Al-Karaji's binomial-coefficient recurrence

How can we find the numbers appearing when a sum of two quantities is multiplied by itself repeatedly? The sources attribute to Abu Bakr al-Karaji a rule generating these coefficients by adding adjacent entries in a table; al-Samawal al-Maghribi transmits the account of an original computation that no longer survives ^{1 2}. Its setting is early-eleventh-century Baghdad algebra, but this account does not establish the particular lost text's exact composition date or writing location ^{1 2}. We describe a bounded improvement in coefficient generation, not the invention of all algebra or the world's first arithmetic triangle ^{1 3}.

A binomial here is a sum of two parts; a coefficient records how often a particular product contributes after like terms are collected, illustrated in the transmitted account by the square's sequence of one, two and one ¹. In modern teaching notation we write(a+b), whose square is a²+2ab+b²; these letters and superscripts are not a transcription of Karaji's manuscript ¹. This page's particular contribution is a method for finding the next power's coefficient list, rather than another account of all the operations on composite expressions already described separately ¹. The accompanying modern image displays six rows, whereas the historical witness describes a table that can be pictured as a triangle lying on its side ^{4 1}.

Operations on unknown quantities did not begin with Karaji: MacTutor places his work within the tradition of al-Khwarizmi and other Arabic algebraists, alongside the arithmetic influence of Diophantus ¹. Al-Samawal's biography also includes Abu Kamil among the authors he studied before developing his own work ⁵. Outside this Arabic sequence, the history of Jaina mathematics describes rules for combinations and permutations, connections between combinations and binomial coefficients, and a triangle in a tenth-century commentary on an older work ³. That is evidence of an earlier or parallel Indian coefficient tradition, not a document showing that the particular commentary reached Karaji ^{3 1}. Similar shapes do not by themselves supply a journey or a translation.

MacTutor reports that Karaji calculated the cube of a sum in al-Fakhri, and the cube of a difference and fourth power of a sum in al-Badi; the general table construction is preserved in al-Samawal's later writing ¹. Adel Anbouba's Dictionary of Scientific Biography entry places Karaji's coefficient calculation after1007, referring to an edition of al-Badi that we have not independently read ². That reference is not treated here as the exact composition date of a complete surviving original ². Al-Samawal admired Karaji but also identified shortcomings and developed improvements; the biographies describe al-Bahir in algebra as a youthful work that may subsequently have been revised ^{5 2}. The preservation story is therefore a later author naming a predecessor within a larger algebraic project, not a documented personal meeting.

Karaji's role in this record is improver: the repeatable generation of coefficients is associated with him within an older tradition, without world-first claims for combinations or triangles ^{1 3}. Ataie-Ashtiani and Simmons explicitly identify the named mathematician and engineer as Muslim, but do not attest his personal religious school ⁶. The separately declared Sunni frame belongs to the museum; Buyid patronage and Baghdad context remain distinct from evidence of individual belief ⁷. Al-Samawal's role here is transmitter: his biography places the initial al-Bahir in his youth and his conversion to Islam in1163, so neither every passage's Islamic-period composition nor a personal Sunni affiliation is assumed ^{5 2}.

The translated testimony reproduced by MacTutor begins with ones and constructs1,2,1 for the square; the extremes represent each part's square, while the middle records the product of the two parts twice ¹. To obtain the next list, retain a one at each extreme, add one and two to obtain three, and then two and one to obtain another three, producing1,3,3,1 ¹. Generating this list does not require dividing a large factorial or independently searching for every coefficient; adjacent addition is the procedure actually described by the witness ¹.

In a modern reconstruction we can name the coefficient of power n at position k as C(n,k), giving the internal recurrence C(n+1,k)=C(n,k−1)+C(n,k), with one at each boundary ¹. This indexing and symbolism explain the rule; they are not the author's words or evidence that he supplied the modern definition of a zeroth-power row ¹. When(a+b)³ is multiplied by(a+b), each interior term receives contributions in two ways: a preceding term multiplied by a and its neighbour multiplied by b, whose coefficients must consequently be added ¹. From1,3,3,1 we therefore obtain1,4,6,4,1 for the fourth power, followed by1,5,10,10,5,1 for the fifth, the last row of the modern illustration ^{1 4}.

Take a=2 and b=3 as numbers selected by the museum for explanation, not a historical example attributed to the author: the square gives4+12+9=25, and the cube8+36+54+27=125 ¹. The fourth-power list gives16+96+216+216+81=625, agreeing with5⁴ ¹. This numerical check illustrates what coefficients do, but does not alone prove the rule for every power; MacTutor distinguishes the early successive reasoning from a rigorous exposition of mathematical induction ¹. The scope here is finite integer powers, not an infinite series for a negative or fractional exponent ^{1 8}.

Symmetry supplies another check on the witness's lists:1,3,3,1 reads identically from either end, as do the displayed fourth- and fifth-power rows ^{1 4}. In the reconstruction, exchanging the names of the two parts reverses term order without changing their sum, making symmetry a teaching check rather than an attribution of a further undocumented theorem ¹. The number of coefficients exceeds the exponent by one because the two boundary terms remain alongside the interior contributions ⁴.

The transition established by the sources read is textual: from Karaji's work to al-Samawal's account in al-Bahir, not a map of dated stations for every manuscript ^{5 2}. Anbouba says that al-Bahir's influence on Arabic algebra is difficult to assess, suggests indirect and restricted influence on al-Kashi's Key of Arithmetic, and describes the book as apparently unknown in the West ². These are the historian's assessments, not a museum proof that no transmission could ever have occurred ². Nor are al-Samawal's travels in Iraq, Syria and elsewhere converted into a confirmed itinerary for this table, since the biography does not associate every journey with a copy of al-Bahir ⁵.

The operational foundation is multiplication and collection of like terms in algebra, described within Karaji's arithmetic of composite expressions ¹. Multiplication supplies the coefficient's meaning, while the recurrence supplies a way to generate the next coefficients; these are not interchangeable names for the whole algebraic achievement ¹. The conceptual diagram records Karaji's general expression arithmetic and earlier algebra as contexts, followed by the particular recurrence and then the witness in al-Bahir ^{1 5}. The Indian tradition is not placed in an invented causal chain: it appears in the preceding historical comparison with the evidential limit made explicit ³.

The nearest documented subsequent construction is al-Samawal's incorporation of the testimony within his broader treatment of operations, equations and summations ⁵. For China, MacTutor describes Jia Xian's table, known from Yang Hui's1261 explanation, including the sixth-power row and the rule adding the two entries above each position ⁹. This is comparable work, not a dependent demonstrated to have borrowed from Karaji ^{9 1}. Pascal's biography connects his triangle work through Wallis to Newton's extension to negative and fractional powers; the passage read supplies no route from Baghdad to that European investigation ⁸. The later extension consequently remains distinct from this record's finite scope.

The effect we can confidently describe concerns the calculator and learner: generating the next power's coefficients by adjacent addition and connecting them to multiplication, rather than presenting only disconnected lists of answers ¹. MacTutor follows the explanation through the fifth power, allowing a reader to continue the pattern ¹. The sources read do not measure reductions in working hours or market costs caused by this procedure, or establish that it became every merchant's daily tool ^{1 2}. General benefits of arithmetic should not be assigned to this particular algorithm without evidence, nor should later mathematical applications be retroactively attributed to Baghdad users.

Al-Samawal's writing preserves material about a lost original calculation; the historical importance is therefore not merely the survival of a shape, but of its attribution and construction account ^{2 1}. MacTutor highlights successive reasoning from one case to the next as an important step towards understanding inductive proof, while denying equivalence with a fully formulated modern induction principle ¹. Knowing the values1,3,3,1 is consequently different from describing a procedure generating further powers and explaining their relationship to multiplication ¹. We do not say that Europe or modern mathematics could not have developed without it; that counterfactual is not supported by the displayed sources.

Sunni museum frame: we present a contributor with documented Muslim identity, preserving predecessors and attribution limits, without assigning a religious school the technical judgement that the triangle began here ^{6 3}. Karaji's documented Muslim identity does not alone establish his personal school, and that distinction remains visible in his person record ⁶.

Academic account: the mediated translation in MacTutor attributes the general construction to Karaji through al-Samawal's testimony, while Anbouba confirms that the original computation is lost; this page does not claim an independent Arabic reading of that original or the al-Bahir edition ^{1 2}. Earlier Indian evidence prevents assigning all binomial-coefficient history to Karaji, just as Pascal's biography explicitly denies that Pascal first studied the triangle ^{3 8}. The more precise question is not which name cancels all others, but which procedure a particular witness attributes to a particular author.

A reader can now see the recurrence in a table of coefficients for nonnegative integer powers, with a single-entry opening row representing the zeroth power in modern presentation ⁴. That instructional row and its symbolic definition are not credited to Karaji's formulation as transmitted in the biography ¹. Separating modern illustration from historical witness allows comparison of mathematically correct rows without confusing numerical validity with documentary authenticity ^{4 2}. Fractional and negative exponents belong to the wider development discussed in Pascal's biography in connection with Newton, not a claim that all such results occurred in the lost book ⁸.

The principal textual witness for this attribution is al-Bahir, while the biographies state that Karaji's original computation is lost ^{5 2}. Anbouba's bibliography lists Istanbul witnesses, including Aya Sofya2718 and Esat Efendi3155, and a modern Ahmad/Rashed edition; we have not inspected these manuscripts' images or their current catalogues ¹⁰. The article itself gives the edition's year as1972 and elsewhere1971, a discrepancy not silently resolved by guessing ¹⁰. The displayed image is a modern work by Conrad Irwin after Drini, dated2007 in the Commons record, not a leaf from any copy of al-Bahir ⁴.

“Pascal's triangle” is a familiar name in modern mathematical biography, but does not imply that the triangle was unknown before him; MacTutor expressly says otherwise ⁸. Comparing Karaji's table with Jia Xian's shows how distinct names preserve different works involving the same shared mathematical arrangement ^{1 9}. No particular appearance in a film, postage stamp or named school textbook is attributed here, because the sources used supply no specific witness for it.

Write1,2,1 on paper and place a one at each end of a new line; between them add1+2 and2+1 to obtain1,3,3,1 ¹. In the modern example these numbers tell you how to collect the results of multiplying a two-part sum three times ¹. If you choose the parts2 and3, the result is8+36+54+27, or125, the same as5×5×5 ¹. Those example numbers are our instructional choice, while al-Samawal attributes the addition rule to Karaji ¹. Remember that the illustration is a modern six-row diagram, not a picture of the scholar or his manuscript ⁴. Can you build the next line without first multiplying every term?

• MacTutor: Karaji, biography and testimony transmitted through al-Samawal ¹.
• MacTutor: al-Samawal, al-Bahir and religious chronology ⁵.
• Anbouba: Dictionary of Scientific Biography, algebra and bibliography ^{2 10}.
• Jaina mathematics and Jia Xian, comparative evidence, not a direct diffusion claim ^{3 9}.
• Pascal, triangle and wider powers ⁸.
• TDV Karaji entry277–278 and peer-reviewed study762 for context and identity ^{7 6}.
• Image record, source and licence, not mathematical history ⁴.