Misc Notes on the History of Algebra

Misc Notes on the History of Algebra… rmiller@greenhillsschool.org Ruth Miller Taken from Unknown Quantity; a Real and Imaginary History of Algebra by John Derbyshire ©2006 by John Derbyshire; Joseph Henry Press, Washington, DC 1) Babylonian Tablets: There were table tablets and problem tablets Table tablets are multiplication and division tablets and there are some formulas a ( a + b) ab = 2) 3) 4) 5) 6) − ( a − b) to help with computation, as the numbers were difficult to use Problem tablets contained problems written in narrative, and included problems with b solutions that essentially used the quadratic formula, though not with negative solutions Ahmes Papyrus Not as algebraic as the Babylonians , but did contain problems like #24; x + x = 15 a Ancient Greeks- tried to solve similar problems Geometrically 200 AD Diophantus; the “Father of Algebra” Begins to use some primitive symbols to represent squares and cubes and to use letters a to represent general numbers- though he is often only concerned with finding one set of numbers that will satisfy a given set of constraints rather than more than one solution i Wrote a book called Arithmetica containing lots of equations, since called Diophantine Equations that fascinated generations, including Pierre de Fermat He had a copy of the book which he wrote in regarding proofs of the assertionsand gradually all of Fermat’s generalizations and proofs were validated, except for one, which became Fermat’s Last Theorem Begins to work with higher degree polynomials, equations with more than one b unknown, and systems of equations i Recall that the Chinese were using matrices to solve systems independently at about the same time Medieval Islamic Scholars, according to Derbyshire (p 56) “give is the word ‘Algebra.’ They begin to focus on equations as worthwhile objects of inquiry in themselves and classified linear, quadratic, and cubic equations according to how difficult it was to solve them with the techniques available.” 800 AD Al-Khwarizmi- contributes the idea of an equation as an object, though he does a not yet use symbols 1000 AD Khayam- still not a lot of symbolic manipulation, but he separates cubics into b categories and finds some general solutions by using geometry 1200 AD Fibonacci puts everything together- Muslim work, Indian, though probably not Chinese, into the best book on math written in 300 years, the Liber abbaci The book foreshadows an interest in the nature of a curve rather than in its solution a The book focuses on analysis- and foreshadows work that will not be brought to fruition b for another 600 years Misc Notes on the History of Algebra… rmiller@greenhillsschool.org Ruth Miller Still no real use of symbolic manipulation, though- the work that started with c Diophantus in this area has been lost Fibonacci popularizes the Indian number system 7) 1494d.Pacioli re-introduces symbols, though they are not as efficient as those of Diophantus 1200 years ago- still lost in an unknown monastery? 8) 1501-1576 Girolamo Cardano- quite a character Wrote a book called Consolation which is widely believed to be the inspiration for Hamlet’s “To be or not to be?” speech, Cardano was an astrologer, gambler (he did a lot of work with probability and combinatorics as well), physician, and a rogue He wrote a book called Arts Magna or The Great Art- also known as the First Book of the Rules of Algebra in which he set down the techniques for solving cubics and quadraticsincluding the notion that all cubics had three solutions and that some could be negative, irrational or even imaginary Unfortunately, there is a lot of evidence that he plagiarized the methods From Chapter 37 of Ars Magna where he wants to find numbers whose product is 40: “Putting aside the mental tortures involved, multiply making 25 − (−15) which [latter] is sophisticated… +15 (5+ −15 ) by (5− −15 ), Hence this product is 40 … This is truly 9) 1560 Bombelli rediscovers Diophantus Arithmetica is published in Latin in 1621, and Fermat owns a copy of this edition 10) 1630 Descartes Algebracizes Geometry and provides a huge bridge between the work of Viete and Cardano Viete had done some great stuff on the solutions to polynomials, but was a Pythagorean “holdover” and looked at everything geometrically Descartes begins to merge Algebra and Geometry- the work done by the Muslims in Algebra and the leftover Geometric work found in Europe  Remember that the “Quadrivium” was the idea of Pythagoras that mathematics was made up of four subjects: Arithmetic, Geometry, Music and Astronomy Pythagoras was so influential, that this configuration became the mathematics curriculum throughout antiquity, the Dark Ages (for whatever mathematics was taught in Europe), and the Middle Ages, on into the Renaissance The Muslims guarded the Geometry, but developed Algebra- Descartes is the guy that finally connected the two Taken from History of Math homework prepared by Gail Kaplan, PhD, for the Towson University History of Math Class Misc Notes on the History of Algebra… rmiller@greenhillsschool.org Ruth Miller Jiuzhang suanshu: Nine Chapters on the Mathematical Art a Chapter Problem 6: There is a square pond with side 10 feet, with a reed growing in the center whose top is foot out of the water If the reed is pulled to the shore, the top just reaches the shore Find the depth of the water and the length of the reed Generalization: There is a square pond with side 2y feet, with a reed growing in the center whose top is a feet out of the water If the reed is pulled to the shore, the top just reaches the shore Find the depth of the water and the length of the reed "Multiply half of the side of the pond by itself; decrease this by the product of the length of the reed above the water with itself; divide the difference by twice the length of the reed above the water This gives the depth Add this to the length of the reed above the water This gives the length of the reed b Chapter Problem 31: There are three classes of grain, of which three bundles of the first class, two of the second, and one of the third make 39 measures Two of the first, three of the second, and one of the third make 34 measures And one of the first, two of the second, and three of the third make 26 measures How many measures of grain are contained in one bundle of each class? Arrange the 3, 2, and bundles of the three classes and the 39 measures of their grain at the right Arrange other conditions at the middle and at the left With the first class on the right column multiply currently the middle column and directly leave out [Perform the same operation with respect to the left column.] Then with what remains of the second class in the middle column, directly leave out c Use the Chinese square root algorithm (Katz, p 119) to find the root of 131,044 Misc Notes on the History of Algebra… rmiller@greenhillsschool.org Indian Mathematics Ruth Miller Sulbasutras “If you wish to turn a circle into a square, divide the diameter into eight parts, and again one of these eight parts into twenty nine parts, of these twenty nine parts, remove twenty 8, and moreover, the sixth part (of the one part left) less the eighth part (of the sixth part)” Misc Notes on the History of Algebra… rmiller@greenhillsschool.org Ruth Miller Discovering Systems of Equations Babylonian Style: The Babylonians often solved systems of equations with the method of false position The idea is similar that of the Egyptians: pick an answer, check, adjust, and solve Let’s apply this idea one step at a time In this problem, a sila measures volume and a sar measures area There are two fields The first one yields sila per sar; the second yields sila per sar The yield of the first field is 500 sila more than that of the second Their combined area is 1800 sar How large is each field Which field is more productive? Explain Let x be the area of the first field and y the area of the second Write a system of equations that models this problem in modern notation Let the first equation will deal with yield and the second with area The scribe would start by assuming the size of each field to be 900 sar (half of 1800) What is the total yield of each field under this assumption? The difference in the yield between the two fields is said to be 500 sila If both of them are 900 sar, what is the difference? Is this the desired result? How far off is it? Let’s adjust our initial guess To solve the problem we want the first field, represented by x, to be (larger, smaller) and the second field, represented by y, to be (larger, smaller) Use the first equation from system from #5 and replace x with x + a and y with y - a This adjustment makes the first field larger and the second field smaller Rewrite this equation in the form x- y+ a=500 From #3 we know our original guess was off by 350 Using #6, this tell us that x- y+ a=500 2  a=500 -  x- y ÷ 3  a=500 -150 a=350 Thus, a = Misc Notes on the History of Algebra… rmiller@greenhillsschool.org Ruth Miller Adjust the original guess for x and y by appropriately increasing and decreasing x= , y= You have now completed the method of the scribe Solve the original system using modern methods What are the advantages of each method? Answer Key There are two fields The first one yields sila per sar; the second yields sila per sar The yield of the first field is 500 sila more than that of the second Their combined area is 1800 sar How large is each field KEY Which field is more productive? Explain first Let x be the area of the first field and y the area of the second Write a system of equations that models this problem in modern notation Let the first equation will deal with yield and the second with area x- y = 500, x+y=1800 3 The scribe would start by assuming the size of each field to be 900 sar (half of 1800) What is the total yield of each field under this assumption? (900) = 600, (900) = 450 3 The difference in the yield between the two fields is said to be 500 sila If both of them are 900 sar, what is the difference? Is this the desired result? How far off is it? difference = 150, desired result = 500, off by 500 – 150 = 350 Let’s adjust our initial guess To solve the problem we want the first field, represented by x, to be (larger, smaller) and the second field, represented by y, to be (larger, smaller) first field to be larger and the second field to be smaller Use the first equation from system from #5 and replace x with x + a and y with y - a This adjustment makes the first field larger and the second field smaller (x+a)+ (y-a)=500 Rewrite this equation in the form x- y+ a=500 From #3 we know our original guess was off by 350 Using #6, this tell us that Misc Notes on the History of Algebra… rmiller@greenhillsschool.org Ruth Miller x- y+ a=500 2  a=500 -  x- y ÷ 3  a=500 -150 a=350 Thus, a = Adjust the original guess for x and y by appropriately increasing and decreasing x= , y= You have now completed the method of the scribe Solve the original system using modern methods What are the advantages of each method? ... Two of the first, three of the second, and one of the third make 34 measures And one of the first, two of the second, and three of the third make 26 measures How many measures of grain are contained... a feet out of the water If the reed is pulled to the shore, the top just reaches the shore Find the depth of the water and the length of the reed "Multiply half of the side of the pond by itself;... less the eighth part (of the sixth part)” Misc Notes on the History of Algebra? ?? rmiller@greenhillsschool.org Ruth Miller Discovering Systems of Equations Babylonian Style: The Babylonians often