Draft: this lesson was written with AI help and has not yet been read by a human editor.
1 What your class asks
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
Curriculum expectations, with their official codes
- B1.2 describe, compare, and order numbers in the real number system (rational and irrational numbers), separately and in combination, in various contexts
- B1.3 estimate and calculate square roots, in various contexts
2 Before you start
A thought to carry into the lesson.
3 The lesson
Between which two whole numbers does the root fall?
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.
Go deeper
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.
Why they did it
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.
Often misunderstood
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.
Think about it
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?
Into the museum
- 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.
Try this
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.
4 Carry it further
Where this lesson leads in the museum.
5 Check yourself
Nothing here is marked or sent anywhere. It is for you.
In your own words: what does Al-Kashi's general numerical root extraction have to do with this lesson?
Compare with the card
Finding a number from its power. In the Key to Arithmetic, al-Kashi treated numerical extraction of roots of general orders, not only square roots. The record says root extraction reverses raising a number to a power: the positive cube root of eight is two because 2³=8, and when a number gives no whole-number answer a procedure needs an approximation and some account of what it guarantees. It calls this development of inherited computation, not the first root ever known. What should an approximation tell you?
In your own words: what does Abu Kamil's algebra with irrational coefficients have to do with this lesson?
Compare with the card
An irrational number inside an equation. The record says Abu Kamil's algebra admitted irrational quantities into the coefficients of equations as well as into their solutions. It separates the two: an equation with integer coefficients such as x²=2 has an irrational positive solution, while in x²+√2·x=4 an irrational quantity sits in the coefficient itself. The second equation is the museum's teaching example, not a line from the book. Say, for each equation, where the irrational quantity appears.
I can now
Notes
- 1. Ontario Ministry of Education, The Ontario Curriculum: Mathematics
- 2. Abu Kamil — Cengiz Aydin, work1
- 3. Abu Kamil's algebra — Dosay1990, p.66
- 4. Abu Kamil — MacTutor, biography
- 5. Abu Kamil — Cengiz Aydin, biography
- 6. Abu Kamil — MacTutor, Book on algebra
- 7. Abu Kamil — MacTutor, surveying discussion
- 8. Al-Kashi — MacTutor, decimals-and-roots
- 9. Al-Kashi — MacTutor, root-extraction