Twitter 
StumbleUpon 
Arithmetization of Algebra
by Meam Wye
Al-Karaji (953 -1029), a Persian Muslim mathematician and engineer, is regarded as the first mathematician who laid the foundation of ‘Arithmetization of Algebra’ i.e. application of elementary arithmetic procedures to algebraic expressions, as is done in modern elementary Algebra, and thus free Algebra from geometric operations. Al-Samawal (12th century), one of al-Karaji's successors, gave the first formal definition of Arithmetization of Algebra as :
... operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.
Al-Karaji wrote several books, some of his well-known works are: ‘Al-Badi' fi'l-hisab’ (Wonderful on calculation), ‘Al-Fakhri fi'l-jabr wa'l-muqabala’ (Glorious on algebra), ‘Al-Kafi fi'l-hisab’ (Sufficient on calculation) and ‘Inbat al-miyah al-khafiya’ (The Extraction of Hidden Waters).
Al-Karaji was first to define the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and he gave rules for products of any two of these. He also provided the fundamental rules of algebraic operations:
(+).(+)=(+) ; (+).(-)=(-) ; (-).(+)=(-) ; (-).(-)=(+)
For polynomials, he formulated various rules:
axm – bxm = (a - b)xm; if a>b
axm – bxm = - (b - a)xm; if a
axm – (- bxm) = (a + b)xm
Al-Karaji provided two methods for the solution of quadratic equations. One geometric and the other purely algebraic; the later involving formation of a complete square followed by extraction of square roots.
The historian of mathematics, F. Woepcke, in ‘Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi’ (Paris
Read in a Reader
January 4, 2010 7:48 AM
As a high school student I certainly wouldn't have used the terms "wonderful" and "glorious" for algebra. With some years, experience, and education behind me, I can more appreciate its beauty.
January 8, 2010 10:04 PM
Like ChrisJ, I would never have considered algebra wonderful in high school, nor would I have in college! :-) Being a wordsmith is what I have always been, not a numbers person. But with the same breath I must also admit to a real fascination with algebra and how it functions on the page and in every day life.
Thanks for an interesting piece.