**
Al-Baghdadi**
is sometimes known as **Ibn Tahir.** His full
name is Abu Mansur Abr al-Qahir ibn Tahir ibn
Muhammad ibn Abdallah al-Tamini al-Shaffi
al-Baghdadi. We can deduce from al-Baghdadi's
last two names that he was descended from the
Bani Tamim tribe, one of the Sharif tribes of
ancient Arabia, and that he belonged to the
Madhhab Shafi'i school of religious law. This
school of law, one of the four Sunni schools,
took its name from the teacher Abu 'Abd Allah
as-Shafi'i (767-820) and was based on both the
divine law of the Qur'an or Hadith and on human
logical reasoning when no divine teachings were
given.

We have a few details of al-Baghdadi's life. He was born and brought up in Baghdad but left that city to go to Nishapur (sometimes written Neyshabur in English) in the Tus region of northeastern Iran. He did not go to Nishapur alone, but was accompanied by his father who must have been a man of considerable wealth, for al-Baghdadi, without any apparent income himself, was able to spend a great deal of money on supporting scholarship and men of learning.

At this time Nishapur was, like the whole of the region around it, a place where there was little political stability as various tribes and religious groups fought with each other. When riots broke out in Nishapur, al-Baghdadi decided that he required a more peaceful place to continue his life as an academic so he moved to Asfirayin. This town was quieter and al-Baghdadi was able to teach and study in more peaceful surroundings. He was certainly considered as one of the great teachers of his time and the people of Nishapur were sad to lose the great scholar from their city.

In Asfirayin, al-Baghdadi taught for many years in the mosque. Always having sufficient wealth, he took no payment for his teachings, devoting his life to the pursuit of learning and teaching for its own sake. His writings were mainly concerned with theology, as we must assume were his teachings. However, he wrote at least two books on mathematics.

One, *Kitab fi'l-misaha,* is
relatively unimportant. It is concerned with the
measurement of lengths, areas and volumes. The
second is, however, a work of major importance
in the history of mathematics. This treatise, *
al-Takmila fi'l-Hisab,* is a work in which
al-Baghdadi considers different systems of
arithmetic. These systems derive from counting
on the fingers, the sexagesimal system, and the
arithmetic of the Indian numerals and fractions.
He also considers the arithmetic of irrational
numbers and business arithmetic. In this work
al-Baghdadi stresses the benefits of each of the
systems but seems to favour the Indian numerals.

Several important results in
number theory appear in the *al-Takmila* as
do comments which allow us to obtain information
on certain texts of al-Khwarizmi which are now
lost. We shall discuss the number theory results
in more detail below, but first let us comment
on the light which the *al-Takmila* sheds
on the problem of why Renaissance mathematicians
were divided into "abacists" and "algorists" and
exactly what is captured by these two names. It
seems clear that those using Indian numerals
used an abacus and were then called "abacists".
The "algorists" followed the methods of al-Khwarizmi's
lost work which, contrary to what was originally
thought, is not a work on Indian numerals but
rather a work on finger counting methods. This
becomes clear from the references to the lost
work by al-Baghdadi.

Let us now consider the number
theory in *al-Takmila.* Al-Baghdadi gives
an interesting discussion of abundant numbers,
deficient numbers, perfect numbers and
equivalent numbers. Suppose that, in modern
notation, *S*(*n*) denotes the sum of
the aliquot parts of *n*, that is the sum
of its proper quotients. First al-Baghdadi
defines perfect numbers (those number *n*
with *S*(*n*) = *n*), abundant
numbers (those number *n* with *S*(*n*)
> *n*), and deficient numbers (those number
*n* with *S*(*n*) < *n*). Of
course these properties of numbers had been
studied by the ancient Greeks. Al-Baghdadi gives
some elementary results and then states that 945
is the smallest odd abundant number, a result
usually attributed to Bachet in the early 17^{th}
century.

Nicomachus had made claims about
perfect numbers in around 100 AD which were
accepted, seemingly without question, in Europe
up to the 16^{th} century. However,
al-Baghdadi knew that certain claims made by
Nicomachus were false. Al-Baghdadi wrote (see
for example [2] or [3]):-

He who affirms that there is only one perfect number in each power of10is wrong; there is no perfect number between ten thousand and one hundred thousand. He who affirms that all perfect numbers end with the figure6or8are right.

Next al-Baghdadi goes on to
define equivalent numbers, and appears to be the
first to study them. Two numbers *m* and *
n* are called equivalent if *S*(*m*)
= *S*(*n*). He then considers the
problem: given *k*, find *m*, *n*
with *S*(*m*) = *S*(*n*) =
*k*. The method he gives is a pretty one.
He then gives the example *k* = 57,
obtaining *S*(159) = 57 and *S*(559) =
57. However, he missed 703, for *S*(703) =
57 as well.

The results that al-Baghdadi
gives on amicable numbers are only a slight
variations on results given earlier by Thabit
ibn Qurra. In modern notation, *m* and *n*
are amicable if *S*(*n*) = *m*,
and *S*(*m*) = *n*. Thabit ibn
Qurra's theorem is as follows: for *n* > 1,
let *p*_{n} = 3.2^{n}
-1 and *q*_{n} = 9.2^{2n-1}
-1. Then if *p*_{n-1}, *p*_{n},
and *q*_{n} are prime, then
*a* = 2^{n}*p*_{n-1}*p*_{n}
and *b* = 2^{n}*q*_{n}
are amicable numbers while *a* is abundant
and *b* is deficient.

**Article
by:** *J J O'Connor*
and *E F Robertson*

**November
1999**