
Aren't hao2 and li2 reverse? --Anon3

 * I suggest someone with good Chinese language backgroud to research deeply into this topic. When China turned to the SI metric system, such as milligram, centimeter, millimeter etc., they use hao and li as a qualifying adjective in front of the unit.  For example, centimeter is called li mi and millimeter is called hao mi etc.  However, I really doubt if those who picked these translations really researched into the original usage of hao and li in Chinese measuring units.  If I remembered correctly from the books I studied a few decades ago.  Hao and li are unit for weight, not quantifiers.  When used as weighing unit, li was 10x hao.  However, when used in a SI quantifier, hao is 10x li.  Hence, I speculate those who made up the translations did it arbituarily with no knowledge of the traditional usage of these two characters.  Of course, all language evolves based on contempory usage, so even a misusge will turn into standard.

 * If you look up an old Chinese dictionary, you may find hao & li very interesting because they have different value depending on context. Li = One thousandth of a Chinese 尺 foot. Li = One hundredth of a 畝 [mu3]. Li = One thousandth of a 兩 [liang3] or tael. Li = (Rate of interest) one tenth of one percent (i.e. one thousandth of a unit).  Hao = a centigram in weight or one-hundredth of a centimeter in length.

 * You will see a consistency if you look closer. The reason that li2 = 1/1000 of 尺 foot was because there is a unit name for 1/10 foot: cun4 寸. fen1 = 1/10 cun4, li2 = 1/100 cun4, hao2 = 1/1000 cun4,...  Similarly, qian2 錢 = 1/10 liang3, fen1 = 1/10 qian2, li2 = 1/100 qian2, hao2 = 1/1000 qian2,...  For small units of measure beyond the named units, the order is always fen1, li2, hao2, si1, hu1, wei1 for 1/10, 1/100, 1/1000, 1/10000, 1/100000, 1/1000000.
 * Felix Wan 2004-06-26 02:32 (UTC)

 * For measurement of land area, mu3 畝 is already the smallest named unit, so fen1 = 1/10 mu3, li2 = 1/100 mu3, hao2 = 1/1000 mu3,...

 * For a centigram and 1/100 of a centimeter, actually the old translation was gong1hao2 公毫. When SI was first introduced to China, the people translated them by adding the prefix gong1, meaning public, to the traditional units. So, kilogram = gong1jin1, hectogram = gong1liang3, decagram = gong1qian2, gram = gong1fen1, decigram = gong1li2, centigram = gong1hao2. Similarly, meter = gong1chi3, decimeter = gong1cun4, centimeter = gong1fen1, millimeter = gong1li2, and 1/100 of a centimeter is naturally gonghao2.

 * For interest rates, there is some complication. If we are talking about annual rates, 1 fen1 = 10% of the principal, 1 li2 = 1%, 1 hao2 = 0.1%, although in modern Chinese, we usually use li2 exclusively.  For monthly rates, 1 fen1 = 1% per month, i.e. 12% per year for simple interest. 1 li2 = 0.1%.  For daily rates, 1 fen1 = 0.1% per day, i.e. about 3.65% per year for simple interest. 1 li2 = 0.01%.

 * In conclusion, although li2 and hao2 have many meanings depending on the situation, the order is consistent within each single system: fen1 = 10 li2, li2 = 10 hao2.

 * Felix Wan 2004-07-08 03:12 (UTC)

 * Another example is zhao discussed in the section above. PRC defined zhao differently from Japan, Taiwan and Hong Kong.  I am wondering if this kind of screw-ups happened during the Cultural Revolution which was an era when knowledgable scholars went to jail and uneducated ideots took over adminstrations and standards committees.

 * See my explanation under zhao4. (I have moved my old answer, dated 2004-06-26, to where it should belong, and made some changes.)
 * Felix Wan 2004-07-08 02:30 (UTC)

 * Did you know that "an apron" used to be "a napron"? Of course current usage wins.  But it would be an interesting fact to mention in dictionary or encycopedia to help future scholar to understand the switch over and why.

 * Kowloonese 05:33, 24 Jun 2004 (UTC)

Numeral 9
I reverted a change regarding the Su Zhou numeral 9.

Many web browsers are displaying the wrong glyph for this character, hence what you are looking at on this page may be wrong, don't trust the display.

 * 9 is a dot on top of a variant of the 〤 (4) symbol
 * (〩 U+3029) is a character with two brush strokes which writes in the stroke order similar to "jiu3 (久 U+4E45)" by combining the first two strokes.
 * 夂 & 夊 are characters with 3 brush strokes.
 * 攵 & 文 each has 4 brush strokes

The bottom line is that all the above are not the same though they all look similar. When the browsers show the wrong glyph, it just make these characters more confusing.

Kowloonese 03:21, 18 Mar 2004 (UTC)

114?
From the article (emphasis mine):

 * However, when more than two digits are involved, the abbreviation usually does not take place except in Japanese. For example, 114 is read as '1' '100' '1' '10' '4', and definitely not '100' '10' '4'. Although '1' '100' '10' '4' is marginally acceptable, it is not common.

I wonder what language you are refering to? 114 in Mandarin Chinese is universally spoken as yi1 bai3 shi2 si4. As a speaker of Chinese, I find the allegation that it should be read as yi1 bai3 yi1 shi2 si4 quite strange, as I have never heard yi1 shi2, ever. This is because shi2 is grammatically not considered a measure word, where as higher units are. Therefore you always say yi1 bai3, but never just bai3. The exception to this is the use of er2 rather than liang3 with bai3 and higher, reflecting their shared status as numerals. liang3 bai3 is said, but it's sort of slang. One would never write it. But shi2 is different; it is never preceded by yi1.

 * It is not true to say shi2 is not a measure word, shi2 and bai3 are used the same way grammetically. e.g. yi1 shi2, er2 shi2 or shi2, er2 shi2 are acceptable.  Kowloonese 19:14, 20 May 2004 (UTC)

Someone ought to change this information, because it's wrong. --Anon

 * Your dialect of Mandarin is not universal. I didn't write your quoted statement, but it is true that Taiwanese Mandarin certainly says yibai-yi-shisi for 114, in fact, I never heard of yibai-shisi. Maybe it's a Mainlander feature or whereever you're raised. Either way, that paragraph should be modified. It is confusing. --Menchi 11:36, 20 May 2004 (UTC)

We say "yi1bai3-yi1shi2si4" not "yi1bai3-shi2si4". the latter one sounds weird. And in Cantonese, when saying 1XX, it's acceptable to omit the yi1, especial when it comes to 1X0 (except the case of 100). --θαλαμηγός (talk) 18:30, May 20, 2004 (UTC)

 * Isn't it what the article say? yi1bai3-shi2si4 is marginally acceptable, but not common.  Though the mainlander said the opposite.  Cantonese and Taiwanese Mandarin preserved many traditional usage of the ancient Chinese language while the Mainlander has introduced many variations probably as side effects of the simplication done decades ago.  Every language evolves over time.  This is just one example that it went to two extremes.  The mainlander never heard of yi1bai3-yi1shi2si4 while Cantonese and Taiwanese never heard of yi1bai3-shi2si4.  I am no authority in this kind of stuff.  Change the sentence as you see fit.  But be neutral to present both extremes.  Kowloonese 19:27, 20 May 2004 (UTC)

 * Hi, I'm the anon poster, sorry about the confusion. It's true that I speak mainland Chinese and am not very familiar with Taiwanese differences.  I also do not speak Cantonese or Minnanhua so I can't exactly comment on those.  However, in the Beijing dialect, we omit the yi1 in front of shi2.  While it certainly true that many non-Mandarin dialects are more conservative (closer to classical Chinese), Mandarin is the language of Beijing.  Consider that in largely Cantonese speaking areas, spoken Mandarin is probably more likely to use some Cantonese grammatical features; similarly for places where Minnanhua is common (Taiwan).  I currently live in Shanghai and the locals here certainly allow their Shanghainese habits to influence their spoken Mandarin.

 * The fact that Beijing dialect is "standard" Mandarin doesn't change the fact that many people do not speak it that way. And actually, even Beijing Mandarin isn't really standard, I guess.  Hebei Mandarin would probably be closer.  But my point is that I'm not trying to argue that I'm right simply because I speak a Beijing dialect.  Especially since this article is about Chinese numerals, and so any Chinese dialect's habits are equally valid.  I think that in this light, we should mention both usages.  It would also be good if we could get some sort of authoritative answer on the usage in Taiwan, and find out whether the use of yi1 shi2 is current because that's how it's said in Taiwanese, or whether some antiquated form of Mandarin used it too.  Very interesting.  Actually I have some Taiwanese coworkers, maybe I'll ask them too.  I know that there are a lot of differences between Mainland mandarin and Taiwanese mandarin that have been officialized; for example I hear Taiwanese pronounce 法 as fa4 a lot, which sounds strange to me (in Beijing and the surrounding countryside we say fa3).  What's interesting about that is that I've been told Taiwanese dictionaries actually list fa4 as the pronunciation of this character, indicating that the government actually considers it to be the correct pronunciation.  So in light of yi1 shi2, is it a similar situation?

 * Sorry to barge in here and claim it was wrong when really, I guess, Mandarin varies quite a bit! I'll pop by again tomorrow and see what you all think :)

 * The whole paragraph is about where the yi1 is commonly omitted and not. Yi1 shi2 is not common, but yi1 bai3 yi1 shi2 and shi2 is.  The leading yi1 sometimes is omitted, however the second yi1 in yi1 bai2 yi1 shi2 is not leading.  Kowloonese 08:03, 21 May 2004 (UTC)

 * I realize this. However, the fact still remains that in the Mandarin spoken in Beijing and in the surrounding area, which is presumably "standard", one does not say yi1 bai3 yi1 shi2 si4.  I believe (though I am not sure) that the use of yi1 bai3 yi1 shi2 si4 in Hong Kong & Taiwan is most likely due to the influence of the Cantonese and Taiwanese dialects.  But I do not know this for sure.  The fact remains that the yi1 is ommitted in Beijing, on which both the official mandarin in Taiwan and the official mandarin on the mainland are based.  While I am certainly willing to believe that the Taiwanese usage differs, I do not think you can make a reasonable case for the Taiwanese usage being the "correct" one, if only because the vast majority of Mandarin speakers are on the Mainland, versus Taiwan's relatively small population.

 * Anyway, mark what you want, but remember that China's 1.3 billion Mandarin speakers are likely to disagree with you on this point.

 * You may be able to make a reasonable argument that the usage you describe was current in Ancient Chinese, as evidenced by the relatively conservative nature of Cantonese and Taiwanese.

Oh, My! you Beijingese would say "yi1bai3-shi2si4"?? surprised to know that! i never say in that way when i am speaking in Mandarin. i always say "yi1bai3-yi1shi2si4" in Mandarin. "yi1" is always not omitted when in the middle of the number. :O --θαλαμηγός (talk) 18:00, May 27, 2004 (UTC)

 * Paper or Plastic? He got a point though.  Mainlanders outnumber traditional Chinese and all language evolves according to popular usage.  Since traditional Chinese is a minority and its written and usage will be forgotten over time.  Blame it on Mao.  Kowloonese 22:27, 27 May 2004 (UTC)

I'm totally confused. I consider myself as a native Beijing dialect speaker, and yibaishisi is definitely what we say in Beijing at least. I'm pretty sure that it is officialized in math textbook for elementary school that I used (in 1990s). To emphasize my point, we say yibai-YI-shisi, for sure, 100%. -Liuyao

 * The traditional Chinese was not confused. Both the wikipedians from Taiwan and Hong Kong claimed that we do "yi bai yi shi si" in Traditional Chinese.  Now two mainlanders disagreed on how mainlanders do it.  Since social values and even culture in the mainland has changed so much away from original Chinese traditions due to the Cultural Revolution and communism, the traditional ways are not the Chinese ways anymore because the tradition keepers are now the minority.  It would be interesting to start an article list the Chinese traditions that are only maintained by non-mainlanders.  Kowloonese 08:07, 18 Nov 2004 (UTC)

Hi, I'm the original Anon poster that caused the confusion. I know this is quite late, but I'd like to retract what I said regarding the pronunciation of 114. Although I currently live in the PRC, I spent a great portion of my youth abroad, and I apparently inherited a somewhat grammatically incorrect speech style in this respect. Sorry about the confusion. *loses face*

That having been said, of course, omitting the yi1 in front of the shi2 is unfortunately quite common, which is one of the reasons that I was able to keep on saying in without anyone noticing. In particular, my Shanghainese coworkers (whose Mandarin I would not describe as standard by a long shot) seem to ommit the yi1 as well. At the time I confered with one of them (a graduate of Beijing People's University, no less) and she didn't seem to think that yishi was standard. But then she majored in History....

Anyway, sorry again. It was immature of me to be so insistant. We bilinguals are wacky -- we can't speak any language properly :)

// Ryan:

1) hehe, "1.3 billion Mandarin speakers" eh? Are you a real 幹部 or just an admirer? Seriously though, is this a real statistic or just another myth of Chinese supernationalism? I mean, no offence, but despite the fact that Mandarin is the official language, I am hard pressed to believe that even half of Mainlanders are fluent. I mean, back in the States we have a pretty good educational system, and a good deal of money, and I sometimes wonder what my fellow country-people are saying. I'm really curious about this, and I hope my snide remarks don't turn anyone off. Something about being in the crosshair of 700 missiles makes one a little sarcastic . . . 2)As a one year Taiwan resident, all I've ever heard here is "yi bai yi shi si," or more like "yi bai yi si si," but the folks here take pride in the accent and I take pride in having adopted it. Moreover, all this talk of "officialness" is really wasted thought; language can't be controlled like that. 3)And finally, with regard to traditional characters, I would be careful about so blithely predicting their downfall. Firstly, reunification is by no means guaranteed. Secondly, the dual existence of simplified and more formal, complicated characters has existed for much longer than the modern period, from what I can tell. In fact, many of the so called "simplified" characters are not new at all and many had been in popular use but not formalized. Finally, one cannot overlook the aesthetic element here. Many people, myself included, find simplified characters useful yet rather ugly. They're great for taking notes, but even some Mainland companies and websites I've seen prefer to use the traditional for thier logos. //

 * Well put, but please sign (use ~ to do it, it will automatically be replaced) :) Sjschen 21:30, 13 December 2005 (UTC)

Notes section format
Some people might disagree with me, but I think that the notes section after the table (starts "Leading '1' can sometimes be...") would be easier to read if the Chinese was put in characters and pinyin with tone marks, e.g. 十shí, rather than the notation it is in now (shi2). At the moment it looks a little bit cramped and a little bit hard to follow. I'm tempted to include the first point, so 一百一十四 rather than '1' '100' '1' '10' '4', but my opinion on that part is probably influenced by my small but sufficient-for-this knowledge of Chinese. Even so, I think 1-100-1-10-4 or something like that would work better (and if that doesn't help, a friendly reminder that the manual of style asks for double quotes to assist the search engine).

Or, maybe, it would work better to write the rules first and then provide several examples after that, rather than have one example for every rule. Thus --Rule 1                                 --Rule 1 --Rule 2                                   Example 1 --Rule 3             rather than         --Rule 2 --Examples                                 Example 2 Example 1                              --Rule 2 Example 2                                Example 3 Example 3

On the other hand, maybe it just needs more white space. This is just a format thing though. I'll check back in a few days to see if anyone's said anything. Neonumbers 13:10, 20 Apr 2005 (UTC)

 * I did most of those. So it is only fair for me to present the reasons why I did it that way. I didn't use 一百一十四 because that require the readers to read the Chinese text.  For people who don't care to read the Chinese text or install the Chinese fonts, they would be happy with a collection of rectangles, then the sentence will not make any sense when you cannot tell one rectangle from another.  I didn't use shi, bai etc. because that also requires the users to remember shi means ten, bai means hundred.  Such linguistic knowledge is not required when we only need to present the structure of the Chinese number, not the pronunciation.  A few paragraphs down, we talked about how a Chinese number is read out loud, we used yi, wan, qian, bai, shi etc. in the illustration.  I chose between '1' '100' '1' '10' '4' verses '1' hundred '1' ten '4' because I tried to map one numeral character to one numerical value instead of multiple English words.  So I used '100' and not 'hundred' to represent 百 and so forth. The reason for doing so is more obvious when you need to mention "ten thousand" as one numeral (万) or "one hundred ten-thousand" as three numerals (一百万). Regarding the examples, I think the rearrangement will make it even harder to read because you don't know which rule is applied in which example.  Kowloonese 00:56, Apr 21, 2005 (UTC)

 * That's all fair enough.
 * How about using tone marks on the pinyin? My personal view is that using numbers instead makes it look messy, and tone marks are more out-of-the-way for those that are just reading for structure (though I know there are some homophones when you disregard tone).  The other thing I don't like is the single quotes - I'm tempted (only tempted) to say that square brackets, say, would work better (thus [1] [100] [1] [10] [4]).  Maybe it's because the single quotes on either side are identical - I can't lay a finger on exactly what.  Anyway, for those that *do* have Chinese fonts, the Chinese characters could be written also, but whether they are or not it is of course imperative that the [1] [100] (or '1' '100') form is there.  For lone numbers (like in "...the basic numeral '10' and...") maybe the quotes can be omitted altogether or written as words.
 * I agree that going all rules, all examples would make a life difficult when I think about it but it's not easy to read as it is now - so there has to be a way to make it so. Maybe going all rules, all examples for one point, then for the next point?  I'll think about it.
 * I'm going to change some of the wording, but I won't change too much. It'll mostly be minor changes (like the deletion of the part about Japanese, as this an article about Chinese), but also a part about following zeroes in the leading ones point and, for the examples thing, I won't rearrange it but I'll try em dashes (" — ") and see how it looks.  Neonumbers 12:37, 23 Apr 2005 (UTC)
 * Oh, and a point about the 114 debate earlier - saying yi bai shi si *is* grammatically incorrect, but then so is wan qi for 17000, and they're both common in some places, so isn't it worth giving them equal status? Neonumbers 12:41, 23 Apr 2005 (UTC)
 * Actually I'm going to re-write that part on a subpage that will exist some time from now when I get round to it: User:Neonumbers/Chinese numerals. It'll be order of points mostly, I think perhaps for a layman they'd be a bit hard to follow given the order.  Neonumbers 12:54, 23 Apr 2005 (UTC)
 * I've put my changes into the article. Neonumbers 13:16, 25 Apr 2005 (UTC)

Table simplification
I simplified the table, sacrificing a lot of the information in it which was either self-explanatory (e.g. 1012 is a trillion) or irrelevant (i.e. parts that aren't strictly about Chinese, e.g. myriad). I also tried to narrow the first three columns to make it look better (:-)) and more followable. I don't think I took anything too important out (some things I reworded and I moved one thing to a table footnote. Please don't move this back into the table; it's too long - if it must be somewhere else, put it somewhere apart from inside the table.) Neonumbers 13:16, 25 Apr 2005 (UTC)

Chinese numerals is/n't a numeral system
There is one thing that bugs me about this article: Someone wrote a sentence saying it "shouldn't be considered a numeral system", yet is in the series "Numeral systems". Surely it must be one or the other. I don't know if the Chinese ever used these numbers as numerals or not, so I'm not going to make a claim as to whether it is or isn't - but surely, it can't be both a numeral system *and* not a numeral system. Right? Neonumbers 10:50, 26 Apr 2005 (UTC)

 * The SuZhou numerals are definitely a numeral system. However, the written numbers are questionable.  When you spell out "Three Thousands" in English or 三千 in Chinese, it is a number for sure, but is it a numeral?  Kowloonese 02:14, 5 November 2005 (UTC)

 * Yeah, this article is just about how to write out numbers in Chinese words, even including regional usages. If you think that this article belongs in the Numeral systems, then we should also equivalently include an article called "French numerals", which lists "un, deux, trois, ...", and another called "Spanish numerals" with "uno, dos, tres, ...", all with regional usages too. If you think that is ridiculous, then you shouldn't have this article on words for Chinese numbers either. The other "numeral systems" that are on Wikipedia are all about systems that are distinct from the regular words, but the numerals here aren't -- they are just about regular Chinese words. This belongs in a dictionary, not Wikipedia. --128.97.245.66 (talk) 21:40, 2 February 2010 (UTC)
 * The article "Latin numerals"(i.e. numbers written using the latin alphabet - one/hundred/six trillion etc.) is a valid article. It is presenting the way in which numbers are used in the language. In certain languages, it will be using a well defined set of symbols, but in other languages, such as Chinese and English, the difference between a number and a word/character is less well defined. Charlesisbozo (talk) 12:37, 30 April 2014 (UTC)

 * It's correct that there is no distinct system for writing Chinese numbers. In general the system used for writing out numbers such as phone numbers and prices is the same as used for transcribing number words. But there is still a lot more to it than that (otherwise this article would be very short). E.g
 * The "financial" or 大寫 characters, used only when writing (but said the same).
 * Special "number words" like 幺, 廿. They are dictionary words but it's good to have an article that collects them and discusses usage
 * Similar for large numbers, small numbers, such as the various scales for large numbers
 * More generally how the characters are used
 * So although there's not a separate system of symbols, distinct from the words, there are many things that are distinct and unique worth having an article on. Maybe the article would be better titled "Chinese numbers", but that would then make it more difficult to find from other "... numerals" articles, and it might be confused with e.g. Numbers in Chinese culture. But in terms of content the article covers the subject fairly well.-- JohnBlackburne wordsdeeds 22:01, 2 February 2010 (UTC)
 * The

7=guai3, etc.
In Taiwan ham radio, we use the military names for 0,1,2,7,and sometimes 9. Add a table. Also I swear 釐米 is different than 公釐. Mention it. say if Arabic numbers were ever written backwards. --User:Jidanni 2006-04-16

How does "極" come??
I've checked a few Japanese website (Unicode version):   and used excite translation:  discovered that "極" does not come from Buddhist texts. However, the sites do not explain the origin of "極" clearly. Now, I ask everybody, is there any relevant information about "極"??220.118.11.13 07:14, 22 June 2006 (UTC) You may be misunderstood, "極" is indeed came from Buddhist texts. QQ 13:39, 29 May 2007 (UTC)

but 零(0) is more commonly used, especially in schools.
HaHa.It's suposed to mean that if your teacher gives you a 0 he has to write 零.Because i don't see for what practical reason you'll prefer a much more complicated symbol.--87.64.17.127 01:19, 19 April 2007 (UTC)

This is simply because they do not know how to input the Chinese 〇 in computer and some old computer don't support this character, it is usually wrongly written as Arabic 0.--刻意(Kèyì) 11:50, 17 June 2010 (UTC)

0 In Ancient China
Did the Chinese numeral system ever develop a concept of zero independently to the Hindu-Arabic numeral system? Or did they only get the concept of zero when they discovered the hindu-arabic system? Zachorious 10:49, 29 August 2007 (UTC)
 * I have written on the Chinese zero at 漢数字. According to the sources I have cited, the concept of zero already appeared in The Nine Chapters on the Mathematical Art, but the character 〇 appeared much later than the first introduction of the Indian zero to China in 718 by Gautama Siddha. - TAKASUGI Shinji (talk) 04:52, 27 May 2008 (UTC)

Tally numerals
Should this article also include tally numerals (i.e. the forms of the numbers 1-5 that use progressive strokes of 正)? Rod (A. Smith) 20:55, 24 October 2007 (UTC)

 * Good idea. I suppose that when those are pronounced (infrequently, I would think), they are spoken just like the usual 1 - 5 numerals, though maybe using 兩. But I have never heard that. Dratman (talk) 01:47, 13 January 2012 (UTC)

other dialects
I can't see why the list only includes cantonese but not other chinese dialects which have completely different pronunciation. Is this some sort of cantonese bigotry? Please add other dialects like shanghainese or sichuanese to the list in order to be comprehensive. --Small potato 19:53, 25 October 2007 (UTC)

 * It would probably be best to remove pronunciation entirely and just link the numerals to the corresponding English Wiktionary entries so interested readers can click through. Rod (A. Smith) 18:50, 26 October 2007 (UTC)

"Buddhist numerals" in large section
In the "large numerals" section, notice the comment:

 * The numerals beyond 載 jí come from Buddhist texts in Sanskrit, but these "Buddhist numerals" have become "ancient usage".

Then try to see what that sentence refers to. In the table, there is no entry for "jí". The character (載) matches the one called "zài", but since it is the largest one in the table, there are no "numerals beyond" to which the sentence could be referring.

Can someone help work this out? Robert Munafo (talk) 05:50, 27 February 2008 (UTC)

Suzhou numerals
I think Suzhou numerals should have its own article. Chinese number gestures got its own article, so Suzhou numerals being more historically significant should get its own article. --Voidvector (talk) 18:53, 26 May 2008 (UTC)
 * I support splitting the Suzhou numerals section to its own article. The subject of the section diverges from the main subject of the article just enough to warrant a separate article. –  j ak s mata  19:08, 27 May 2008 (UTC)

An OR comment on "Buddhist large numbers" and "Small numbers"
This is original research and so can't be included in the article, but I couldn't help but point out that the Buddhist mathematicians who concluded that there were 1052 "grains of sand in the Ganges river" were so far off base that even with the craziest estimates of parameters they must have been lousy at arithmetic to think this was remotely plausible. In fact even if a grain of sand is only 10 microns across (which is more of a very fine silt than sand), 1052 such grains would weigh around 1034 kilograms, which is not only vastly larger than the mass of the entire Earth, it is more than a thousand times larger than the mass of the entire solar system. So like the Maya, they seem to have been OK at using a positional system to generate stupendously large numerals, but had little actual understanding of the size of these numbers in context.

By the way, can anyone explain why these large numbers go up by factors of 104? Based on Indian tradition (e.g. the lakh) I would have thought (or rather, wildly guessed) powers of 102 more likely in a Buddhist text. (I note, as pure speculation, that if someone has gotten this factor wrong and it is actually factors of 102, or in other words each of these large numbers has effectively been squared, then the "grains of sand in the Ganges river" number actually works out quite close to a reasonable estimate, so perhaps the ancient Buddhists weren't quite so bad at maths, just their translators were!)

Finally, what on earth can the "Ancient" Chinese have wanted with a prefix that meant 10-12? Not even the most foamy-mouthed nationalist can pretend that anyone prior to the twentieth century could measure anything with sufficient precision to make such a prefix meaningful. Which I guess leads one to wonder, how confident are we, and what references are there to support, the claim that in "Ancient" times it had precisely this meaning? I emphasise "precisely" -- after all, the SI prefixes "micro" and "nano" are Ancient Greek, but to the Ancient Greeks they just meant "small" and "dwarf", they didn't have precise metrological meanings. (And what is meant by "Ancient"? Normally in Chinese history, that means prior to the Qing, which began in 221 BCE. But the oldest known mathematical treatise in Chinese is the Suàn shù shū, which is no older than 202 BCE.) -- 203.20.101.203 (talk) 05:35, 4 August 2009 (UTC)
 * The only one of your questions I can answer is why they go up by factors of 104: In American English, we use names for numbers incrementing by 103, namely, ones, thousands, millions, billions, etc. In contrast, Chinese counting increments similarly, but by 104. Their named numbers would be, ones, myriads 萬 (1,0000), myriad myriads 億 (1,0000,0000), etc.
 * My knowledge of the subject is too limited to answer your other questions. –  j ak s mata  13:40, 4 August 2009 (UTC)
 * As for 10−12: maybe it was just for fun, playing with numbers, and like the Mayans' usage, it didn't have anything to do with the real world? Double sharp (talk) 03:49, 28 September 2015 (UTC)

Actually I think somebody was joking with this section haha. You'll have to know Chinese to notice, but the words for 1060 and above are silly. "那由他" sounds like a pun on "哪有他" which is a slang way of saying, "pffft, that doesn't exist". "不可思意" does means unimaginable, but in the sense like, "Did you know your neighbor is Brad Pitt?" "不可思意！(That's unimaginable!)" "无量" is like "无穷" which means infinity. And "大数" is just funny because 大 = big and 数 = number, so it's a literal term that I've never heard anybody use in China. The juxtaposition of "infinity" and then "big number" is the punch line because it calls to mind an image of young scholars sitting around, and having run out of legitimate ideas, decided to make crap up to screw with future generations. My guess is that this table was copied from an online joke off of a Chinese website. --174.34.144.211 (talk) 03:35, 24 April 2010 (UTC)
 * A large number of these phrases that are commonly used today originate from Buddhism. Today, although 不可思意 is used in everyday speech during events such as "holy shit, I just saw a ten-foot crocodile!" "really? 不可思意!", they do have their alternate uses. (Note to other editors: WP:NOTCENSORED; my words not to be refactored by anyone) For example, back in antiquity, 萬 meant scorpion, and not ten thousand. Also, I'm assuming you haven't studied Buddhism before. The Japanese use the same numbering system (那由他 = 1060) as well, but phrases such as 那由他 and 不可思議 aren't even used in everyday Japanese speech (as figures of speech like in Chinese). I think you should assume good faith on behalf of other editors; else, prove me wrong. --  李博杰   | —Talk contribs email 10:41, 6 September 2010 (UTC)
 * Additionally, there is much more than the handful provided on the English Wikipedia article. Refer to 中文数字 for a more detailed list. The numbers come from the Avatamsaka Sutra (華嚴經). There are also articles on the Chinese Wikipedia about 那由他 and 阿僧祇, as well as relevant English Wiktionary entries. Korean numerals also lists the same numbers. --  李博杰   | —Talk contribs email 10:54, 6 September 2010 (UTC)
 * I noticed that (I've been curious about these Chinese numerals). But the given magnitudes go up double-exponentially... (appear to be based on the formula 107 x 2 n ) where do these few that go up like myriad powers come from? mike4ty4 (talk) 09:59, 7 September 2010 (UTC)
 * Dunno, but the very same myriad-power words appear in Japanese in the Jinkōki (though it makes 1068 無量大数, as opposed to Chinese 1068 無量, 1072 大數). Double sharp (talk) 03:47, 28 September 2015 (UTC)

Chinese dialects
Currently the article seems to be centred along describing the numerals based on how they are pronounced in Standard Mandarin (i.e. all figures have their readings given in Pinyin). If I were to create a new section regarding the pronounciation of Chinese numbers in other dialects, would I be met with support or opposition? If favourable, I can start on it. --  李博杰   | —Talk contribs email 10:37, 6 September 2010 (UTC)
 * Looking at it I'm not sure it's needed for all dialects, as for most of them I think the interest is quite small and would be better served by the pages for the dialects themselves. The one exception is Cantonese, which is heard at least as often as Mandarin outside of China. But rather than a new section it could be added to the existing tables, for example by adding an additional column next to the the Pinyin one. This is close to the usage in other articles where Pinyin and Cantonese Romanisation are often both given after Chinese names.-- JohnBlackburne wordsdeeds 11:21, 6 September 2010 (UTC)
 * We could use the table(s) at 中文数字 and 漢数字 as a rough guide; but I'm afraid that doing it in such a manner may take up too much space and/or make it look cluttered. Keep in mind that we'll need an extra column for English explanations (one more column than on the ZH and JA Wikipedias). I was just thinking of listing the big five: Mandarin, Cantonese, Wu, Min and Hakka; the Chinese Wikipedia has a bopomofo column, but I don't think that's necessary on the EN Wiki (it's for people from Taiwan who can't read Hanyu Pinyin).

 * I was initially thinking of making a table in a new section that could be collapsed and hidden with the { {hidden begin}} and { {hidden end}} tags, in a similar format to List of numbers in various languages (but with more than just 0-10; as a minimum should include 百, 千, 万 and 亿; also, it would be more preferable if the numbers went down vertically). --  李博杰   | —Talk contribs email 11:39, 6 September 2010 (UTC)

Military numerals
Taken from 漢数字. Apparently they are also used in aviation and railways within China, not only in the military. If someone can confirm its validity, that would be wonderful. Regards, --  李博杰   | —Talk contribs email 12:09, 6 September 2010 (UTC)

Please clarify why this is a number system and not direct translation?
"The Chinese character system can be classified as part of the language, but it still counts as a number system." <- Could somebody elaborate on this claim? --Frozenport (talk) 07:22, 25 November 2010 (UTC)

Great article
I just want to comment that this is a marvelous article. I cannot imagine finding this wealth of detail anywhere else. The article could be expanded and published as a very nice little book, but it is so much easier and more practical to find it all right here in Wikipedia. — Preceding unsigned comment added by Dratman (talk • contribs) January 13, 2012

Court Dialect (Mandarin), Cantonese, and Fujian (Hokkien) pronounciations of numbers
The Middle Kingdom: A Survey of The... Chinese Empire and Its ..., Volume 1 By Samuel Wells Williams

http://books.google.com/books?id=nINu15k9FuMC&pg=PA495&dq=they+are+here+introduced+with+their+pronunciation+in+three+dialects&hl=en&ei=wL9zTsjmOYTe0QGyicirDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC4Q6AEwAA#v=onepage&q=they%20are%20here%20introduced%20with%20their%20pronunciation%20in%20three%20dialects&f=false

Rajmaan (talk) 14:57, 15 October 2014 (UTC)

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万亿 and 亿亿 as simplified forms of 兆 and 京
This doesn't make sense. Just because the PRC uses 万亿 and 亿亿 instead of 兆 and 京 doesn't mean that we can accept them as Simplified forms, especially since some other countries using Simplified still use 兆 (such as Singapore). By that logic the entire Simplified row could be replaced with combinations of 万 and 亿 (I've seen 亿亿亿 used in a published work, for example). Moreover, the table compares four different systems that use the same ancient number names, and thus keeping the table as it was suggests that 万亿 could mean 10^6 and 亿亿 could mean 10^7, which is wrong. Rather, 万亿 and 亿亿 should be discussed as part of the note at the bottom of the table.

Considering that the table is in fact misleading as mentioned above, I've gone ahead and changed it with an added note.Arcorann (talk) 11:52, 1 February 2018 (UTC)

I didn`t change back, but just to clarify that Singapore adopts the standards of the People’s Republic of China and uses simplified characters, that means Singapore uses 兆 for 10^6 rather than 10^12.UU (talk) 11:33, 14 October 2018 (UTC)

This is my first time using the talk page, so apologies if I'm doing something wrong. I saw the article, and I wanted to say that in addition to the use of 万亿 in Taiwan, I've seen 百万 used quite frequently on the Mainland as an alternative to 兆 as well. 16tonweight (talk) 23:31, 5 July 2020 (UTC)

Template removed
I removed this template "cleanup lang|date=October 2023", because it claims that it is necessary to specify the "language" of the non-English content, but this is not true, because the content consists of characters, which do not have a "language", any more than Г or α does. Imaginatorium (talk) 07:05, 12 December 2023 (UTC)
 * If we assume that is the case (it doesn't seem to be entirely true—there are plenty of characters on the page that are clearly specifically representing Chinese, such as in the first table. moreover, the thing that makes logographs different from letters is that they often "have a language", as they correspond to morphemes or words), it is necessary to tag the text as und-Hani, which is the ISO 639 code representing an unspecified language written in Han characters, or Han characters in and of themselves. I actually created the template to do this easily. Without further specification, as HTML documents all English Wikipedia articles are tagged as being wholly in English—for some reason, it rather bothers me that this is quite often not their declaration does not tell the whole truth about their contents.  Remsense  留  08:49, 12 December 2023 (UTC)

Republic of China Taiwanese characters for metric prefixes 10^-27 and 10^-30 are missing
The metric prefixes ronto and quecto are missing their respective Chinese character in the Republic of China (Taiwan) table for SI Prefixes. Uuruuseiyo (talk) 20:13, 21 December 2023 (UTC)
 * Uuruuseiyo, do you know what they are and where a reliable source is? Remsense  留  23:18, 22 December 2023 (UTC)
 * I don’t know which is why I’m asking. Also there doesn’t seem to be a reference linked to the prefixes that were just added since 2022. Uuruuseiyo (talk) 10:30, 23 December 2023 (UTC)
 * I added both with an RS. The original ROC quetta prefix was wrong, BTW. Double sharp (talk) 15:30, 14 April 2024 (UTC)
 * thank you! Remsense  诉  16:27, 14 April 2024 (UTC)

"弐" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=%E5%BC%90&redirect=no 弐] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. Utopes (talk / cont) 09:20, 25 January 2024 (UTC)

title: Talk:Chinese numerals


text:
Mastaba

A mastaba (,  or  ), also mastabah or mastabat) is a type of ancient Egyptian tomb in the form of a flat-roofed, rectangular structure with inward sloping sides, constructed out of mudbricks or limestone. These edifices marked the burial sites of many eminent Egyptians during Egypt's Early Dynastic Period and Old Kingdom. Non-royal use of mastabas continued for over a thousand years.

The word mastaba comes from the Arabic word مصطبة (maṣṭaba) "stone bench". The Ancient Egyptian name was pr-Djt, meaning "house of stability", "house of eternity", or "eternal house".

History
The afterlife was centralized in the religion of ancient Egyptians. Their architecture reflects this, most prominently by the enormous amounts of time and labor involved in building tombs. Ancient Egyptians believed that the needs from the world of the living would be continued in the afterlife; it was therefore necessary to build tombs that would fulfill them, and be sturdy enough to last for an eternity. These needs would also have to be attended to by the living.

Starting in the Predynastic era (before 3100 BCE) and continuing into later dynasties, the ancient Egyptians developed increasingly complex and effective methods for preserving and protecting the bodies of the dead. They first buried their dead in pit graves dug from the sand with the body placed on a mat, usually along with some items believed to help them in the afterlife. The first tomb structure the Egyptians developed was the mastaba, composed of earthen bricks made from soil along the Nile. It provided better protection from scavenging animals and grave robbers. The origins of the mastaba can be seen in Tarkhan, where tombs would be split into two distinct portions. One side would contain a body, oriented in a north-south position, and the other would be open for the living to deliver offerings. As the remains were not in contact with the dry desert sand, natural mummification could not take place; therefore the Egyptians devised a system of artificial mummification. Until at least the Old Period or First Intermediate Period, only high officials and royalty were buried in these mastabas.

Structure
The term mastaba comes from the Arabic word for "a bench of mud". When seen from a distance, a flat-topped mastaba does resemble a bench. Historians speculate that the Egyptians may have borrowed architectural ideas from Mesopotamia, since at the time they were both building similar structures.

The above-ground structure of a mastaba is rectangular in shape with inward-sloping sides and a flat roof. The exterior building materials were initially bricks made of the sun-dried mud readily available from the Nile River. Even after more durable materials such as stone came into use, the majority were built from mudbricks. Monumental mastabas, such as those at Saqqara, were often constructed out of limestone. Mastabas were often about four times as long as they were wide, and many rose to at least 30 ft in height. They were oriented north–south, which the Egyptians believed was essential for access to the afterlife. The roofs of the mastabas were of slatted wood or slabs of limestone, with skylights illuminating the tomb. The above-ground structure had space for a small offering chapel equipped with a false door. Priests and family members brought food and other offerings for the soul, or ba, of the deceased, which had to be maintained in order to continue to exist in the afterlife. The construction of mastabas was standardized, with several treatments being common for masonry.

Mastabas were highly decorated, both with paintings on the walls and ceilings, and carvings of organic elements such as palm trees out of limestone. Due to the spiritual significance of the color, it was preferable to construct mastabas from white limestone. If this was not available, the yellow limestone or mudbrick of the tomb would be whitewashed and plastered. Mastabas for royalty were especially extravagant on the exterior, meant to resemble a palace.

Interior
A mastaba was essentially meant to provide the ba with a house in the afterlife, and they were laid out accordingly. Some would be used to house families, rather than individuals, with several burial shafts acting as "rooms". The burial chambers were cut deep, into the bedrock, and were lined with wood. A second hidden chamber called a serdab (سرداب), from the Persian word for "cellar", was used to store anything that may have been considered essential for the comfort of the deceased in the afterlife, such as beer, grain, clothes and precious items. The mastaba housed a statue of the deceased that was hidden within the masonry for its protection. High up the walls of the serdab were small openings that would allow the ba to leave and return to the body (represented by the statue); Ancient Egyptians believed the ba had to return to its body or it would die. These openings "were not meant for viewing the statue but rather for allowing the fragrance of burning incense, and possibly the spells spoken in rituals, to reach the statue". The statues were nearly always oriented in one direction, facing the opening. The serdab could also feature inscriptions, such as the testament and mortuary cult of the owner. More elaborate mastabas would feature open courtyards, which would be used to house more statues and allow the dead to perform rites. Over time, the courtyards grew into magnificent columned halls, which served the same purposes. These halls would typically be the largest room in the mastaba, and they could be used for sacrifices of livestock. Larger mastabas also included a network of storerooms, which the presiding phyle would use to maintain the mortuary cult of the mastaba's owner. Generally, there would be five of these storerooms, used by the living to store equipment needed for performing rites; unlike the serdab, they were not meant to be used by the deceased. These lacked any form of decoration, again distinguishing their function from that of the rest of the tomb. Due to the great expense of adding a complex of storerooms, these were only constructed in the largest of mastabas, for the royal family and viziers.

Architectural evolution
The mastaba was the standard type of tomb in pre-dynastic and early dynastic Egypt for both the pharaoh and the social elite. The ancient city of Abydos was the location chosen for many of the cenotaphs. The royal cemetery was at Saqqara, overlooking the capital of early times, Memphis.

Mastabas evolved over the early dynastic period (c. 3100-2686 BCE). During the 1st Dynasty, a mastaba was constructed simulating house plans of several rooms, a central one containing the sarcophagus and others surrounding it to receive the abundant funerary offerings. The whole was built in a shallow pit above which a brick superstructure covering a broad area. The typical 2nd and 3rd Dynasty (c. 2686–2313) mastabas was the 'stairway mastaba', the tomb chamber of which sank deeper than before and was connected to the top with an inclined shaft and stairs. Many of the features of mastabas grew into those of the pyramids, indicating their importance as a transitory construction of tombs. This notably includes the exterior appearance of the tombs, as the sloped sides of the mastabas extended to form a pyramid. The first and most striking example of this was Djoser's step pyramid, which combined many traditional features of mastabas with a more monumental stone construction.

Even after pyramids became more prevalent for pharaohs in the 3rd and 4th Dynasties, members of the nobility continued to be buried in mastaba tombs. This is especially evident on the Giza Plateau, where at least 150 mastaba tombs have been constructed alongside the pyramids.

In the 4th Dynasty (c. 2613 to 2494 BCE), rock-cut tombs began to appear. These were tombs built into the rock cliffs in Upper Egypt in an attempt to further thwart grave robbers. Mastabas, then, were developed with the addition of offering chapels and vertical shafts. 5th Dynasty mastabas had elaborate chapels consisting of several rooms, columned halls and 'serdab'. The actual tomb chamber was built below the south-end of mastaba, connected by a slanting passage to a stairway emerging in the center of a columned hall or court.

Mastabas are still well attested in the Middle Kingdom, where they had a revival. They were often solid structures with the decoration only on the outside.

By the time of the New Kingdom (which began with the 18th Dynasty around 1550 BC), "the mastaba becomes rare, being largely superseded by the independent pyramid chapel above a burial chamber".

Non-royal

 * Mastaba of Hesy-Re
 * Mastaba of Kaninisut
 * Mastaba of Seshemnefer
 * Mastaba S3503
 * Mastaba S3504

Royal

 * Buried Mastaba (Sekhemkhet)
 * Mastabat al-Fir'aun (Shepseskaf)
 * Mastaba of Neferefre (Neferefre)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Mastaba


text:
System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. For example,
 * $$\begin{cases}

3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\frac{1}{2}y-z=0 \end{cases}$$ is a system of three equations in the three variables $x, y, z$. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the ordered triple $$(x,y,z)=(1,-2,-2),$$ since it makes all three equations valid.

Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.

Very often, and in this article, the coefficients and solutions of the equations are constrained to be real or complex numbers, but the theory and algorithms apply to coefficients and solutions in any field. For other algebraic structures, other theories have been developed. For coefficients and solutions in an integral domain, such as the ring of integers, see Linear equation over a ring. For coefficients and solutions that are polynomials, see Gröbner basis. For finding the "best" integer solutions among many, see Integer linear programming. For an example of a more exotic structure to which linear algebra can be applied, see Tropical geometry.

Trivial example
The system of one equation in one unknown
 * $$2x = 4$$

has the solution
 * $$x = 2.$$

However, most interesting linear systems have at least two equations.

Simple nontrivial example
The simplest kind of nontrivial linear system involves two equations and two variables:

 * $$\begin{alignat}{5}

2x &&\; + \;&& 3y &&\; = \;&& 6 & \\ 4x &&\; + \;&& 9y &&\; = \;&& 15&. \end{alignat}$$

One method for solving such a system is as follows. First, solve the top equation for $$x$$ in terms of $$y$$:

 * $$x = 3 - \frac{3}{2}y.$$

Now substitute this expression for x into the bottom equation:

 * $$4\left( 3 - \frac{3}{2}y \right) + 9y = 15.$$

This results in a single equation involving only the variable $$y$$. Solving gives $$y = 1$$, and substituting this back into the equation for $$x$$ yields $$x = \frac{3}{2}$$. This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on elementary algebra.)

General form
A general system of m linear equations with n unknowns and coefficients can be written as
 * $$\begin{cases}

a_{11} x_1 + a_{12} x_2 +\dots + a_{1n} x_n = b_1 \\ a_{21} x_1 + a_{22} x_2 + \dots + a_{2n} x_n = b_2 \\ \vdots\\ a_{m1} x_1 + a_{m2} x_2 + \dots + a_{mn} x_n = b_m, \end{cases}$$ where $$x_1, x_2,\dots,x_n$$ are the unknowns, $$a_{11},a_{12},\dots,a_{mn}$$ are the coefficients of the system, and $$b_1,b_2,\dots,b_m$$ are the constant terms.

Often the coefficients and unknowns are real or complex numbers, but integers and rational numbers are also seen, as are polynomials and elements of an abstract algebraic structure.

Vector equation
One extremely helpful view is that each unknown is a weight for a column vector in a linear combination.

x_1\begin{bmatrix}a_{11}\\a_{21}\\ \vdots \\a_{m1}\end{bmatrix} + x_2\begin{bmatrix}a_{12}\\a_{22}\\ \vdots \\a_{m2}\end{bmatrix} + \dots + x_n\begin{bmatrix}a_{1n}\\a_{2n}\\ \vdots \\a_{mn}\end{bmatrix} = \begin{bmatrix}b_1\\b_2\\ \vdots \\b_m\end{bmatrix} $$ This allows all the language and theory of vector spaces (or more generally, modules) to be brought to bear. For example, the collection of all possible linear combinations of the vectors on the left-hand side is called their span, and the equations have a solution just when the right-hand vector is within that span. If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique. In any event, the span has a basis of linearly independent vectors that do guarantee exactly one expression; and the number of vectors in that basis (its dimension) cannot be larger than m or n, but it can be smaller. This is important because if we have m independent vectors a solution is guaranteed regardless of the right-hand side, and otherwise not guaranteed.

Matrix equation
The vector equation is equivalent to a matrix equation of the form $$ A\mathbf{x} = \mathbf{b} $$ where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries.

$$ A= \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix},\quad \mathbf{x}= \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix},\quad \mathbf{b}= \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix}. $$ The number of vectors in a basis for the span is now expressed as the rank of the matrix.

Solution set
A solution of a linear system is an assignment of values to the variables x1, x2, ..., xn such that each of the equations is satisfied. The set of all possible solutions is called the solution set.

A linear system may behave in any one of three possible ways:
 * 1) The system has infinitely many solutions.
 * 2) The system has a unique solution.
 * 3) The system has no solution.

Geometric interpretation
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set.

For three variables, each linear equation determines a plane in three-dimensional space, and the solution set is the intersection of these planes. Thus the solution set may be a plane, a line, a single point, or the empty set. For example, as three parallel planes do not have a common point, the solution set of their equations is empty; the solution set of the equations of three planes intersecting at a point is single point; if three planes pass through two points, their equations have at least two common solutions; in fact the solution set is infinite and consists in all the line passing through these points.

For n variables, each linear equation determines a hyperplane in n-dimensional space. The solution set is the intersection of these hyperplanes, and is a flat, which may have any dimension lower than n.

General behavior
In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns. Here, "in general" means that a different behavior may occur for specific values of the coefficients of the equations. In the first case, the dimension of the solution set is, in general, equal to n − m, where n is the number of variables and m is the number of equations.
 * In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an underdetermined system.
 * In general, a system with the same number of equations and unknowns has a single unique solution.
 * In general, a system with more equations than unknowns has no solution. Such a system is also known as an overdetermined system.

The following pictures illustrate this trichotomy in the case of two variables:
 * {| border=0 cellpadding=5

The first system has infinitely many solutions, namely all of the points on the blue line. The second system has a single unique solution, namely the intersection of the two lines. The third system has no solutions, since the three lines share no common point.
 * width="150" align="center" | [[Image:One Line.svg|120px]]
 * width="150" align="center" | [[Image:Two Lines.svg|120px]]
 * width="150" align="center" | [[Image:Three Lines.svg|120px]]
 * align="center"| One equation
 * align="center"| Two equations
 * align="center"| Three equations
 * }
 * align="center"| Three equations
 * }

It must be kept in mind that the pictures above show only the most common case (the general case). It is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines intersect at a single point).

A system of linear equations behave differently from the general case if the equations are linearly dependent, or if it is inconsistent and has no more equations than unknowns.

Independence
The equations of a linear system are independent if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence.

For example, the equations
 * $$3x+2y=6\;\;\;\;\text{and}\;\;\;\;6x+4y=12$$

are not independent — they are the same equation when scaled by a factor of two, and they would produce identical graphs. This is an example of equivalence in a system of linear equations.

For a more complicated example, the equations
 * $$\begin{alignat}{5}

x &&\; - \;&& 2y &&\; = \;&& -1 & \\ 3x &&\; + \;&& 5y &&\; = \;&& 8 & \\ 4x &&\; + \;&& 3y &&\; = \;&& 7 & \end{alignat}$$ are not independent, because the third equation is the sum of the other two. Indeed, any one of these equations can be derived from the other two, and any one of the equations can be removed without affecting the solution set. The graphs of these equations are three lines that intersect at a single point.

Consistency
A linear system is inconsistent if it has no solution, and otherwise, it is said to be consistent. When the system is inconsistent, it is possible to derive a contradiction from the equations, that may always be rewritten as the statement 1.

For example, the equations
 * $$3x+2y=6\;\;\;\;\text{and}\;\;\;\;3x+2y=12$$

are inconsistent. In fact, by subtracting the first equation from the second one and multiplying both sides of the result by 1/6, we get 1. The graphs of these equations on the xy-plane are a pair of parallel lines.

It is possible for three linear equations to be inconsistent, even though any two of them are consistent together. For example, the equations
 * $$\begin{alignat}{7}

x &&\; + \;&& y &&\; = \;&& 1 & \\ 2x &&\; + \;&& y &&\; = \;&& 1 & \\ 3x &&\; + \;&& 2y &&\; = \;&& 3 & \end{alignat}$$ are inconsistent. Adding the first two equations together gives 3x + 2y = 2, which can be subtracted from the third equation to yield 1. Any two of these equations have a common solution. The same phenomenon can occur for any number of equations.

In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent.

Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there is an infinitude of solutions. The rank of a system of equations (that is, the rank of the augmented matrix) can never be higher than [the number of variables] + 1, which means that a system with any number of equations can always be reduced to a system that has a number of independent equations that is at most equal to [the number of variables] + 1.

Equivalence
Two linear systems using the same set of variables are equivalent if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one. It follows that two linear systems are equivalent if and only if they have the same solution set.

Solving a linear system
There are several algorithms for solving a system of linear equations.

Describing the solution
When the solution set is finite, it is reduced to a single element. In this case, the unique solution is described by a sequence of equations whose left-hand sides are the names of the unknowns and right-hand sides are the corresponding values, for example $$(x=3, \;y=-2,\; z=6)$$. When an order on the unknowns has been fixed, for example the alphabetical order the solution may be described as a vector of values, like $$(3, \,-2,\, 6)$$ for the previous example.

To describe a set with an infinite number of solutions, typically some of the variables are designated as free (or independent, or as parameters), meaning that they are allowed to take any value, while the remaining variables are dependent on the values of the free variables.

For example, consider the following system:
 * $$\begin{alignat}{7}

x &&\; + \;&& 3y &&\; - \;&& 2z &&\; = \;&& 5 & \\ 3x &&\; + \;&& 5y &&\; + \;&& 6z &&\; = \;&& 7 & \end{alignat}$$ The solution set to this system can be described by the following equations:
 * $$x=-7z-1\;\;\;\;\text{and}\;\;\;\;y=3z+2\text{.}$$

Here z is the free variable, while x and y are dependent on z. Any point in the solution set can be obtained by first choosing a value for z, and then computing the corresponding values for x and y.

Each free variable gives the solution space one degree of freedom, the number of which is equal to the dimension of the solution set. For example, the solution set for the above equation is a line, since a point in the solution set can be chosen by specifying the value of the parameter z. An infinite solution of higher order may describe a plane, or higher-dimensional set.

Different choices for the free variables may lead to different descriptions of the same solution set. For example, the solution to the above equations can alternatively be described as follows:
 * $$y=-\frac{3}{7}x + \frac{11}{7}\;\;\;\;\text{and}\;\;\;\;z=-\frac{1}{7}x-\frac{1}{7}\text{.}$$

Here x is the free variable, and y and z are dependent.

Elimination of variables
The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows:
 * 1) In the first equation, solve for one of the variables in terms of the others.
 * 2) Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown.
 * 3) Repeat steps 1 and 2 until the system is reduced to a single linear equation.
 * 4) Solve this equation, and then back-substitute until the entire solution is found.

For example, consider the following system:
 * $$\begin{cases}

x+3y-2z=5\\ 3x+5y+6z=7\\ 2x+4y+3z=8 \end{cases}$$ Solving the first equation for x gives $$x=5+2z-3y$$, and plugging this into the second and third equation yields
 * $$\begin{cases}

y=3z+2\\ y=\tfrac{7}{2}z+1 \end{cases}$$ Since the LHS of both of these equations equal y, equating the RHS of the equations. We now have:
 * $$\begin{align}

3z+2=\tfrac{7}{2}z+1\\ \Rightarrow z=2 \end{align} $$ Substituting z = 2 into the second or third equation gives y = 8, and the values of y and z into the first equation yields x = −15. Therefore, the solution set is the ordered triple $$(x,y,z)=(-15,8,2) $$.

Row reduction
In row reduction (also known as Gaussian elimination), the linear system is represented as an augmented matrix

 * $$\left[\begin{array}{rrr|r}

1 & 3 & -2 & 5 \\ 3 & 5 & 6 & 7 \\ 2 & 4 & 3 & 8 \end{array}\right]\text{.} $$ This matrix is then modified using elementary row operations until it reaches reduced row echelon form. There are three types of elementary row operations:
 * Type 1: Swap the positions of two rows.
 * Type 2: Multiply a row by a nonzero scalar.
 * Type 3: Add to one row a scalar multiple of another.

Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.

There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are Gaussian elimination and Gauss–Jordan elimination. The following computation shows Gauss–Jordan elimination applied to the matrix above:
 * $$\begin{align}\left[\begin{array}{rrr|r}

1 & 3 & -2 & 5 \\ 3 & 5 & 6 & 7 \\ 2 & 4 & 3 & 8 \end{array}\right]&\sim \left[\begin{array}{rrr|r} 1 & 3 & -2 & 5 \\ 0 & -4 & 12 & -8 \\ 2 & 4 & 3 & 8 \end{array}\right]\sim \left[\begin{array}{rrr|r} 1 & 3 & -2 & 5 \\ 0 & -4 & 12 & -8 \\ 0 & -2 & 7 & -2 \end{array}\right]\sim \left[\begin{array}{rrr|r} 1 & 3 & -2 & 5 \\ 0 & 1 & -3 & 2 \\ 0 & -2 & 7 & -2 \end{array}\right] \\ &\sim \left[\begin{array}{rrr|r} 1 & 3 & -2 & 5 \\ 0 & 1 & -3 & 2 \\ 0 & 0 & 1 & 2 \end{array}\right]\sim \left[\begin{array}{rrr|r} 1 & 3 & -2 & 5 \\ 0 & 1 & 0 & 8 \\ 0 & 0 & 1 & 2 \end{array}\right]\sim \left[\begin{array}{rrr|r} 1 & 3 & 0 & 9 \\ 0 & 1 & 0 & 8 \\ 0 & 0 & 1 & 2 \end{array}\right]\sim \left[\begin{array}{rrr|r} 1 & 0 & 0 & -15 \\ 0 & 1 & 0 & 8 \\ 0 & 0 & 1 & 2 \end{array}\right].\end{align}$$ The last matrix is in reduced row echelon form, and represents the system, ,. A comparison with the example in the previous section on the algebraic elimination of variables shows that these two methods are in fact the same; the difference lies in how the computations are written down.

Cramer's rule
Cramer's rule is an explicit formula for the solution of a system of linear equations, with each variable given by a quotient of two determinants. For example, the solution to the system
 * $$\begin{alignat}{7}

x &\; + &\; 3y &\; - &\; 2z &\; = &\; 5 \\ 3x &\; + &\; 5y &\; + &\; 6z &\; = &\; 7 \\ 2x &\; + &\; 4y &\; + &\; 3z &\; = &\; 8 \end{alignat}$$ is given by

x=\frac {\, \begin{vmatrix}5&3&-2\\7&5&6\\8&4&3\end{vmatrix} \,} {\, \begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix} \,} ,\;\;\;\; y=\frac {\, \begin{vmatrix}1&5&-2\\3&7&6\\2&8&3\end{vmatrix} \,} {\, \begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix} \,} ,\;\;\;\; z=\frac {\, \begin{vmatrix}1&3&5\\3&5&7\\2&4&8\end{vmatrix} \,} {\, \begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix} \,}. $$ For each variable, the denominator is the determinant of the matrix of coefficients, while the numerator is the determinant of a matrix in which one column has been replaced by the vector of constant terms.

Though Cramer's rule is important theoretically, it has little practical value for large matrices, since the computation of large determinants is somewhat cumbersome. (Indeed, large determinants are most easily computed using row reduction.) Further, Cramer's rule has very poor numerical properties, making it unsuitable for solving even small systems reliably, unless the operations are performed in rational arithmetic with unbounded precision.

Matrix solution
If the equation system is expressed in the matrix form $$A\mathbf{x}=\mathbf{b}$$, the entire solution set can also be expressed in matrix form. If the matrix A is square (has m rows and n=m columns) and has full rank (all m rows are independent), then the system has a unique solution given by

 * $$\mathbf{x}=A^{-1}\mathbf{b}$$

where $$A^{-1}$$ is the inverse of A. More generally, regardless of whether m=n or not and regardless of the rank of A, all solutions (if any exist) are given using the Moore–Penrose inverse of A, denoted $$A^+$$, as follows:

 * $$\mathbf{x}=A^+ \mathbf{b} + \left(I - A^+ A\right)\mathbf{w}$$

where $$\mathbf{w}$$ is a vector of free parameters that ranges over all possible n×1 vectors. A necessary and sufficient condition for any solution(s) to exist is that the potential solution obtained using $$\mathbf{w}=\mathbf{0}$$ satisfy $$A\mathbf{x}=\mathbf{b}$$ — that is, that $$AA^+ \mathbf{b}=\mathbf{b}.$$ If this condition does not hold, the equation system is inconsistent and has no solution. If the condition holds, the system is consistent and at least one solution exists. For example, in the above-mentioned case in which A is square and of full rank, $$A^+$$ simply equals $$A^{-1}$$ and the general solution equation simplifies to
 * $$\mathbf{x}=A^{-1}\mathbf{b} + \left(I - A^{-1}A\right)\mathbf{w} = A^{-1}\mathbf{b} + \left(I-I\right)\mathbf{w} = A^{-1}\mathbf{b}$$

as previously stated, where $$\mathbf{w}$$ has completely dropped out of the solution, leaving only a single solution. In other cases, though, $$\mathbf{w}$$ remains and hence an infinitude of potential values of the free parameter vector $$\mathbf{w}$$ give an infinitude of solutions of the equation.

Other methods
While systems of three or four equations can be readily solved by hand (see Cracovian), computers are often used for larger systems. The standard algorithm for solving a system of linear equations is based on Gaussian elimination with some modifications. Firstly, it is essential to avoid division by small numbers, which may lead to inaccurate results. This can be done by reordering the equations if necessary, a process known as pivoting. Secondly, the algorithm does not exactly do Gaussian elimination, but it computes the LU decomposition of the matrix A. This is mostly an organizational tool, but it is much quicker if one has to solve several systems with the same matrix A but different vectors b.

If the matrix A has some special structure, this can be exploited to obtain faster or more accurate algorithms. For instance, systems with a symmetric positive definite matrix can be solved twice as fast with the Cholesky decomposition. Levinson recursion is a fast method for Toeplitz matrices. Special methods exist also for matrices with many zero elements (so-called sparse matrices), which appear often in applications.

A completely different approach is often taken for very large systems, which would otherwise take too much time or memory. The idea is to start with an initial approximation to the solution (which does not have to be accurate at all), and to change this approximation in several steps to bring it closer to the true solution. Once the approximation is sufficiently accurate, this is taken to be the solution to the system. This leads to the class of iterative methods. For some sparse matrices, the introduction of randomness improves the speed of the iterative methods. One example of an iterative method is the Jacobi method, where the matrix $$A$$ is split into its diagonal component $$D$$ and its non-diagonal component $$L+U$$. An initial guess $${\bold x}^{(0)}$$ is used at the start of the algorithm. Each subsequent guess is computed using the iterative equation:
 * $${\bold x}^{(k+1)} = D^{-1}({\bold b} - (L+U){\bold x}^{(k)})$$

When the difference between guesses $${\bold x}^{(k)}$$ and $${\bold x}^{(k+1)}$$ is sufficiently small, the algorithm is said to have converged on the solution.

There is also a quantum algorithm for linear systems of equations.

Homogeneous systems
A system of linear equations is homogeneous if all of the constant terms are zero:
 * $$\begin{alignat}{7}

a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = \;&&& 0 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = \;&&& 0 \\ &&      &&            &&              &&  &&   \vdots\;\  &&&  \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = \;&&& 0. \\ \end{alignat}$$ A homogeneous system is equivalent to a matrix equation of the form
 * $$A\mathbf{x}=\mathbf{0}$$

where A is an m × n matrix, x is a column vector with n entries, and 0 is the zero vector with m entries.

Homogeneous solution set
Every homogeneous system has at least one solution, known as the zero (or trivial) solution, which is obtained by assigning the value of zero to each of the variables. If the system has a non-singular matrix ($det(A) ≠ 0$) then it is also the only solution. If the system has a singular matrix then there is a solution set with an infinite number of solutions. This solution set has the following additional properties: These are exactly the properties required for the solution set to be a linear subspace of Rn. In particular, the solution set to a homogeneous system is the same as the null space of the corresponding matrix A.
 * 1) If u and v are two vectors representing solutions to a homogeneous system, then the vector sum u + v is also a solution to the system.
 * 2) If u is a vector representing a solution to a homogeneous system, and r is any scalar, then ru is also a solution to the system.

Relation to nonhomogeneous systems
There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:
 * $$A\mathbf{x}=\mathbf{b}\qquad \text{and}\qquad A\mathbf{x}=\mathbf{0}.$$

Specifically, if p is any specific solution to the linear system Ax = b, then the entire solution set can be described as
 * $$\left\{ \mathbf{p}+\mathbf{v} : \mathbf{v}\text{ is any solution to }A\mathbf{x}=\mathbf{0} \right\}.$$

Geometrically, this says that the solution set for Ax = b is a translation of the solution set for Ax = 0. Specifically, the flat for the first system can be obtained by translating the linear subspace for the homogeneous system by the vector p.

This reasoning only applies if the system Ax = b has at least one solution. This occurs if and only if the vector b lies in the image of the linear transformation A.

title: System of linear equations