Al-Karajī, Math Olympiad Trainer Academy

Al-Karajī was born on 13 April 953 in Baghdad and died around 1029. His major mathematical works were written in Baghdad, where his principal algebra treatise Al-Fakhrī was composed and dedicated to a Baghdad ruler. Later in his career he left Baghdad for what MacTutor describes only as the "mountain countries," where he appears to have shifted his attention toward practical engineering, including well drilling.

Principal treatises

Al-Fakhrī, his principal algebra treatise. Defines monomials \(x, x^2, x^3, \ldots\) and \(1/x, 1/x^2, 1/x^3, \ldots\); gives multiplication rules for them; defines arithmetic on polynomials ("composite quantities"); gives a square root extraction rule (restricted to positive coefficients); computes \((a+b)^3\).
Al-Badīʿ, computes \((a-b)^3\) and \((a+b)^4\).
A third work, referred to in the MacTutor references as The Sufficient book of al-Karajī and in the wider secondary literature as al-Kāfī fī al-ḥisāb ("The Sufficient in Arithmetic"). The MacTutor main text provides the title and limited detail about its contents, so this entry records the title alone.

The sum of cubes identity

Al-Karajī states and demonstrates the identity

\(1^3 + 2^3 + \cdots + n^3 \;=\; (1 + 2 + \cdots + n)^2\).

MacTutor gives the identity in the form \(\sum i^3 = (\sum i)^2\), and records that Al-Karajī demonstrated it explicitly for \(n = 10\). This is one of the small handful of closed form summation identities that still opens olympiad problem sets in number theory and combinatorics.

An early inductive argument

MacTutor's language on this is worth quoting exactly:

Although this is not induction proper, it is a major step towards understanding inductive proofs.

What Al-Karajī actually does is verify a statement for small values , \(n = 1\), then \(n = 2\) from the case \(n = 1\), and so on up to about \(n = 5\), and then assert that the process continues indefinitely. A modern proof would establish the step from \(n\) to \(n+1\) for every case. The underlying idea, that each case rests on the one before it, is unmistakably the ancestor of the technique every olympiad student eventually uses.

Diophantine problems, extended

Al-Karajī worked directly with the first three books of Diophantus's Arithmetica, in the Arabic translation by Qusṭā ibn Lūqā (rendered by MacTutor as "ibn Liqa," c. 870). He incorporated many of Diophantus's problems and, per Franz Woepcke's analysis cited by MacTutor , generalised them and posed new ones of his own. This is the specific lineage of the number theoretic problem style olympiads still use. A family of equations is reduced to a general method.

On priority claims

MacTutor's page describes Al-Karajī as one "who can be regarded as the first person to free algebra from geometrical operations and replace them with the type of operations which are at the core of algebra today," and records that this interpretation is due first to Franz Woepcke and later endorsed by Roshdi Rashed. Both attributions are named. We report the characterisation as MacTutor does: a substantive claim about the direction of algebraic thinking, attributed to identifiable modern historians.

Sources

MacTutor History of Mathematics, University of St Andrews, al-Karajī biography (birth date 13 April 953, Baghdad base, treatise names, sum of cubes identity, induction language, Diophantus extension via Qusṭā ibn Lūqā, Woepcke and Rashed attributions). https://mathshistory.st-andrews.ac.uk/Biographies/Al-Karaji/
Encyclopædia Britannica, al-Karajī biography (secondary verification of dates, Baghdad, treatise titles Al-Fakhrī and Al-Badīʿ). https://www.britannica.com/biography/al-Karaji
Roshdi Rashed, "L'induction mathématique: al-Karajī, as-Samaw'al," Archive for History of Exact Sciences, vol. 9 (1972), pp. 1 to 21, the specific modern study cited on the induction question (indexed via Semantic Scholar). https://www.semanticscholar.org/paper/L%27induction-math%C3%A9matique%3A-al-Karaj%C4%AB%2C-as-Samaw%27al-Rashed/fc3c72d557fd6fd904f768fef11170b0ee470cad