When I considered what people generally want in calculating, I found that it always is a number -Calculation by Restoration and Reduction ABU JAFAR MUHAMMAD IBN MUS AL-KHWARIZMI The Inventor of Algebra al-Khwarizmi Abu Jafar Muhammad ibn Mus alKhwarizmi amassed mathematical knowledge from diverse cultures during his studies at the House of Wisdom to develop the techniques and practicality of Algebra, which in turn revolutionized

mathematical thinking throughout the world. Historical Background al-Khwarizmi relied heavily on mathematical texts from other cultures and regions The pre-existing network of knowledge within the Islamic Empire Islamic culture united distant regions which created a melting pot of new ideas

Islamic Empire 622 AD Prophet Muhammad ignited the spread of Islam religion He attracted pilgrim followers as he fled from Mecca Emphasized of toleration and equality The passion and egalitarian character of early Islam fired the

imagination and won the devotion of many who came across it for the first time Umayyad Dynasty Controlled empire between 661 and 750 Encouraged a degree of religious tolerance Non-Muslims were discriminated and lacked social mobility Expanded the Islamic empire into central Asia and India, Spain and Northern African Abbasid Dynasty

Took over in AD 750 Strongly encouraged religious tolerance and cultural diversity The Abbasid empire restored ancient ties among historic centers of civilization across a huge landmass United regions of Islamic empire Baghdad

Established in 752 New capital of under Abbasid Dynasty Located along Tigris River Access to trade routes Vibrant multi-cultural center

Demand for intellectual and scientific exchange Knowledge of papermaking from China in 751 House of Wisdom Founded between 813833 in Baghdad Made up of translation bureau, library, academy of scholars, and intellects from across the empire Safeguard invaluable manuscripts within the empire

Translated foreign classics from Indian, Persian, and Greek scholars in Arabic Arabic became universal language for scientific inquiry Admiral, Alcohol, Almanac, Chemistry, Cotton, Mattress, Syrup al-Khwarizmis Life Lived between 780-850 AD Family possibly originates from Khwarizm east of the

Caspian sea in modern date Uzbekistan Evidence that he was from Zoroastrian descent Possibility that he converted to Islam to improve social status Appointed as a member of House of Wisdom in 820 Practical Mathematics Zij al-Sindhind Earliest extent knowledge of Islamic Astronomy

Set of planetary table Used mathematics to calculate horoscopes Islamic faith needed precise astronomical calendars Geography/Cartography Mapped city locations of Asia and Africa Improved previous approximations of locations based on latitude and longitude The direction of Mecca for the use of prayer Hisab al-jabr wal-muqabala

Calculation by Restoration and Reduction 813-833 CE Dedicated to Caliph al-Mamun the first systematic treatment of the general subject of algebra as distinct from the theory of numbers No symbols or written equations throughout his work

Mathematical processes are expressed using words 3 sections; Theoretical, Mensuration, Legacies Theoretical Explained the theoretical methods of algebra It brought together common geometrical methods with a new method of algebra and arithmetic.

Made solving problems of inheritance much simpler and easier It demonstrated the multiplication of unknown numbers. Six different types of equations He reduced all equations to six main types

Roots equal squares: bx = ax2 Roots equal numbers: bx = c Squares equal numbers: ax2 = c Squares and roots equal numbers: ax2 + bx = c Roots and numbers equal squares: bx + c = ax2 Squares and numbers equal roots: ax2 + c = bx a, b, c all positive integers Examples

Al-jabr 2x + 5 = 8 - 3x 5x + 5 = 8 Al-muqabala 5x + 5 = 8 5x + 5 5 = 8 5 5x = 3 (Roots equal numbers) Al-jabr (9/4)x +1/8 = 3 + (5/8)x 18x + 1 = 24 + 15x Al-muqabala 18x + 1 = 24 + 15x 18x - 15x + 1 1 = 24 1 + 15x 15x 3x = 23 Al-Khwarizmis Solution X2 + 10X = 39 One square, and ten roots of the same, amount to thirty-nine dirhams; that is to say, what must be the square which, when increased by ten of its own roots, amounts to thirty-nine? The solution is this: you halve the number of the roots, which in the

present instance yields five. This you multiply by itself; the product is twenty-five. Add this to thirtynine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from half the number of the roots, which is five; the remained is three. This is the root of the square which you sought for; the square itself is nine. Modern Algebraic solution X2 + 10X = 39 1. 2. 3. 4. 5. 10/2 5 52 25 (X+5)2 = 39 + 25 = 64

X+5=8 X=3 Modern Geometric Solution Mensuration Al-Khwarizmi gave equation for finding the area and circumference of a circle. He also gave a fairly accurate estimate of He also gives equations for computing volumes of cones, pyramids, and truncated pyramids.

Pythagorean theorem: The theorem states that for a right triangle, the square of the hypotenuse c is equal to the sum of the squares of the two short sides a and b. Legacies and Inheritances In Islamic law it is straightforward on how an estate is distributed among family members. It becomes complicated;

When a certain amount is bestowed to a stranger. When the deceased has debt. When the deceased has bestowed some land to a mother, spouse, sisters, brothers, and a stranger. Example of Legacies Problem A man dies, leaving his mother, his wife, and two brothers and two sisters by the same father and mother with himself; and he bequeaths to a stranger oneninth of his capital. Treatise on Hindu Reckoning Arithmetic

Also called Algorithmi de numero indorum (Calculation with Indian Numerals) Written around 825 CE after Algebra. No extant text left, only Latin translations. Why did he write this?

Concerning the numbering of the Indians by means of IX symbols in their universal system of numbering, for the sake of its ease and brevity, so that this work, to be sure, might as the smallest, and whatever there is in its as result of multiplication and division, also addition and subtraction, etc. He wanted to make practical mathematics as easy as possible, and Hindu numerals offered a much easier way to solve problems in comparison with previous methods. Hindu Numerals cont..

Introduced Hindu numerals 1,2,3,4,5,6,7,8,9 and 0 Introduced the positional value system But not decimal fraction He showed how easy it was to write large numbers with zero instead of the previous method. Showed how to add, subtract, multiply, and divide with these new numerals and fractions. Controversy

Scholars question the originality of Al-Khwarizmis work Argue that Al-Khwarizmi merely complied and organized the principles of algebra from the mathematic work of other cultures Exposure to Influences at House of wisdom Greek Geometrical Influences

He had access to Greek texts in the House of Wisdom Scholars argue that first 10 proofs in Book II of the elements were proved independent of each other This emphasizes the the power of the method of geometrical algebra However, lack of evidence that algebra was directly influenced by Greek geometry

Al-Khwarizmis technique for a geometrical solution prove that Al-Khwarizmi remained outside the sphere of Greek influence Mesopotamian Influences Exposed Mesopotamian influences at the House of Wisdom No concrete evidence to connect Indian or Babylonian texts with al-Khwarizmi

Mathematicians ruled out the direct influence from Indian algebra based on distinct differences in quadratic bases Scholars argue a connection between Babylonian equations that used geometry and al-Khwarizmis second degree equations Like the cuneiform tablets we looked at in class Only hypothetical speculation of a link between Babylonian mathematics and alKhwarizmi Arabic Culture

Influenced by texts of Greek, Indian, local Arabic mathematics Development of Arabic sciences lead to demand for comprehensive translations from ancient texts Arab mathematicians and scientists were not directly searching for ideas tosteal from ancient texts A discovery within mathematics would return Arab scholar back to ancient Greek texts, which were then translated, edited and improved upon Different perspective about originality

Controversy Conclusion Scholars agree: It was the first systematic treatment of the general subject of algebra as distinct from the theory of numbers Al-Khwarizmi was definitely influenced by previous cultures mathematic texts He independently possessed the creativity and genesis to fuse diverse

aspects from multiple cultures of mathematics to establish the foundation of algebra Foundation for future developments in mathematics Al- Khwarizmis Influences Al- Khwarizmis work revolutionized future mathematical thinking Algebra, Hindu numerals, and astronomical theories People were able to use Al-Khwarizmis

past works and combine them with different cultures to develop new ideas Europe Kitab al-Jabr wal Muqabalah It was from this work that Europe later learned the branch of mathematics known as algebra Caliph al-Mamun Encouraged me to compose a work on algebra confining it to the fine and important parts of its calculation for

practical purposes Hindu-Arabic Numerals De Numero Indorum Al-Khwarizmis Hindu-Arabic numerals became digits of the western world Europe adopted the HinduArabic numerals as they were

used in Spain Western Europe adopted the numerals including the zero sign from Muslim Spain and Northern Africa Landed in Italy New notation came to be known as that of Al- Khwarizmi Origins of the Names Treatise on Hindi Numerals translated in Latin to Algoritmi de Numero Indorum (Al- Khwarizmi

Concerning the Indian Art of Reckoning) Al- Khwarizmi Alchoarismi Algorismi algorismus Algorisme Algoritmi Algorism Algorithm Algorithm A procedure for solving a mathematical problem in a finite number of steps that often involves repetition of an operation Influence of the Zij

Ibn Muadh al-Jayyani The leading Andalusian astronomer Tabulae Jahenen Set of astronomical tables directly based on Al-Khwarizmis Zij al-Sindhind Able to take Khwarizmis work and further developed mathematical thinking for practical purposes Adapted Zij for the longitude of Jaen, his hometown, and he took into account the equinoxes Hisab al-Khataayn

Rule of Double false position Popular method when solutions to linear equations were difficult Rule brought to Europe by the Arabs and is found in the earlier works of Al-Khwarizmi Arabic Influences

Abu Kamil ibn Aslam 850-930 AD Quotes directly from Al-Khwarizmis works Copies many of his problems and examples Hebrew Mishnat ha-Middol 1092-1145 AD Abraham bar Hiyya and Abraham ibn Ezra Oldest mathematical text in Hebrew Gives practical rules for mensuration Section on the solution of quadratic equations with geometric analogues This section was strongly influenced by al-Khwarizmis

algebra Descartes Geometry was inspired by what Descartes called the true mathematics He proved geometric basis for algebraic operations

This technique had already been done by Al-Khwarizmi Descartes work produced great progress in algebra and other mathematics Work gave rise to a new branch of mathematics called analytical geometry Fibonacci Considered one of the greatest mathematicians of all time

Wrote The Book on Calculation, the first work in Christian Europe on algebra and geometry The fifteenth and final chapter of Fibonaccis influential book, Liber Abbaci, depended heavily on ideas of al-Khwarizmi Spread the numerals of Khwarizmi through Europe with his statement, The nine Indian numerals are 9,8,7,6,5,4,3,2,1. With these nine and 0, any desired number can be written. Questions