I'd be particularly interested in who first used the name in a European language and whether it was used in a non-European language such as Arabic, Persian, or an Indian language before that.

[Edit 2010/01/22: Thanks to everyone who responded. It took me a few days to check Jonas Meyer's references. (The discussion of the CRT is on pp 175-176 of Part III of Wylie's book.) As JM said, they seem to narrow the appearance of the name in a European language to 1853-1929, which is hundreds of years later than I expected, and it now wouldn't be so surprising if it first appeared in English, maybe even in Dickson's book. So,

Question: Are there any European languages in which the CRT has a name that is not a direct translation of "Chinese remainder theorem"?

One more point: Wylie says,

'In examining the productions of the Chinese one finds considerable difficulty in assigning the precise date for the origin of any mathematical process; for on almost every point, where we consult a native author, we find references to some still earlier work on the subject. The high veneration with which is has been customary for them to look upon the labours of the ancients, has made them more desirous of elucidating the works of their predecessors than of seeking fame in an untrodden path; so that some of their most important formulae have reached the state in which we now find them by an almost innumerable series of increments. One of the most remarkable of these is the Ta-yen, "Great Extension," a rule for the resolution of indeterminate problems. This rule is met with in embryo in Sun Tsze's Arithmetical Classic under the name of Wuh-puh-chi-soo, "Unknown Numerical Quantities," where after a general statement in four lines of rhyme the following question is proposed: ...

In tracing the course of this process we find it gradually becoming clearer till towards the end of the Sung dynasty, when the writings of Tsin Keu-chaou put us in full possession of the principle, and enable us to unravel the meaning of the above mysterious assemblage of numerals....'

The Song dynasty apparently ended in 1279, which gives an interval of several hundred years. So, it seems that the name Chinese Remainder Theorem is not completely unreasonable, since according to Wylie, it's not clear when the general form was discovered, or at least might not have been at the time the theorem got its name. ]

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    $\begingroup$ When I taught elementary number theory, I joked that if we are to use a cavalier name like the "Chinese Remainder Theorem", then maybe we should call the Euclidean Algorithm the "European Algorithm" instead. $\endgroup$ – Greg Kuperberg Jan 22 '10 at 6:38
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    $\begingroup$ @Greg: Because Euclid worked in Alexandria? $\endgroup$ – Gerald Edgar Aug 15 '10 at 12:22
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    $\begingroup$ What is this theorem called in modern Chinese? $\endgroup$ – Gerald Edgar Aug 15 '10 at 12:25
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    $\begingroup$ @Gerald Edgar, the title of the article on the Chinese Wikipedia (zh.wikipedia.org/wiki/…) is 中国剩余定理, which is basically a translation of "Chinese remainder theorem." However, the article also gives eight older names: 「韓信點兵」、「孫子定理」、「鬼谷算」、「隔墻算」、「剪管術」、「秦王暗點兵」、「物不知數」, which I (roughly and hesitantly) translate as "Han Xin's inspection of troops", "Sun Tzu's Theorem", "Guigu (Wang Xu)'s Counting", "Counting by partitioning", "Cut tube technique", "Qin Shi Huang inspecting troops in the dark", and "Objects with unknown number". $\endgroup$ – j.c. Mar 16 '11 at 19:43
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    $\begingroup$ A propos of @GregKuperberg's and Gerald Edgar's comments, I'm herewith rechristening the Euclidean Algorithm the African Algorithm. $\endgroup$ – benblumsmith Aug 15 '16 at 21:35

The book A History of Mathematics: An Introduction by Victor J. Katz says:

"...probably the most famous mathematical technique coming from China is the technique long known as the Chinese remainder theorem. This result was so named after a description of some congruence problems appeared in one of the first reports in the West on Chinese mathematics, articles by Alexander Wylie published in 1852 in the North China Herald, which were soon translated into both German and French and republished in European journals..." (page 222)

This seems to suggest that the name "Chinese Remainder Theorem" was introduced soon after Wylie's 1852 article.

But the book Historical Perspectives on East Asian Science, Technology, and Medicine, edited by Alan Kam-leung Chan, Gregory K. Clancey and Hui-Chieh Loy says:

"A. Wylie introduced the solution of Sun Zi's remainder problem (i.e. "Wu Bu Zhi Shu") and Da-Yan Shu to the West in 1852, and L. Matthiessen pointed out the identity of Qin Jiushao's solution with the rule given by C. F. Gauss in his Disquisitiones Arithmeticae in 1874. Since then it has been called the Chinese Remainder Theorem in Western books on the history of mathematics."

This is ambiguous, as it is not clear whether the author means that the name "Chinese Remainder Theorem" came into use after 1852 or after 1874. But the phrasing does suggest that the name came into use before 1929.

In 1881, Matthiessen published the following article:

L. Matthiessen. "Le problème des restes dans l'ouvrage chinois Swang-King de Sum-Tzi et dans l'ouvrage Ta Sen Lei Schu de Yihhing." Comptes rendus de l'Académie de Paris, 92 :291-294, 1881.

But does the name "Chinese Remainder Theorem" ("le théorème chinois des restes") appear in this article?

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    $\begingroup$ I haven't found the name in Matthiesen's article. Its review in the Jahrbuch talks about the "Chinese method for solving systems of linear congruences". $\endgroup$ – Franz Lemmermeyer Feb 15 '10 at 11:08
  • $\begingroup$ Thanks. It seems quite possible the name first appeared in 1929 and in English. $\endgroup$ – Marko Amnell Feb 17 '10 at 0:33
  • $\begingroup$ Somehow having Gauss in the same sentence as 1874 strikes me as an anachronism. If "in 1874" refers to "Matthiessen pointed out", then [those authors] might want to move it a bit up for clarity. Of course it is to late for that now, and you only cited it... $\endgroup$ – Marc van Leeuwen May 8 '14 at 16:52

This is not a complete answer.

The website "Earliest Known Uses of Some of the Words of Mathematics" is made for just this kind of question, but it doesn't give any evidence that its example is really the first use. All it says is:

The term CHINESE REMAINDER THEOREM is found in 1929 in Introduction to the theory of numbers by Leonard Eugene Dickson [James A. Landau].

I thought it might be relevant that in Dickson's History of the theory of numbers, it says on page 57:

The rule became known in Europe through an article, "Jottings on the science of Chinese arithmetic," by Alexander Wylie, a part of which was translated into German by K.L. Birnatszki.

Like most sources, Dickson is giving the history of the spread of the idea and not the origin of the name. Wylie's article from 1853 can be read in full here (as reprinted in 1897); it includes an exposition of the Chinese work without using the name "Chinese Remainder Theorem". This could indicate that the name was not yet in use in 1853, but that's still a long way from 1929.

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    $\begingroup$ Excellent. That helps a lot. I had no idea the name was (apparently) such a recent thing. I see the phrase "Chinese problem of remainders" as a section heading in Dickson, but I haven't been able to find "Chinese remainder theorem" there, though I'm sure how much to make of the difference. Also, for some reason, the Google books link for Wylie doesn't give me any text, maybe because I'm in Australia. I'll have to go to the library on Monday. $\endgroup$ – JBorger Jan 16 '10 at 6:47
  • $\begingroup$ "Chinese Remainder Theorem" does appear in Dickson's number theory book as Landau claims, but not in his history book. Google says that Wiley's book is public domain in the U.S.A., but perhaps not in Australia; I'll take the slight risk of making the pdf available until there are objections: docs.google.com/…. It is a big file which I'm not taking time to break up. The arithmetic article starts on page 440 of the pdf. $\endgroup$ – Jonas Meyer Jan 16 '10 at 10:04
  • $\begingroup$ I got the file. Thanks! Also thanks for pointing out my confusion about the Dickson books. Now I don't have to go to the library for Wylie, but I do for the second Dickson! :) $\endgroup$ – JBorger Jan 17 '10 at 0:03
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    $\begingroup$ Incidentally, the full text of Dickson's history books (3 volumes) can be read or downloaded at archive.org . $\endgroup$ – Brendan McKay Aug 10 '13 at 1:20
  • $\begingroup$ Links to at least one version of Dickson's volumes on archive.org, as referenced by @BrendanMcKay: v1 v2 v3. $\endgroup$ – LSpice Dec 23 '19 at 23:13

The theorem is attributed to the mathematician Sun Tzu, also known as Sun Zi, and not to be confused with the military strategist Sun Tzu (of Art of War fame). People think Sun Tzu lived circa 400 AD.

See e.g.



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    $\begingroup$ Anweshi had something about these in a previous version, but he perhaps overindulged my nitpicking and removed it. Seeing it reposted shows me my error in stifling interesting related info. $\endgroup$ – Jonas Meyer Jan 16 '10 at 4:10
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    $\begingroup$ Although this is not the complete answer, the identity of the Chinese mathematician who is thought to have first proved the theorem has to be relevant to the question! $\endgroup$ – Pete L. Clark Jan 16 '10 at 4:10
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    $\begingroup$ No worries. (This space unintentionally not left blank.) $\endgroup$ – Pete L. Clark Jan 16 '10 at 4:25
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    $\begingroup$ en.wikipedia.org/wiki/Sun_Tzu_%28mathematician%29 $\endgroup$ – Jonas Meyer Jan 16 '10 at 19:13
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    $\begingroup$ Why is CRT attributed to Sun Zi? This certainly appears in Sun Zi Suan Jing, but there are also other stories related to CRT. Actually, CRT is often known as the "Han Xin Dian Bing" problem in China (Han Xin, a Han Dynasty general, counting the number of soldiers), and Han Xin lives some time around ~200 BC. The credibility of such folklore can be questioned though. $\endgroup$ – Ho Chung Siu Jan 22 '10 at 10:16

Wikipedia says that the theorem appears in Fibonacci's Liber Abaci (1202). So that could be the first European instance where this name is used(though wikipedia does not say anything about what name was used, as Jonas Meyer notes below).

I suggest that you check into Weil's overview of the history of number theory from Hammurapi to Legendre. That book must surely contain something.

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  • $\begingroup$ This seems not very related to the question. The first part is relevant to why someone would give it the name it has, but none of it has a clear relevance to the actual history of the name. $\endgroup$ – Jonas Meyer Jan 16 '10 at 3:06
  • $\begingroup$ Except for the advice about Weil's book, that may indeed have something good. $\endgroup$ – Jonas Meyer Jan 16 '10 at 3:07
  • $\begingroup$ @Jonas Meyer. Thanks. Shortened the answer accordingly. $\endgroup$ – Anweshi Jan 16 '10 at 3:10
  • $\begingroup$ Wikipedia does not say that "it" appears in Liber Abaci, because what is being asked is not where the theorem appears but where the name appears. Wikipedia says nothing about where the name first appeared. $\endgroup$ – Jonas Meyer Jan 16 '10 at 3:22
  • $\begingroup$ Ok, very well. Your objection is valid. However I am reluctant to further shorten my answer to just one sentence. So let it stand! $\endgroup$ – Anweshi Jan 16 '10 at 3:25

The Chinese Remainder Theorem first appears in "Sun Zi's Calculation Classic" between the 3rd and 5th century AD (http://en.wikipedia.org/wiki/Sun_Tzu_(mathematician)).

There is a website about the Chinese Remainder Theorem (http://www.cut-the-knot.org/blue/chinese.shtml), where the author refers to a similar puzzle described by Indian mathematician Brahmagupta in 598 AD. I think it's possible that the Chinese Remainder Theorem became well-known early on among mathematicians elsewhere in Asia after Sun Zi published his book.

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