Powers and roots

Write 3 × 3 × 3 × 3 as a power. Then ask which number multiplied by itself gives 49, and how you would check your answer.

Look at how al-Karaji handled the powers of one unknown quantity. The record starts from distinguishing its successive powers: the unknown, its square, its cube and further powers, and also their reciprocals ². Multiplying two of them needs a rule that names the resulting power, rather than a new geometrical diagram for each operation ². In modern teaching notation x² times x³ gives x⁵; the record says this is its own illustration, not a manuscript transcription ². MacTutor stresses that he approached this law without defining x⁰=1, so the complete modern formulation is not credited to him ².

The records read here preserve no sentence in which al-Karaji says why he worked on these operations. They show his setting: his principal mathematical books are placed in Baghdad, and TDV connects al-Fakhri with Fakhr al-Mulk's patronage in the Buyid setting ^2 3. For al-Kashi, the record says his Key to Arithmetic addresses calculation useful for astronomy, surveying, architecture, accounting and trade; these are purposes reported for the book, not evidence that a particular workshop used a particular example ⁴.

As reported by MacTutor, Woepcke's and Rashed's interpretations emphasize algebraic operations becoming less dependent on geometrical representation, an important direction toward autonomous algebraic arithmetic ². The account it reports from Crossley stresses numerical examples, limits of demonstration and retained Euclidean theorems in equation solving ². TDV likewise notes geometrical proof in al-Fakhri ³. The record presents these readings without resolving them: new operational organization and older forms of justification can coexist in one body of work.

One reading sees algebra becoming less dependent on drawings; another stresses that Euclid's theorems stayed in the proofs. Which would you give more weight to as evidence of a new kind of mathematics, a new rule for naming powers or a proof that still draws a figure, and why?

• al-Karaji's polynomial arithmetic: the rule for powers, its limits and the historians' readings.
• al-Kashi's general numerical root extraction: roots of general orders, with the earlier work the record keeps visible.
• al-Karaji: the mathematician and engineer, and the doubts about his name.
• al-Kashi's Key to Arithmetic: the teaching book the root method comes from.