Al-Khwarizmi

Just outside Khiva's Ata Darwaza (Old Gate), bathed in the afternoon light, is the statue of a bearded man pensive over a scroll. I ask the young Uzbek lady who trips up to me with her kid sister in order to get a photo taken if they know who it is. They can manage 'Aksakal' (greybeard) but are not sure of much else.

Poor Al is having a not-so-good couple-of-decades. From being co-Hero of the Uzbek SSR (along with Ali-Sher Navoiy), his spot as torch-bearer has been usurped by Tamerlane (and Bobur Mirza in Ferghana.) Every third person in this country seems to be named Ulugh Beg. Meanwhile, his declining star in Uzbkeistan, combined with his birth having had been in a Persian-speaking family of Khiva, has emboldened Iran to start an international Khwarizmi Award “in memory of Abu Jafar Mohammad Ibn Mousa Khwarizmi, the great Iranian Mathematician and Astronomer (770-840 C.E)”; his association with al-Mamun's Baghdad lets the Britannica Childrens' Encyclopedia call al-Khwarizmi an Arab Mathematician in its one-line entry; while The Dictionary Of Scientific Biography claims that his title al-Majūsī indicates that his ancestry belongs the old 'Magian' (Zoroastrian) religion after all.

Sometime after 800 CE, al-Khwarezmi seems to have traveled to India, coming in contact with the mathematical work of Brahmagupta (598-668 CE.) Interestingly, Brahmagupta hailed from the peripheries of the classical Indic civilization – he was born and raised in the kingdom of the Bhils (Bhillamalla, today Bhinmal in the hills near Jalore in Southern Rajasthan) – though he moved to Ujjain later in his life. The radical concepts outlined by Brahmagupta in his Brahmasphutasiddhanta are fortune (positive) quantities, debt (negative) quantities, and zero:

The sum of two fortune quantities is fortune
The sum of two debt quantities is debt
The sum of zero and a debt quantity is debt
The sum of zero and a fortune quantity is fortune
The sum of zero and zero is zero.
The sum of a fortune and a debt is their difference; or, if they are equal, zero
In subtraction, the less is to be taken from the greater, fortune from fortune
In subtraction, the less is to be taken from the greater, debt from debt
When the greater however, is subtracted from the less, the difference is reversed
When fortune is to be subtracted from debt, and debt from fortune, they must be added together
The product of a debt quantity and a fortune quantity is debt
The product of a debt quantity and a debt quantity is fortune
The product of two fortune, is fortune.
fortune divided by fortune or debt by debt is fortune
fortune divided by debt is debt. debt divided by fortune is debt
A fortune or debt number when divided by zero is a fraction with the zero as denominator
Zero divided by a debt or fortune number is zero
Zero divided by zero is zero.

In the Arab world, numbers had been represented according to a system known as huruf al jumal (letters for calculating) or abjad  (abbreviating the first four numbers 1 = a, 2= b, 3 = ja, 4 = d). The numbers from 1 to 9 were represented by the first 9 letters; then 10 to 90 by the next nine letters (viz. 10 = y, 20 = k, 30 = l, 40 = m, ...), subsequently 100 to 900 by the next letters (100 = q, 200 = r, 300 = sh, 400 = t, ...). The 28^th Arabic letter was used to represent 1000.

Al-Khwarizmi would go on to extend the ideas of Brahmagupta (though he could not stomach Brahmagupta's negative numbers, nor negative roots) becoming one of the first mathematicians to not only use the concept of zero as a number, but also use zero as a symbol in positional base notation. This idea was met with great skepticism. Most could not at first accept that attaching this “worthless nothing” to another number would somehow increase the number's value tenfold; but Ibn-Sina mentions being taught the Indian system of counting when he was around 10 (c. 997 CE) by visitors to his father's house, so it would seem the Arab world was slowly converting to the al-Khwarizmi's Indian notation over 200 years.

Leonardo of Pisa (c. 1170-1250 CE), also known as Leonardo Bonacci or Leonardo Fibonacci, was the son of a bureaucrat. As the director of a Pisan trading colony in Algeria, Bonacci Sr. understood the quality of Arab schooling. As a result, Leonardo was sent first to study, and then to work, in various Arabic cities. From this immersion, Fibonacci deemed the Hindu-Arabic number system to be the superior vehicle for calculation. Around 1225 CE, Fibonacci produced Liber Abbaci (Book of the Abacus), Pracitca Geometriae (Practice of Geometry) and Liber Quadratorum (Book of Square Numbers), introducing al-Khwarizmi's notation to Europe, where it again struggled for centuries before gaining acceptance. (The Fibonacci numbers named after him had been known in India for 500 years.)

Al-Khwarezmi also contributed to the abstract systemization of linear and quadratic equations in his Kitāb al-mukhtaṣar fī hīsāb al-jabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing.)

The al-jabr (restoration, completion) and al-muqābala (balancing) aspects can be illustrated thus:

Take an equation to be solved: x² + 10x = 39
Complete (the square) by adding 25 to 39, and balance it by adding 25 to the other side as well, i.e. x² + 10x + 25 = 39 + 25
Thus (x + 5)² = 64, i.e. x + 5 = 8
So x = 3.
(Note the absence of the second possibility, x + 5 = -8 or x = -13.)

Needless to say, from the word al-jabr we get algebra; and from al-Khwarizmi we get algorithm.

Here is the Algorithm Sunset/Moonrise (executive summary: minute 4:30.)