Tenths, hundredths and equal amounts

This lesson asks you to read, represent, compare and order decimal numbers up to thousandths, and to show how fractions and decimals name the same amounts. The museum offers one early account of decimal fractions; the tools, the drawings and most of the practice your class brings. ¹

Look at the expectations: they ask for equivalences, shown with tools and drawings. Write a fraction you can split in half, write each half as a decimal, and compare the places you used. A hundred grid or a number line can show that your two names mark the same amount.

The record offers a teaching comparison: a quarter is twenty-five hundredths, and it is also fifteen sixtieths ². These numbers are the record's own modern explanation, not a line from al-Uqlidisi's book. Equal amounts do not make the two writing systems the same: each decimal place divides the previous unit by ten, while sixty-based places use sixty ². The question is therefore not only whether an answer is correct, but which scale supports further calculation ². Saidan distinguishes giving a root on the decimal scale from converting its result into sixty-based notation ².

The records read here preserve no sentence in which al-Uqlidisi says why he wrote on fractions. They show how he describes his own preparation: as the biography reports, his introduction describes travel, reading and learning from arithmeticians he met ³. The record calls this transmitted testimony, not an independently established itinerary or complete teacher list ³. The biography also stresses the practical value the author attributes to Indian arithmetic ³.

It is easy to think that al-Uqlidisi invented the Indian numerals or today's decimal point. The record says the story is neither: it concerns operations on fractional parts and a sign that preserves the units position, with historians disagreeing over how complete the concept was ³. That sign is not the dot that separates whole and fractional parts in our modern image ^{4 2}. And writing whole numbers in base ten does not by itself establish the same treatment of fractions ³.

A quarter is twenty-five hundredths and also fifteen sixtieths: one amount in two scales. Which scale would you choose for carrying a calculation below one, and why?

• al-Uqlidisi's decimal-fraction methods: the units sign, repeated halving and the disagreement about how complete the idea was.
• al-Uqlidisi: what is known of the author, and what is not.
• Algorism: the Indian place-value arithmetic that came before his book.

Read the record on al-Uqlidisi's decimal-fraction methods. Write one half, then half of that, first as fractions and then as decimals, and shade each on a hundred grid.