Lecture 9 Al Uqlidisi's 'Arithmetic'.

From 'The Aritmetic' of Al Uqlidisi,
Chapter 3, Halvong where fractions occur,and doubling: all methods.
Edited by A. A. Saidan

The halving which yields a fraction occurs when the units cell is odd. For halving that, when it is so, and for drawing it, there are methods. One is on the principle of derivation of numbers, and one is on the principle of degrees and minutes.

We have shown how to halve a degree [...]. If the degree is halved ten times that gives:

  • 00.
  • 00
  • 03
  • 30
  • 55
  • 15

To check the correctness of that we double. If the degree is regained it is right.

That is all about halving and doubling the degree. If more than that is drawn, like degrees, it will lead to this. For finally we shall have one or an odd number.

In what is drawn on the principle of numbers, the half of one in any place is 5 before it. Accordingly, if we have an odd number we set the half as 5 before it, yhe units place being marked by a sign ' above it to denote the place. The units place becomes the tens to what is before it. next we halve the 5 as it is the custom in halving whole numbers. The units place becomes hundreds in the second time of halving. So it goes always.

For example, we want to halve 19 five times. We say: one half of 9 is 4 and 1/2; we set the half as 5 before the 4; next we halve the 10. We mark the units place. That becomes 9'5. Now we halve the 5 and the 9; we get 4'75. We halve that and get 2'375, the units place being thousands to what is before it, for if we want to state what we have got, we say that halving has led to 2 and 375 of one thousand. We halve that to get 1'1875. We halve a fifth time to get 0'59375, which is 59375 of one hundred thousand.

We relate that by saying one half, and half of one eighth and one fourth of one eighth.

If we want to retrace that, we start as usual in such cases. It comes back to what it was, but whenever a zero comes at the beginning, we drop it because we do not need it. The 19 is regained.

As for halving not according to the habit of hie who says that half of 1 is 1/2, but of he who says that it is 1 of 2, the drawing of half the 1 is by drawing it as 1 of 2 below it; for every further halving, he doubles the 2, and so on. Thus to halve 19 five times we say: half the 9 is 4 1/2, half the 10 is 5. Draw it as 9 1/2.

To halve that again, we halve the 9 and double the 2. For the unit is now four parts and the fraction we now have is one fourth, being one of four. When we add it to the half of 9, we get 4 3/4.

We halve that by doubling the lower 4 and halving the upper 4. We get 2 3/8. We halve it again and get 1 3/16. The fifth time we get 0 19/32. This is because the half which comes from the top is equivalent to half the bottom. The unit has become 32 parts of which we have 16 and 3. We find the ratio is 19/32 the same as stated before.

We retrace by halving the bottom; it becomes 16. The 19 is greater than that; we thus elevate 16 units one place on top in the place of 0 and leave 3 above the 16; we get 1 3/16. Now we double the 1 and halve the 16, then double the 2 and halve the 8. We get 4 3/4. We double the upper 4 and halve the lower 4 and of the 3 elevate 2 into 1. We get 9 1/2.

We double it and regain the 19.

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Arabic mathematics

Greek mathematics did not finish with Archimedes. There are major developments in:
  • Ptolemy (c. 150): astronomy, tables of sines and tangents, ideas of function.
  • Diophantos (c. 250): algebra, systems of equations, Diophantine problems.
  • Apollonius (c.-200) and Pappus (c. 300): plane and solid geometry.

But the next major advances in the development of Western mathematics accompanied the Islamic conquest of Syria, Egypt, Arabia, Persia, India, Central Asia, N. Africa, Spain in the period 630--700.

After this, the Islamic empire split into Eastern and Western wings, the Eastern being the Damascus caliphate, based on Baghdad, and the Western based on Cordoba.

Under the enlightened rulers Haroun al Rashid and al--Mamun , scholars in Baghad created a University-like institution the Beit Ha' Hokhma, House of Knowledge and gathered and translated into Arabic Greek, Babylonian, Indian and possibly Chinese mathematics and astronomy. For exampe the Indian numerals were introduced about 770.

The most famous mathematician of the next generation was Al Khwarizmi (c. 800) whose legacy includes the words and concepts algebra, algorithms, zero, cipher. His work includes algorithms for the four operations, solution of quadratics, Egyptian fractions, and was translated into Latin c. 1100 and spread through the Christian world.

Around 950, a minor mathematician Abu 'lHasan al Uqlidisi (the Euclidean) of Damascus, wrote an 'Arithmetic' which was only re-discovered about 1960. It is the earliest extant work on Indian numerals, emphasising written algorithms as against counting boards: they are permanent, you can locate errors, you can leave and return, and so on.

He describes three calculation methods, sexagesimal (scientific), decimal (business) and common fractions.

Decimal fractions are suitable for both exact and approximate calculations.

Al Uqlidisi first writes down the result in sexigesimal of halving one degree ten times. [CHECK IT!] He notes that this can be checked by ten multiplications.

He then shows how to halve 19 ten times in decimal notation, introducing for the first time the equivalent of the decimal point. Finally he shows how to do the operations using common fractions.

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For a biography of Al Uqlidisi, click here.

For more on Al Khwarizmi, see here or here.

To return to Table of Contents, click here. For next Lecture , click here. Last update: 28 June 2000

Author: Phill Schultz, schultz@maths.uwa.edu.au