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History of premodern Algebra
Medieval Algebra was part of arithmetic, together with rule of three, single false position and double false position. Unlike those methods, where calculations are performed on known numbers, in algebra unknowns are named and operated on to set up and solve equations. Conceptually medieval algebra differs greatly from modern one.

al-Khwārazmī
Al-Khwarazmi dedicated his book to caliph Al-Ma'mun. It is the oldest complete book on algebra extant in Arabic. Two other book were written at the same time one by ibn Turk, of which only fragments survive and by Sanad Ibn Ali which is lost .The book is completly rhetorical showing largely oral nature of medieval Islamic civilization.

Like method of double false position, algebra was a method for numerical problem solving, which most likely circulated among merchants and practitioners,as part of "sub-scientific tradition" Al-Khwarazmi took up this heritage and wrote a book on it, introducing more "scientific" elements into it. The book is divided in three parts. First "algebra proper", gives rules of algebra with 39 work-out problems. Second covers rule of three and mensuration. Third, which cover about half of the book solves inheritance problems. The powers of the unknown are given specific names in Arabic. These are jidhr(root) or shay(thing) for first power equivalent to x, mal for second power or x^2, and units are usually counted in dirhams, which is silver coin. The words jidhr and mal are also used in enunciation part, in non-algebraic scence, as sum of money for mal and "square root" for jidhr. The solutions of 39 problems by algebra in the first part constitutes 3 stages:

Stage 1: The unknown is named and operations in the enunciation of the problem are performed to state the equation.

Stage 2: Equation is simplified to one of sex type.

Stage 3: The equation is solved using prescribed procedure.

Al-Khwarazmi qualify six types of equations. There are simple equations:

Type 1: mals equal roots ($$ax^2 = bx$$)

Type 2: mals equal numbers ($$ax^2 = c$$)

Type 3: roots equal numbers ($$bx = c$$)

And composite equations:

Type 4: mals and roots equals numbers( $$ax^2 + bx = c$$)

Type 5: mals and numbers equals roots ($$ax^2 + c = bx$$)

Type 6: mals equals roots and numbers( $$ax^2 = bx + c$$ )

Modern equal sign is used in variety of situations, like for operations like x(x+ 7) = x^2 + 7x and for equations x^2 -2x + 1 = 0.Here we use equality for both stating the result of operation and for equation. This was not the case in medieval algebra as those two equalities were treated differently

As stated earlier operations stated in enunciation of the problem were performed before stating the equation. In one of the problem Al-Khwarazmi states the following operation: "so multiply ten less a thing by itself. So it yields a hundred and a mal less twenty things(or 100 + x^2 -20x in modern notation)". Here the verb adrib(multiply) express the operation being performed, while the verb yakun("yields") announce the result of operation. The outcome verbs vary according to operation. For operation of addition the the good example is Abu Kamil's problem 19, where he just calculated "10 things less a mal" (10x - x^2) and "so you add(zada) to it ten less a thing, so it yields ten dirhams and nine things less a mal(10 + 9x -x^2). Here is the verb zada express the operation addition being performed. The corresponding verbs for subtraction is usually talqi(from laqiya) and the outcome of the operation is baqiya. For division the name of the operation being performed is Qasama("to divide, part") and the outcome verb is kharaja("to emerge"). For example for division Abu Kamil writes: "So you divided(qasamta) the larger part by smaller part. So it resulted in(kharaja) four".

In Arabic algebra equations are distinguished from operations. Operations are expressed with two verbs like: "So you multiply things by itself, so it yields mal($$x\times x = x^2$$) ". By contrast equations are fixed relations expressed in terms of algebraic powers: "55 dirhams and mal equals(tadil, from adala) 27 dirhams and 11 things($$55 + x^2 = 27 + 11x$$) " Equations are always stated using a single verb adala. The verb adala is never used to state the result of operation. This strict distinctions of verbs between operations and equations was maintained throughout entire medieval period.

Equations in modern algebra are linear combinations of powers of $$x$$, that is scalar multiplications with additions and subtractions. That is not the case in medieval algebra, there equation is merely a collection or aggregation od dirhams, things and mals.Modern algebraic expression like $$10x$$ pronounced "ten ex". x is not made plural because 10 is scalar multiple, that is $$10x$$ is a product of 10 and x. This is not the case in medieval algebra, here in "ten things"(10x) "things" is plural. Here one speaks of "ten things" the same as "ten apples" or "ten oranges". Here "ten" is not a number multiplied by "thing" but rather it tells us how many things we have. "Ten things" and "ten mals" is merely a collection of ten objects. All numbers in medieval algebra are cordinal numbers. One can have nine dirhams, nine things or nine mals, but "nine" cannot stand on it's own. It must be nine of something.

The distinction between operations and equations is seen in verbs used for addition. For example Abu Kamil begins his solution with "It's rule is that you make your mal a thing. So you add to it seven dirhams, so it yields a thing and seven dirhams". Here "Add"(Zada), is a operation, while "and"(wa) is a conjunction. In other words thing and seven dirhams($$x + 7$$) is a result of operation. It is a collection of 8 objects of two kinds, one is things the other is dirham. Similarly in another problem Abu-Kamil writes:"So cast away two things from twenty dirhams($$20 - x$$). So there remains twenty dirhams less two things". Here cast away(laqiya) is the operation, while less(illa) is negative conterpart to "and". Medieval mathematicians thought of expressions like "six mals less two things"($$6x^2 -2x$$) as incomplete or diminished six mals, like an "apple less two bites".

Polynomials in medieval algebra are not linear combination of $$x$$, but rather a mere collection of various algebraic powers with no operations present. For example a "polynomial' "A hundred, three things and two mals"($$100 + 3x + x^2$$) is a collection of 104 objects of three different kinds with no operation between them. All operations in enunciation were performed before setting up an equation.