When it comes to history, discussions become difficult.
Sir Joseph Needham (dec. March, 1995) wrote on p. 53 of volume 3 of his work: Science and Civilization of China, Cambridge University Press, 1959.
.... By some obscure play upon words, it was thought that the Europeans admitted that their algebra had come from the East, and although the intense nationalism of some later Chinese writers in this respect has been much castigated by 19-th century Europeans, the fact remains that whatever future research may reveal about transmission, algebra was just as essentially Indian and Chinese as geometry was Greek. Actually, there is some evidence of transmissions from the Arabs to the Chinese in the 13-th and 14-th centuries, and much more from the Chinese earlier to India and Europe. A great deal of careful historical work will be needed before any final conclusions can be drawn.
One should now refer to p. 1 of this treatise:
There is a very large literature on the history of mathematics in East Asia, though unfortunately (for most Westerners) by far the greater part of it is in the Chinese and Japanese languages. Those who are debarred from this original material inevitably to the well-known histories of mathematics in Western languages, such as those of Cantor, Loria, Cajori, D. E. Smith, and Karpinski. Cantor's famous work is now old (1880); he had to rely on the still earlier translations of E. Biot, while his other main source was a paper by Biernatzki of 1856. This, however, was a mere translation of the work of Wylie of 1852--an excellent account which can still be read with profit today. The most succinct modern description of Chinese mathematics in historical context is that of Cajori, while the fullest is that in Smith, who arranged his two volumes choronologically in the first and according to subject in the second. The work of all these scholars was vitiated by the fact that none of them possessed sufficient Chinese to permit any first-hand contact with the texts themselves. This criticism bears least forcibly upon D. E. Smith, who himself spent some time in China and Japan, made collections of mathematics books there, and had the advantage of intimate collaboration with Asian mathematicians, notably Mikami Yoshio. Such collaboration is also evident in the distinguished recent contribution of A. P. Yushkevitch in Russian. .....
For those who have access to the book:
Robert Temple, The Genius of China, 3,000 years of science, discovery, and invention, Simon and Schuster, 1986.
There is an introduction by Needham plus a preface by Temple about Needham. Unlike most of the western historians, Needham was alreay a distinguished Biochemist when he became interested in China. At the age of 37, he began learning Chinese, by 1942 (5 years later), he was sent to Chunking, China, and literally scoured the countryside to talk to scientists, engineers, and collecting books and materials which were shipped back to Cambridge university. Needham applied the scientific method to his investigations and checked against all kinds of sources. A diligent reader can check Needham's sources. This is in sharp contrast with a number of western texts on history of mathematics where the author furnished very few references and the readers are forced to accept the claims of the author as *truth*.
Since Needham concentrated on China, but gives Smith a good rating (D. E. Smith's two volumes on History of Mathematics are reprinted by Dover Publications), one may go to vol. 2, p. 378, where he discusses the Nature of Algebra. Here, I quote:
If by algebra, we mean the science which allows us to solve the equation ax^2 + bx + c = 0, expressed *in these symbols*, then the history begins in the 17-th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say algebra begins with the Alexandrian School or a little earlier; and if we say that we should class as algebra any problem that should now solve by algebra (even though it was at first solved by mere guessing or by some cumbersome arithmetic process), then the science was known about 1800 BC, and probably still earlier.
One can now be immersed in Smith's treatment. However, one should keep in mind that some of the rough estimates on dates related to the ages of some of the Chinese texts as given in Smith may be a bit too early. A more reliable source is the book:
Chinese Mathematics, A concise history, by Li Yan and Du Shiran, translated by J. N. Crossley and A. W-C. Lun, Oxford Science Publication, 1987.
In any event, D. E. Smith provides quite a bit of information on the evolution of various arithmetic operations.
As an example of the problem of dating, the earliest Chinese text that survived is Chou Pei Suan Ching. It was primarily a book on astronomy. But it contains a picture and a dialogue describing the theorem of Pythagoras. One can argue: the Babylonians had recorded many Pythagorean triples; thus, they *must* have known the theorem of Pythagorus. There is another scenario: they could have discovered that certain pairs of integers have the properties that the sum of their squares were perfect squares and then discovered a rule that would generate such pairs. Unless they drew some pictures involving right triangles, we would not be on safe ground to make the claim that they knew the theorem of Pythagoras. At this point, one faces the problem of dating the text. After citing Needham (who tentatively set the date around 300 BC), Boyer's History of Mathematics went ahead and listed Chou Pei Suan Ching in the chronological table as (1100 BC ?). In contrast, Yan Li's Chinese Mathematics estimated the surviving edition of Chou Pei Suan Ching to between 100 BC to 100 AD (see p. 27 for a description of the cross checking process).
Incidentally, the original Chinese edition of Li Yan's book first appeared around 1960. However, the English translation incorporated some of the archeological discoveries in the 1970's. These seem to have confirmed a number of "myths" or "folklores".
I have to confess that I do not know any of the Chinese texts *first hand*. From all the secondary sources and from reading the work and bibliography of Needham, it seems safe to put more trust in Needham's work than the others. In history, there are no "absolute" proofs. For an interesting account on the priority of the theorem of Pythagoras, there is the little pamphlet:
Frank J. Swetz and T. I. Kao, Was Pythagoras Chinese? The Penn. State Univ. Press, and NCTM. 1977.