From Al-Jabr to Algebra Author(s): Philip Maher Source: Mathematics in School, Vol. 27, No. 4, History of Mathematics (Sep., 1998), pp. 14-15 Published by: The Mathematical Association Stable URL: http://www.jstor.org/stable/30211868 . Accessed: 07/04/2014 11:22 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
[email protected]. . The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access to Mathematics in School. http://www.jstor.org This content downloaded from 173.69.19.125 on Mon, 7 Apr 2014 11:22:59 AM All use subject to JSTOR Terms and Conditions AL-JABR from ALGEBRA to by Philip Maher When you now teach (and when you first studied) do your students now (and did you then, as a student) understand what's going on when you solve a quadratic equation, say x2 +10x - 39 = 0 (1) by completing the square, thus: x2 + 10x- 39 = (x + 5)2- 25 -39 (2) so that in (1) we have (x + 5)2 = 64, whence x = 3 or -13. Of course, your students, providing they are competent in algebra, can check that (x + 5)2 - 25 equals x2 + 10x; and, more generally, that (x + b)2 - b2 equals to x2 + 2bx. But, as well as giving no idea as to why the terminology 'completing the square' is used, this verification seems curiously uncon- vincing because it is, so to speak, after the event: it doesn't give an understanding of what is actually going on; there is, for instance, no geometric understanding. For such a geo- metric understanding you have to gaze a long way back; back, in fact, to the work of Al-Khwarizmi. Abu Jafar Muhammad ibn Musa Al-Khwarizmi was born around AD 780 in Khwarizm (Persia) and died around AD 850. Al-Khwarizmi was famed as a scientist and mathe- matician in what we now call Persia: in about AD 820 he was appointed astronomer and head of the library of the House of Wisdom in Baghdad. Motivated by astronomy he produced the first textbook on arithmetic, his Algorithmo de Numero Indorum (the Latin translation of his Arabic original of which no copies survive): in it he codified the various Indian number systems (e.g. the Brahmin (AD 100) and Bakhsali (AD 200-400), the four operations on integers and fractions and the extraction of square roots. His wonderful demonstration of "completing the square" (of which more soon) is contained in his other, possibly still greater, achievement, the first textbook on what we now call algebra, his Hisab Al-Jabr w'alMuqabala (variously translated as 'Calculation by Restoration and Reduction' and 'Calcula- tion by Completion and Cancellation'). The actual term algebra comes from the operation Al-Khwarizmi called 'Jabr'. This refers to the step in which an equation (using modem notation) such as x - 4 = 10 becomes x = 14 where the left hand side of the first equation, in which x is diminished by 4, is 'restored' or 'completed' back to x. 'Muqabala', which just means cancelling (whereby an equation x2 + 4 x2 + x becomes x = 4) led, on the other hand, to no familiar modem terminology. 14 Al-Khwarizmi's principal achievement in algebra was his discovery of how to solve quadratic equations by completing the square. Al-Khwarizmi divided the study of quadratic equations into six types but, for simplicity, we consider just one example, actually the very example he considered and the equation this article started with, namely x2 + 10x - 39 = 0 (in modem notation). Using the operation of Jabr this equation is the same as x2 + 10x = 39. (3) In discussing this equation Al-Khwarizmi calls the coefficient of x, here 10, the 'root' (quite unlike modem terminology). He writes 'The manner of solving this type of equation is to take one half of the root. Now the root in this problem is 10. Therefore, take 5 which multiplied by itself gives 25, an amount which you add to 39, giving 64, having taken then the square root of this which is 8, subtract from it the half of the root 5, leaving 3.' Al-Khwarizmi gives two, not one, vivid convincing, geomet- ric justifications of the above algorithms. Here they are. In the first, you start with a square of side x; adjoin to it two identical rectangles, each of length 5 and width x so as to form an L shape whose base and height are each x + 5, and then superimpose on it a big square of side x + 5. The area of the L shape is x2 + 5x + 5x, that is, x2 + 10x which, by our original equation (3), must equal 39; the area of the big square, viz. (x + 5)2, is 25 more units than the area of the L shape and so must equal 39 + 25 = 64. Thus, (x + 5)2 = 64 and so, taking positive square roots, x + 5 = 8, that is, x = 3. ---x-, - 5 5 5 5 Are25 25 sx+5 L 5 No t t Area x x 39 S --*X- 5 x+5 In Al-Kharizmi's other geometrical justification, the small square of side x is centred in the middle of a big square, a gain of side x + 5. Here, the area of the four corner squares is 4 x 2 x 2 ' =25 and the area of the cross shape is Mathematics in School, September 1998 This content downloaded from 173.69.19.125 on Mon, 7 Apr 2014 11:22:59 AM All use subject to JSTOR Terms and Conditions x2 + 4(2 1 x) = x 2 + 10x, so again the area of the big square is 64, whence x = 3. xx 222 21 2 a t t 2 2 ex S X+5 Area x+5 Area 39 25 Observe that Al-Khwarizmi's geometrical demonstration al- lows him to obtain only positive solutions and that he does not use modern algebraic notation (Inevitably so: negative numbers and modern algebraic notation didn't come into mathematics until centuries later.) I submit that teaching 'completing the square' geometri- cally and presenting something of Al-Khwarizmi's achieve- ment is valuable for a variety of reasons. First, it is obviously valuable to teach mathematics realis- tically, that is, as a creation/discovery of the human mind rather than to present mathematics as though it were the product of some ahistorical void. Apart from being educa- tionally unmotivated, this approach is intrinsically incoher- ent: mathematics must have been produced by someone. What's more, this view of mathematics-as a creation/ discovery of the human mind-is liberating for your pupils: it allows them to realise that mathematics has the potential of being created by them. It is, moreover, crucial to recognise that the mathematical achievements of the Middle East are insufficiently recognised in the West: partly, because of obvious linguistic reasons; partly, and paradoxically, because the very magnitude of those achievements occurred precisely when mathematics in the West was, comparatively, moribund (9th-12th centu- ries). (The intellectual trajectories of mathematics within the West and the East are diametrically different, a point whose consequences are explored in Maher and in Towndrow (1997)). To be aware of this Middle Eastern mathematical tradition, of which Al-Khwarizmi's beautiful geometric proofs afford access, is clearly salutary-and especially healthy for those of your students of Middle Eastern origin. However, whatever your background and whoever your students are, I am sure you will agree that Al-Khwarizmi's geometrical proofs of completing the square are beautiful. M References Maher, P.J. The Past is Another Country. Submitted. Towndrow, M. 1997 Pre-Renaissance Middle, and Far, Eastern Mathematics and the Public Understanding of Science, Faculty of Technology Technical Report, Middlesex University, TR1.97, pp.97-102. Author Philip Maher, Middlesex University, Queensway, Enfield, Middlesex EN3 4SF. e-mail: philip
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