Al-Al-KhwarizmiKhwarizmiThe The founderfounder of of AlgebraAlgebra

Brief Brief BiographyBiography Al-Khwarizmi was born in Al-Khwarizmi was born in

Baghdad about 780 a.C Baghdad about 780 a.C and died in 850. He and died in 850. He studied at the House of studied at the House of Wisdom, which it was Wisdom, which it was founded by the Caliph Al-founded by the Caliph Al-Mamun, where Greek Mamun, where Greek philosophical and philosophical and scientific works were scientific works were translated. His tasks translated. His tasks there involved the there involved the translation of Greek translation of Greek scientific manuscripts and scientific manuscripts and he also studied, and he also studied, and wrote on, algebra, wrote on, algebra, geometry and astronomy. geometry and astronomy.

Al-Al-KhwarizmiKhwarizmi

Certainly al-Certainly al-Khwarizmi worked Khwarizmi worked under the under the patronage of Al-patronage of Al-Mamun and he Mamun and he dedicated two of dedicated two of his texts to the his texts to the Caliph: the treatise Caliph: the treatise about Algebra and about Algebra and the one about the one about Astronomy. Astronomy.

The treatise about The treatise about Algebra “Algebra “Hisab al-Hisab al-jabr w'al-muqabala”jabr w'al-muqabala” was the most was the most famous and famous and important of all of important of all of Al-Khwarizmi's Al-Khwarizmi's works. It is the title works. It is the title of this text that of this text that gives us the word gives us the word "algebra" and it is "algebra" and it is the first book to be the first book to be written about written about Algebra. Algebra.

The Hisab al-jabr w’almuqabala

The purpose of the book tells us that Al-The purpose of the book tells us that Al-Khwarizmi intended to teach. Indeed only the Khwarizmi intended to teach. Indeed only the first part of the book is a discussion of what first part of the book is a discussion of what we would today recognise as Algebra. we would today recognise as Algebra. However it is important to realise that the However it is important to realise that the book was intended to be highly practical and book was intended to be highly practical and that Algebra was introduced to solve real life that Algebra was introduced to solve real life problems that were part of everyday life in problems that were part of everyday life in the Islam empire at that time. Early in the the Islam empire at that time. Early in the book Al-Khwarizmi describes the natural book Al-Khwarizmi describes the natural numbers and is important to understand the numbers and is important to understand the new depth of abstraction and understanding new depth of abstraction and understanding here.here.

Having introduced the natural Having introduced the natural

numbers…numbers…

Al-Khwarizmi introduces the Al-Khwarizmi introduces the main topic of this first section main topic of this first section of his book, namely the of his book, namely the solution of equations. His solution of equations. His equations are linear or equations are linear or quadratic and are composed of quadratic and are composed of units, roots and squares.units, roots and squares.

For example, to Al-For example, to Al-Khwarizmi a unit was a Khwarizmi a unit was a

number, a root was number, a root was xx, ,

and a square was and a square was x x 22. . However, although we However, although we shall use the now familiar shall use the now familiar algebraic notation in this algebraic notation in this presentation to help the presentation to help the reader understand the reader understand the notions, Al-Khwarizmi's notions, Al-Khwarizmi's mathematics is done mathematics is done entirely in words with no entirely in words with no symbols being used. symbols being used.

The six forms …The six forms … He first reduces an equation (linear or He first reduces an equation (linear or

quadratic) to one of six standard forms: quadratic) to one of six standard forms: 1. Squares equal to roots.1. Squares equal to roots.

2. Squares equal to numbers.2. Squares equal to numbers.3. Roots equal to numbers.3. Roots equal to numbers.4. Squares and roots equal to numbers; 4. Squares and roots equal to numbers;

e.g.e.g. x x22 + 10 + 10 xx = 39. = 39.5. Squares and numbers equal to roots; 5. Squares and numbers equal to roots;

e.g.e.g. x x22 + 21 = 10 + 21 = 10 xx..6. Roots and numbers equal to squares; 6. Roots and numbers equal to squares;

e.g. 3 e.g. 3 xx + 4 = + 4 = xx22. .

… … and the solution!and the solution!

Al-Khwarizmi then shows how to solve Al-Khwarizmi then shows how to solve the six standard types of equations. the six standard types of equations. He uses both algebraic methods of He uses both algebraic methods of solution and geometric methods. For solution and geometric methods. For example to solve the equationexample to solve the equation

X X 22 + 10 x = + 10 x = 3939

He writes :He writes : ““... a square and 10 roots are equal to ... a square and 10 roots are equal to 39 units. The question therefore in this 39 units. The question therefore in this type of equation is about as follows: type of equation is about as follows: what is the square which combined what is the square which combined with ten of its roots will give a sum with ten of its roots will give a sum total of 39? The manner of solving this total of 39? The manner of solving this type of equation is to take one-half of type of equation is to take one-half of the roots just mentioned. Now the the roots just mentioned. Now the roots in the problem before us are 10. roots in the problem before us are 10. Therefore take 5, which multiplied by Therefore take 5, which multiplied by itself gives 25, an amount which you itself gives 25, an amount which you add to 39 giving 64. Having taken add to 39 giving 64. Having taken then the square root of this which is 8, then the square root of this which is 8, subtract from it half the roots, 5 subtract from it half the roots, 5 leaving 3. The number three therefore leaving 3. The number three therefore represents one root of this square, represents one root of this square, which itself, of course is 9. Nine which itself, of course is 9. Nine therefore gives the square. “therefore gives the square. “

In modern notation, one of Al-Khwarizmi's example equations is x2 + 10x = 39.

Al-Khwarizmi's solution is then:

(x+5)2 = 39 + 25 = 64

x + 5 = sqrt{64} = 8

x = 8 - 5 = 3

x2 = 9

Al-Khwarizmi demonstrates this solution with Al-Khwarizmi demonstrates this solution with a square AB, the side of which is the desired a square AB, the side of which is the desired root x. On each of the four sides, he root x. On each of the four sides, he constructs rectangles, each having 2.5 as constructs rectangles, each having 2.5 as their width. So, the square together with the their width. So, the square together with the four rectangles is equal to 39. To complete four rectangles is equal to 39. To complete the square EH, Al-Khwarizmi adds four times the square EH, Al-Khwarizmi adds four times the square of 2.5, or 25. So the area of the the square of 2.5, or 25. So the area of the large square EH is 64, and its side is 8. large square EH is 64, and its side is 8.

Thus, the side x of the original square AB is 8 Thus, the side x of the original square AB is 8 - 5 = 3 - 5 = 3

Al-Khwarizmi also presents a simpler, similar method which constructs rectangles of breadth 5 on two sides of the square AB. Then, the total area of the square EH is x2 + 10x + 25 = 39 + 25 = 64, which yields the same result x = 3 or x2 = 9

Thanks for listening! Agnese & Erica