Roots, and numbers that never end

This lesson asks you to describe, compare and order rational and irrational numbers, and to estimate and calculate square roots. The museum offers two accounts, one of root extraction by approximation and one of an irrational quantity inside an equation; the estimating, ordering and calculating are for your class to do. ¹

Choose a number that is not a perfect square and find the two consecutive whole numbers between which its square root lies, then refine the estimate in decimals. Say how you know your new estimate is better than the last, and where you would place the root on a number line among the rational numbers.

A coefficient is a quantity multiplying an unknown or one of its powers ². Cengiz Aydin explains that Abu Kamil's algebra admitted irrational quantities into equation coefficients as well as solutions ². The record shows a problem from Melek Dosay's modern paper, page 66: two sums, each made of the unknown and a square root, one of 3 and one of 2, are multiplied, and the product is 20 ³. The record says it read the example on an image of the modern paper, not in Abu Kamil's manuscript, and that the notation is modern ³.

Very little personal information survives about Abu Kamil, and the records read here preserve no sentence stating why he extended coefficients ^{4 5}. They show whom he acknowledges, since MacTutor records Abu Kamil acknowledging al-Khwarizmi's authority ⁴, and what his book contains: three parts, on quadratic equations, on algebra applied to regular pentagons and decagons, and on indeterminate problems ⁶. A separate surveying book, MacTutor says, gave rules and examples without proofs for government surveyors ⁷.

Al-Kashi is sometimes credited as the first inventor of decimal fractions and of the root methods in his book. MacTutor qualifies both claims while keeping his achievement ⁸. It reports Luckey's comparison of his procedure with a special case of the later Ruffini and Horner methods, and Rashed's qualification that earlier procedures existed in the school of al-Karaji, particularly al-Samawal ⁹. The record therefore calls it development and integration of inherited computation ⁹.

Luckey compared al-Kashi's method with Horner's, and Rashed pointed to earlier procedures in al-Karaji's school. The record says that a structural comparison and proof that a later author read an earlier text are different claims. What would you need to find before saying a later author learned from an earlier one, and why?

• al-Kashi's general numerical root extraction: the method, its predecessors and the sources the record could not read.
• Abu Kamil's algebra with irrational coefficients: the difference between an irrational coefficient and an irrational solution.
• al-Kashi: the mathematician at Samarkand and his other work.
• Abu Kamil: what little is known of his life.

Read about al-Kashi's root extraction and Abu Kamil's algebra with irrational coefficients. Pick an imperfect square root, approximate it, and order it between two rational numbers.