Where do our decimal fractions come from?

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

Write a decimal number with three places after the point and name what each place stands for. What changes after the units place?

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.

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.

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.

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?

• 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.

Read Al-Uqlidisi's decimal-fraction methods. Take the sequence of numbers the record gives and order it from least to greatest.