Invention of Algebra by Arab Mathematicians. Alex Gearty, Shane Becker, Lauren Ferris

Transcription

1 Invention of Algebra by Arab Mathematicians Alex Gearty, Shane Becker, Lauren Ferris

2 The Algebra of Squares and Roots

3 The Hindu-Arabic Numeral System - Here we see the evolution of the Brahmi system as it develops into how we visualize numbers today. - The Brahmi system, developed in around the 3rd century AD, is said to be the ancestor of Arabic (Arabe) and Indian (Sanskrit) numeral systems.

4 The Evolution of Galley Division - Galley division was the operation system of doing long division in the time period of the modern Arabic numeral system. - Can you do 3, by galley division?

5 Muḥammad ibn Mūsā al-khwārizmī CE The Condensed Book on the Calculation of al-jabr and al-muqabala More on this in a minute! One of the first scholars in the House of Wisdom founded by the caliph al-wathiq in 847. House of Wisdom: The intellectual center during the Islamic Golden Age. He believed math should be practical not theoretical. What people generally want in calculating is a number. He wanted his texts to be a manual for solving equations. He dealt with the square (of the unknown), the root of the square (the unknown itself), and the absolute number (the constant in the equation).

6 al-khwārizmī s Introduction to the Texts That fondness for science, by which God has distinguished the Imam al-ma mun, the Commander of the Faithful, that affability and condescension which he shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and al-muqabala, confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned.

7 al-jabr The origin of what we call Algebra. Restoring and refers to the operation of transposing a subtracted quantity on one side of an equation to the other side where it becomes an added quantity. This is where the world algebra came from, when words were translated into Latin, al-jabr was not and now we have Algebra. al-muqabala Comparing and refers to the reduction of a positive term by subtracting equal amounts from both sides of the equation.

8 al-jabr Conversion of: 3x+2 = 4-2x To: 5x+2 = 4 al-muqabala Conversion of: 5x+2 = 4 To: 5x = 2

9 al-khwārizmī s 6 Types of Equations: 1. Squares are equal roots (ax^2=bx). 2. Squares are equal to numbers (ax^2=c). 3. Roots are equal to numbers (bx=c). 4. Squares and roots are equal to numbers (ax^2+bx=c). 5. Squares and numbers are equal to roots (ax^2+c=bx). 6. Roots and numbers are equal to squares (bx+c=ax^2).

10 More on al-khwārizmī: al-kwarizmi changed the focus of quadratic equation solving away from the actual finding of sides of squares into that of finding numbers satisfying certain conditions. One more definition: Root not as a side of a square but as anything composed of units which can be multiplied by itself, or any number greater than unity multiplied by itself, or that which is found to be diminished below unity when multiplied by itself.

11 One thing to note... Arabic mathematicians did not deal with negative numbers at all! The types of equations listed previously only have positive solutions. Our typical form of ax^2+bx+c=0 would not make sense to al-kwarizmi because if all the coefficients are positive, the roots cannot be. For the first three equations, the solutions are fairly straightforward. The only thing to note is that 0 is never considered an option.

12 Squares and roots are equal to numbers. Let s look closer at one of these types of equations: al-kwarizmi did not use variables to represent ax^2+bx=c. He used words. What must be the square which, when increased by ten of its own roots, amounts to thirty-nine? This would represent x^2+10x=39. Which we will look closer at in a minute!

13 Al-Khwarizmi s proof of x²+10x=39 How can we justify this problem geometrically, using the formula below?

14 The Work of ibn Turk: While little is known about ibn Turk s background His contributions to algebra were massive. He used al-khwarizmi s types of equations 1, 4, and 5. Squares are equal roots (ax^2=bx). Squares and roots are equal to numbers (ax^2+bx=c). Squares and numbers are equal to roots (ax^2+c=bx). He went into further detail of the geometric proofs of the above equations. And specifically, regarding the 5th type of equation noted that, There is the logical necessity of impossibility in this type of equation when the numerical quantity is greater than [the square of] half the number of the roots. This is now used to explain the middle term in a foiled equation.

15 The Algebra of Polynomials

16 Al-Karaji Worked in Baghdad around year 1000 Wrote many works in mathematics, and studied topics in engineering In the first decade of the 11th century he composed al-fakhri which translates to The Marvelous Aim: The determination of unknowns starting from the knowns Mathematicians before him worked with powers up to 3 (Diophantus), but he was the first to understand that powers extend indefinitely Each power x^n was defined recursively as x times the previous power-- this worked for reciprocals as well al-karaji recognized there exists an infinite sequence of proportions 1 : x = x : x^2 = x^2 : x^3 This discovery allowed him to establish general procedures for +, -, x monomials and polynomials Limits: division only with monomials due to the lack of negative numbers / verbal expression Algorithm for calculating square roots of polynomials only worked sometimes

17 Al-Samaw al Biography ( ) Born in Baghdad to well-educated Jewish parents who encouraged him to study mathematics and medicine Traveled to other parts of the Middle East to study mathematics Wrote his major work in mathematics -- Al-Bahir -- at 19 years old Eventually, his interests transferred to medicine and he became a successful physician and wrote medical texts When he was 40, he converted from Judaism to Islam and wrote an autobiography explaining why Became a famous source of Islamic attack on Jews

18 Work of Al-Samaw al First to introduce negative coefficients If we subtract an additive number from an empty power [0x^n - ax^n], the same subtractive number remains; if we subtract the subtractive number from an empty power [0x^n - (-ax^n)], the same additive number remains. If we subtract an additive number from a subtractive number, the remainder is their subtractive sum; if we subtract a subtractive number from a greater subtractive number, the result is their subtractive difference; if the number from which one subtracts is smaller than the number subtracted, the result is their additive difference.

19 Work of Al-Samaw al This allowed addition and subtraction of polynomials by adding like terms For multiplication however, we need laws of exponents Mathematicians before him has used the law in essence, but never explicitly stated it They expressed the product of a square in a cube as a square-cube Al-Samaw al decided the law could be expressed using a table of columns Columns represent a different power of either a number or an unknown In initial explanation he used numbers under the headers --- not an x variable Used it to explain (x^n)(x^m) = x^(n+m) The distance of the order of the product of the two factors from the order of one of the two factors is equal to the distance of the order of the other factor from the unit. If the factors are in different directions then we count (the distance) from the order of the first factor towards the unit; but, if they are in the same direction, we count away from the unit Polynomial division was done using a similar chart Each column stands for a power of x or (1/x), but the numbers in each column represent the coefficients of the various polynomials involved in the division process

20 Works Cited: A History of Mathematics, An Introduction. Victor J. Katz. Third Edition.