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

2. Egyptian Algebra
Earliest finding from the Rhind Papyrus – written approx 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
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 ( ).
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-jabr w’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 ( ) – 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.
Nicolas Chuquet
1484
42 p31 égault 100
Vander Hoecke
1514
4 Se + 3 Pri dit is ghelijc 10
F.Ghaligai
1521
4  e 3c° - 10 numeri
Jean Buteo
1559
4  p 3 p [ 10
R.Bombelli
1572
 p  equals á 10
Simon Stevin
1585
4  + 3  egales 10
François Viète
1590
4Q + 3N aequatus sit 10
Thomas Harriot
1631
4aa + 3a === 10
René Descartes
1637
4ZZ + 3Z  10
John Wallis
1693
4XX + 3X = 10

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 Galois (French, 1811-1832) Cayley (British, 1821-1895)
British mathematicians explored
vectors, matrices, transformations, etc.
Galois (French, )
Developed the concept of a group (set of operations with a single operation which satisfies three axioms)
Cayley (British, )
Developed the algebra of matrices
Gibbs (American, )
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. Questions?
Class Activity

39. Works Cited http://www.cut-the-knot.org/language/symb.shtml

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

41. THE END.