Powers and roots

This lesson asks you to identify and represent perfect squares and determine their square roots, and to evaluate and express repeated multiplication of whole numbers using exponential notation. The museum offers two glimpses: al-Karaji's naming of the successive powers of an unknown with a rule for multiplying them, and al-Kashi's extraction of roots; perfect squares, exponential notation for numbers and the contexts of everyday life are for your class to bring. ¹

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.

Read Al-Karaji's polynomial arithmetic and then Al-Kashi's general numerical root extraction. Choose a number with a whole-number square root and one without, and for each say how you would find or bound its root and how you would check it.