The History of Algebra

1. The History of Algebra By Megan Bell & Tonya Willis

2. Egyptian Algebra • Earliest finding from the Rhind Papyrus – written approx. 1650 B.C. • Solve algebra problems equivalent to linear equations and 1 unknown • Algebra was rhetorical – use of no symbols • Problems were stated and solved verbally • Cairo Papyrus (300 B.C.) – solve systems of 2 degree equations

3. Babylonian Algebra • Babylonians were more advanced than Egyptians • Like Egyptians, algebra was also rhetorical • Could solve quadratic equations • Method of solving problems was rhetorical, taught through examples • No explanations to findings were given • Recognized on positive rational numbers

4. Greek Algebra • The Greeks originally learned algebra from Egypt as indicated in their writings of the 6th century BCE. Later they learned Mesopotamian geometric algebra from the Persians. They studied number theory, beginning with Pythagoras (ca 500 BCE), continuing with Euclid (ca 300 BCE) and Nicomachus (ca 100 CE). The culmination of Greek algebra is the work of Diophantus in the 3rd century CE.

5. Fundamental Limitations of Greek Algebra: • 1. Negative and complex solutions of equations were rejected as “absurd” or “impossible”. • 2. They realized that irrational numbers existed but refused to consider them as numbers. They went to great lengths to avoid them.

6. 3. They did geometric algebra Mesopotamian style. That is, a given algebraic problem is translated into a geometric one which is solved by a geometric construction. This limits the variety of algebraic problems studied. Note that this method does not deal with specific numbers thereby avoiding irrational numbers. • 4. They gave specific examples with rational solutions to illustrate general procedures.

7. Diophantine Algebra • Represents the end of a movement among Greeks away from geometrical algebra to a system of algebra that did not depend on geometry • Diophantus – Greek mathematician from Alexandria • Often considered the “father of Algebra”

8. Wrote series of books “Arithmetica” – features work on solutions of algebraic equations to theory of numbers • 189 problems in “Arithmetica” were all solved by a different method • Some of his writings from this series are still lost • No general method to his solutions • Accepted only positive rational roots

9. Diophantus was the first Greek mathematician to recognize fractions as numbers • His discoveries led to what we know today as “Diophantine Equations” and “Diophantine Approximations” • Furthermore, he introduced the syncopated & symbolic styles of writing

10. Interesting Fact • We know essentially nothing of his life and are uncertain about the date at which he lived (200-284). • The only know detail of his age was a phrase written by Metrodorus which stated:

11. “his boyhood lasted 1/6th of his life, he married after 1/7th more, his beard grew after 1/12th more, and his son was born 5 years later, the son lived to half his father’s age and the father died 4 years after the son” • Translates to: he married at 26 son died at 42, so Diophantus died at 84.

12. Hindu Algebra • Records in mathematics dates back to approx. 800 B.C. • Most mathematics was motivated by astronomy & astrology • Introduced negative numbers to represent debt

13. When solving problems they only stated steps – no proof or reasoning was provided • First to recognize that quadratic equations have two roots • Known for invention of decimal system which we use today

14. Arabic Algebra • Al-Khwarizmi wrote “al-jabrw’al-muqabala” translation “restoration and compensation”(source of the word algebra – mistranslation from ‘Al-jabr’ to Latin ‘Algebra’) • Quadratic equations, practical geometry, simple linear equations, and application of mathematics to solve inheritance problems • Algebra was entirely rhetorical • Could solve quadratic equations

15. Abu Kamil (born 850) forms an important link in the development of algebra between al-Khwarizmi and al-Karaji. Despite not using symbols, but writing powers of x in words, he had begun to understand what we would write in symbols as xn.xm = xm+n. • Remark: symbols did not appear in Arabic mathematics until much later.

16. Al-Karaji (born 953) is seen by many as the first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today. He was first to define the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and to give rules for products of any two of these. He started a school of algebra which flourished for several hundreds of years.

17. Omar Khayyam (born 1048) gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.  • Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work: • If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared.

18. European Algebra • Algebra was still largely rhetorical, slightly syncopated • Solution to cubic and quartic equations • Negative numbers were known, but not fully accepted • No one could solve 5th degree equation • Algebraists were called “cossists” and algebra was called “cossic art”

19. Major Breakthrough – 16th Century • Viete (1540-1603) – lawyer, French mathematician, astronomer, & advisor to King Henri III & IV • Often called the father of “modern algebra” • Focused on algebraic equations in his mathematical writings

20. Introduced letters for both constants and unknowns • First algebraic notations were introduced in his book “In artem analyticam isagoge” • Translates to “Introduction to the analytic art”

21. Now mathematicians were able to write equations with more than one unknown • Thomas Harriot (1620’s) • 5aaa + 7bb • Pierre Herigone (1634) • 5a3 + 7b2 • James Hume (1636) • 5aⁱⁱⁱ +7bⁱⁱ • Rene Descartes (1637) • 5a³ + 7b²

22. Descartes • Symbolic algebra reached full maturity • Used lowercase letters from end of alphabet as unknowns • Used lowercase letters from beginning of alphabet for constants • Introduced ax + by=c, still use today to describe equation of line • Work led to introduction of reciprocals, roots of powers, limits of roots of powers, and exponents.

23. Stages of Algebra • The development of algebra progressed through 3 stages: • Rhetorical – no use of symbols, verbal only • Syncopated – abbreviated words • Symbolic – use of symbols, used today

24. Rhetorical Algebra • 1650 BCE-200 CE • Early Babylonian and Egyptian algebras were both rhetorical • In Greece, the wording was more geometric but was still rhetorical. • The Chinese also started with rhetorical algebra and used it longer.

25. Syncopated Algebra • 200 CE-1500 CE • Started with Diophantus who used syncopated algebra in his Arithmetica (250 CE) and lasted until 17th Century BCE. • However, in most parts of the world other than Greece and India, rhetorical algebra persisted for a longer period (in W. Europe until 15th Century CE).

26. Aryabhata& Brahmagupta • Ist century CE from India • Developed a syncopated algebra • Ya stood for the main unknown and their words for colors stood for other unknowns

27. Symbolic Algebra • Began to develop around 1500 but did not fully replace rhetorical and syncopated algebra until the 17th century • Symbols evolved many times as mathematicians strived for compact and efficient notation • Over time the symbols became more useable and standardized

28. Below is a table of various forms in which the modern day equation 4x2 + 3x = 10 might have been written by different mathematicians from different countries and at different times.

29. Types of Algebra • Algebra is divided into two types: • Classical algebra – equation solving • Abstract/Modern algebra – study of groups • Classical algebra has been developed over a period of 4,000 years, while abstract algebra has only appeared in the last 200 years.

30. Classical Algebra • Finding solutions to equations or systems of equations • i.e. finding roots or values of unknowns • Uses symbols instead of specific numbers • Uses arithmetic operations to establish procedures for manipulating symbols

31. Abstract Algebra • In the 19th century algebra was no longer restricted to ordinary number systems. Algebra expanded to the study of algebraic structures such as: • Groups • Rings • Fields • Modules • Vector spaces

32. The permutations of Rubik’s Cube have a group structure; the group is a fundamental concept within abstract algebra.

33. 19th century • British mathematicians explored vectors, matrices, transformations, etc. • Galois (French, 1811-1832) • Developed the concept of a group (set of operations with a single operation which satisfies three axioms) • Cayley (British, 1821-1895) • Developed the algebra of matrices • Gibbs (American, 1839-1903) • Developed vectors in three dimensional space

34. Subject Areas Under Abstract Algebra: • Algebraic number theory - study of algebraic structures to algebraic integers • Algebraic topology – study of topological spaces • Algebraic geometry – study of algebra and geometry combined

35. Algebraic Number Theory • Study of algebraic spaces related to algebraic integers • Accomplished by a ring of algebraic integers O in a algebraic number field K/Q • Studies the algebraic properties such as factorization, behavior of ideals, and field extensions

36. Algebraic Topology • Study of qualitative aspects of spatial objects • Surfaces, spheres, circles, knots, links, configuration spaces, etc. • Also viewed as the study of “disconnectives” • Interpreted as a hole in space • Example: We live on the surface of a sphere, but locally it is difficult to distinguish this from living on a flat plane

37. Algebraic Geometry • Combines techniques of abstract algebra with the language and the problems of geometry • Areas of study in are algebraic sets, systems of polynomial equations, plane curves (lines, circles, parabolas, ellipses, hyperbolas, and cubic curves) • Study of special points such as singular points, inflection points, and points at infinity

38. Class Activity Questions?

39. Works Cited • http://www.cut-the-knot.org/language/symb.shtml • http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEYQFjAD&url=http%3A%2F%2Forion.math.iastate.edu%2Flhogben%2Fclasses%2FAlgebraicSymbolism.ppt&ei=dFqdT6ybCcf_ggeJzvyHBg&usg=AFQjCNE4fBVsQ6dv0f10gy-BwcpFj1x7VA&sig2=VofVfnEeueRDP1rtVqpOEQ • http://www.math.yorku.ca/Who/Faculty/Kochman/M4400/GreekAlgebra.pdf • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html • http://en.wikipedia.org/wiki/Timeline_of_algebra • http://www.ucs.louisiana.edu/~sxw8045/history.htm • http://fabpedigree.com/james/mathmen.htm • http://www.algebra.com/algebra/about/history/ • http://personal.ashland.edu/~dwick/courses/history/math_bios.pdf • http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html#toc

40. Works Cited (cont.) • Waerden, B. L. Van Der. A History of Algebra. German: Springer-Verlad Berlin Heidelberg, 1985. Print. • Waerden, B. L. Van Der. Geometry and Algebra in Ancient Civilizations. German: Springer-Verlad Berlin Heidelberg, 1985. Print. • Clawson, Calvin C. The Mathematical Traveler: Exploring the Grand History of Numbers. New York: Plenum, 1994. Print. • Sesiano, Jacques. An Introduction to the History of Algebra: Solving Equations from Mesopotamian Times to the Renaissance. Rhode Island: American Mathematical Society, 2009. Print.

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