There are two types of Algebra, “Modern Algebra” and “Classical Algebra”. Classical algebra was related to equation solving and abstract algebra. Abstract Algebra deals with study of groups, rings, and fields. Classical algebra has been developed over a period of 4000 years, whereas abstract algebra has only appeared in the last 200 years. Since algebra grows out of arithmetic, recognition of new numbers, irrational, zero, negative numbers, and complex numbers, it is an important part of mathematics history. The development of algebraic notation progressed through three stages: the rhetorical or verbal stage, the syncopated stage or where abbreviated words were used, and third being the symbolic stage, which we use and are familiar with.

 
Most of the information recorded of ancient Egyptian algebra dating back to 1850 BC, which is based on the Rhind papyrus, which is now located in the British Museum, dates back to 1850 BC and states for example: Divide 100 loaves among 10 men including a boatman, a foreman, and a doorkeeper, who receive double portions. What is the share of each? (Boyer, 112)
In ancient Egyptian times algebra was rhetorical, which has no symbols. The problems were stated as above and solved verbally. They gave no formula and stated no methodology, but only presented the specific problem. Egyptian algebra was slowed down due to their cumbersome method of handling fractions. (Eves, 275)

The mathematics of the Babylonian period, which was from 1800 – 1600 BC, made significant advances than that of Egyptian mathematics. The Babylonians excellent numeration system helped them lead to the development of algebra. Even though the Babylonians had very few, they did have some use of symbols, but like the Egyptians, Babylonian algebra was mainly rhetorical. The procedures used to solve problems were taught through examples and no reasons were given for their procedure. (Kline, 302)

like the Egyptians they recognized only positive rational numbers, although they did find approximate solutions to problems which had no exact rational solution. Advanced application of arithmetic leads to algebra, and the Babylonians made significant beginnings in this area. They could solve linear and quadratic equations, dealing with only positive numbers, systems of two equations and two unknowns, and some special kinds of equations of higher degree. But their arithmetic and algebra remained very elementary by modern standards (Katz, 73-74) . The Greeks contributed much to new applications beyond what the Babylonians had already done, but the Babylonian basis of mathematics was inherited by the Greeks. The Greeks based their mathematics on reason. The Greeks greatest contribution was in applying deductive reasoning and describing general procedures.

Recognition Of Rational Number
Another contribution of the Greeks was the recognition that rational numbers are insufficient for even such elementary needs as measuring lengths of lines. A geometric formulation of irrational or incommensurable arose from this realization. During this time, the classical period 600-300 BC, the Greeks did not recognize the existence of irrational numbers, which obscured the problems created by representing quantities as geometrical magnitudes. Both the Babylonians and Egyptians made significant discoveries in geometry. They were aware of the relationship that we call the Pythagorean Theorem. The first proofs in algebra were to be associated with geometric reasoning. (Burton, 256-257)

Introduction of Negative Numbers
The successors of the Greeks in the history of algebra were the Hindus of India. Astronomy and astrology motivated most of Hindu mathematics. Their record in mathematics dates back to around 800 BC, but only became significant after influenced by Greek achievements. The Hindus introduced negative numbers to represent debts. The Hindus, unlike the Greeks, regarded irrational roots of numbers as numbers and also developed procedures for operating with irrational numbers. For them there was no impediment to the acceptance of irrational numbers, and later generations followed their lead uncritically until the nineteenth century mathematicians established the real number system on a sound basis. Since they allowed negative numbers in their solutions, of what we know today as the quadratic equation, they could combine the various cases considered by Diaphanous into a single rule, and had a procedure for solving quadratics. The Hindus were the first to show an awareness of the fact that roots occurred in pairs, and occasionally even admitted negative and irrational roots as solutions. This was an enormous help in algebra, and Hindu mathematics has been much praised for taking this step. (Boyer, 299-300)
The Hindus made progress in arithmetic as well as algebra. They developed some symbolism, which was enough to classify Hindu algebra as almost symbolic and certainly more so than the syncopated algebra of Diophantus. Only the steps in the solutions of their problems were stated. They did not explain or give reasons for their steps. (Kline, 335)
 

Arab Culture and Algebra
Like the Hindus, the Arabic culture worked freely with irrational. However, they did not accept negative numbers and because of this decision, they decided not to work with them in spite of what they learned from the Hindus. The Arabic culture took over and improved the Hindu number symbols and the idea of positional notation. These numerals and the algorithm for operating them were transmitted to Europe around 1200 and are used throughout the world today.  In algebra, the Arabic culture contributed first of all the name. The word "algebra" comes from the title of a textbook in the subject, Hisab al-jabr w' al mugabala, written about 830. The algebra of this culture was entirely rhetorical even though Diophantus and the Hindus had taken us to the second and third stages of classical algebra.

 
Omar Khayyam (1050-1130) made significant contributions to the solution of cubic equations by geometric methods involving the intersection of conic. The most valuable contribution was the preservation of Greek learning through the Middle Ages, and it is through their translations that much of what we know today about Greeks became available. (Katz, 105) Niccolo Fontana Tartaglia and Girolamo Cardano are intertwined in the history of algebra. Tartaglia, a nickname, was known as the stammered, which is what “Tartaglia” means. This was due to an injury to his jaw and palate received from a solider when he was a young boy, which made speaking difficult for the rest of his life. It seems that from this point on that Tartaglia was untrusting and in general had a bad disposition. He was self taught in mathematics and earned his living as a teacher in Verona and Venice. (Tartaglia, 465-466)

Solution Of Cubic Equation
The first person to algebraically solve cubic equations was Del Ferro. However, the only person he told of this discovery was his student Fior. Fior boasted of being able to solve cubic and a debate between Tartaglia and Fior was arranged. Each was to write 30 questions for the other to solve. Fior submitted 30 questions of the cosa and cubic type believing that his opponent would be unable to solve them. Tartaglia submitted a variety of questions of which Fior was able to work very few. However, inspiration hit Tartaglia the night of the debate on a method for solving cubic and cosa problems. With this new method he solved all of Fior’s problems in less than two hours. Tartaglia was declared the winner.(Van der Waerden, 663-664)
Cardano, or Cardan in Latin, was the illegitimate son of a lawyer in Milan. His father was extremely good in mathematics and was consulted on geometry by Leonardo da Vinci. Cardano was taught mathematics by his father, but enrolled in Pavia University to study medicine. Due to the outbreak of war Pavia University closed and Cardano moved to the University of Padua to finish his schooling. He was said to be a “brilliant student but, outspoken and highly critical, Cardan was not well liked.” Despite this fact he was elected rector of the university.

After receiving his doctorate in medicine he applied to the College of Physicians in Milan, but was repeatedly rejected with his illegitimate birth given as the reason. It is more likely however, that the reputation of his attitude is what kept him out of the college. After years of building support though a near miraculous medical career, Cardano was finally admitted to the college (Ore, 71). It was after this that Cardano approached Tartaglia whom he had heard about due to the success of his debate with Fior. After many months and much convincing, Cardano got Tartaglia to reveal his method for solving cubics. The method turned out to be a substitution of "u-v" for "x" in the equation: x3 + px = q. It was well known at this time that all cubics could be reduced to three types of equations:

(1)x3 + px = q
(2) x3 = px + q
(3) x3 + q = px (Van der Waerden, 211)

The method was revealed on the condition that Cardano would never publish it, and he would write it down only in code so no one would be able to read it in the event of his death. However, Tartaglia began to regret telling Cardano his method, and turned away from all correspondence with him (Tartaglia, 236-237).

In 1545 Cardano published Ars Magna, his greatest mathematical work. Together with his assistant Ferrari, they had make remarkable progress in finding proofs of all cases of the cubic and solving the quartic equation. With solving the (2) equation, a problem arises for Cardano and Ferrari. When it is solved, the solution is:

(4) x = u + v = 3 ( 1/2 + w) + 3 ( 1/2q - w) with,

(5) w = ((1/2q)2 - (1/3p)3)

In the equation (5) it is possible to get a negative under the square root sign that still create suitable answer in the equation (4). These negative square roots we now call imaginary numbers. In generalizing an explanation for this phenomenon Cardano is the first to introduce complex numbers into algebra, though he did this reluctantly (Van der Waerden, 421-423).
However, the findings of Cardano were based on Tartaglia’s method. The reason that Cardano felt justified in publishing this book was that he discovered Tartaglia was not the first to solve the cubic, Del Ferro was the first, and both are fully credited in the book. Also, Tartaglia had only found a method and no proof, for solving only the (1) equation (Cardano, 393).
When Tartaglia found this out his dislike of Cardano turned to hatred. He published a book telling the story of the oath that he had taken, and how he had blatantly gone against his word, adding some personal insults for good measure. However, Cardano was a leading figure in the mathematical, medical, and literary worlds and Tartaglia’s assault did little damage to him.

A debate was then set up between Ferrari and Tartaglia, like a duel of the minds. In 1548 the contest took place. After the first day it was apparent that Ferrari had a better understanding of cubic and quartic equations than Tartaglia, and in the middle of the night Tartaglia slipped away so the duel would be left unresolved. In doing this, Ferrari was left the victory. Tartaglia’s reputation was seriously damaged from this incident. Despite their success in the world of mathematics, both suffered greatly in their personal lives, Tartaglia more before the incident, and Cardano after the incident. (Tartaglia, 465-466)

The contributions of Francois Viete seem to mark the beginnings of modern algebra. Born in 1540 in France, he studied law at the University of Poitiers receiving his degree in 1560. He began publishing his Canon Mathematicus, Seu ad Triagnula cum Appendicibus, in 1571, but his most significant contribution to mathematics came in 1591. In his book, In Artem Analyticam Isagoge, he introduces the first systematic algebraic notation. He uses both plus and minus signs in operations, and letters to represent quantities. Vowels represented unknowns and consonants represented known in this work.(Viete, 401-402)
For the quadratic equation, bx2 + dx = z, he writes, "B in A Quadratum, plus D plano in A, aequari Z solido." (Van der Waerden, 343) Where A and B, A = x in modern notation, are line segments, D is a plane area, and Z is a volume. The goal of this book was actually to revive the analytical method of Pappos explained in his Collection and combine it with the method of Diophantos. Also, Viete stated that in the quadratic equation the unknown and the coefficient preceding the unknown variable squared are linear, the coefficient preceding the unknown to the first power is a plane and the known variable that the equation is set equal to is a volume. Viete stated a rule of operation called “antithesis.” This rule allows transfer of terms from one side of an equation to the other side. This corresponds to what the Arabic Algebraists call “al-jabr.” (Van der Waerden, 256-257)
Al-Khwarizmi is responsible for some methods of algebraic manipulation that we use today, and that Viete used in his work. The "moving" of a term from one side of an equation to the other is an example of these methods. These methods are called "al-jabr." This word is also the originating point of the term "algebra." Al-Khwarizimi's name itself was originating point for the term "algorithm," the procedures used in algebra. (Jones, 230)

Rene Descartes is famous for giving us our current algebraic notation. He introduced this notation at the beginning of his work La Geometrie, in which he explains the principles of analytic geometry (Van der Wareden, 645). This is a part of Descartes great work Discours de la Methode. His use of symbols is using letters at the beginning of the alphabet to represent known quantities and letters at the end of the alphabet to represent unknown quantities. Descartes takes the focus away from things needing to be explained geometrically, and forces a move towards the discovery of analytical processes as opposed to the synthetically building of answers as in the past.

Conclusion
The Babylonians and the Egyptians used almost not symbolism, stating both the problems and the solutions rhetorically. Neither gave formulas and stated no general methodology, but only presented specific problems, presumably to be used as patterns for solving other patterns for solving other problems. They had no concept of proof, and offered not plausible argument that might convince one of the correctness of a procedure. Mathematics before the Greeks, was not a distinct discipline at this period, but a tool in the form of disconnected simple rules which answered questions of practical importance. It was not pursued for its own sake. After the decline of the Roman Empire, India became the temporary center of mathematical research. The most important contributions of the Hindus were the decimal place system, the introduction of zero and negative numbers, and the development of algebra.