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 read, represent, compare and order rational numbers, including decimal numbers to thousandths. The museum offers an early treatment of decimal fractions; negative fractions and the other contexts are for your class to bring. 1
Curriculum expectations, with their official codes
- B1.3 read, represent, compare, and order rational numbers, including positive and negative fractions and decimal numbers to thousandths, in various contexts
2 The lesson
What does each place in a decimal number stand for?
Write a decimal number with three places after the point and name what each place stands for. What changes after the units place?
Go deeper
What does a mark have to protect? The record's examples show it. Saidan mentions dropping final zeros when the halving is reversed by multiplication, and the record adds that this does not allow a reader to delete a zero that keeps a place inside a number: 0.50 equals 0.5, but 0.05 does not 2. A second example in the record: five tenths can be read as fifty hundredths and combined with twenty-five hundredths to make 0.75 2. The record says these numbers illustrate the place-value principle Saidan discusses and are not an example from the manuscript 2.
Why they did it
The records read here preserve no sentence in which al-Uqlidisi states his purpose, but they show the shape of his book. Its first part explains numerals, place value and operations; the second gathers earlier methods in Indian form; the third answers questions beginning "why" or "how" 3. From this the biography infers experience teaching beginners, without naming a school or a formal appointment 4. Collecting earlier sources forms part of the author's own account of his work 5.
Two views
Saidan sees conscious use of fractions and of their sign, while Rashed distinguishes implicit procedures from an explicit general theory 2 4. As reported by MacTutor, Rashed emphasizes the absence of an explicit extension of powers of ten through their reciprocals after defining the zero power 4. The record says this does not deny knowledge of the numeral zero: the criticism concerns a theoretical relationship between places 4. The museum decides nothing between them.
Think about it
Saidan sees conscious use of decimal fractions, and Rashed asks for an explicit general theory. If you can add 0.05 and 0.5 correctly but cannot explain why the places line up as they do, which of the two descriptions fits what you can do, and why?
Into the museum
- al-Uqlidisi's decimal-fraction methods: repeated halving, the units sign and the two readings of the book.
- al-Uqlidisi: what is known of the author, and what is not.
- Algorism: the Indian place-value arithmetic that his book inherited.
Try this
Read Al-Uqlidisi's decimal-fraction methods. Take the sequence of numbers the record gives and order it from least to greatest.
3 Carry it further
Where this lesson leads in the museum.
4 Check yourself
Nothing here is marked or sent anywhere. It is for you.
In your own words: what does Al-Uqlidisi's decimal-fraction methods have to do with this lesson?
Compare with the card
A half is five tenths. A book composed by al-Uqlidisi at Damascus in 341 AH (952–953 CE) contains an early direct treatment of decimal fractions within Indian place-value arithmetic. Saidan gives the repeated-halving sequence 13, 6.5, 3.25, 1.625, 0.8125, and describes a stroke over the units position that the author insists should be used. The record stresses that this is not the invention of the Indian numerals or of today's decimal point. Why does it help to mark the units place?
I can now
Notes
- 1. Ontario Ministry of Education, The Ontario Curriculum: Mathematics
- 2. Al-Uqlidisi — A. S. Saidan's DSB biography, decimal-fraction discussion
- 3. Al-Uqlidisi — A. S. Saidan's DSB biography, four parts
- 4. Al-Uqlidisi — MacTutor biography, Biography
- 5. Al-Uqlidisi — A. S. Saidan's DSB biography, opening biography; accomplishments