It is generally believed that algebra originated from the early arithmetic used in Babylonia around 1800-1600 b.c. Clay tablets bearing mathematical and algebraic tables show that the Babylonians had a rudimentary understanding of algebra and were able to solve simple equations. While the ancient Egyptians and Greeks also dabbled in mathematics and basic arithmetic, they were not as advanced in their knowledge. By 150 b.c., the use of algebra and mathematics, particularly among the Romans, became an important basis of astronomical study. One of the leaders in this area was the Greek mathematician Diophantus. Although very little is known about his life and even the exact period in which he lived (ca. 210-290 a.d.) is uncertain, his work with algebraic equations, symbolism and indeterminate solutions was of great importance to future generations. In fact, equations having integer solutions are still commonly referred to as Diophantine equations in his honor.
Middle Eastern scholars were greatly impressed by Diophantus' works, particularly al-Khwarizmi, a renowned Arab mathematician responsible for expanding algebra and transmitting Diophantus' ideas to the Western world. Around 852 he wrote al-Kitab al jabr w'al-muqubala, which means "The Rules of Restoration and Reduction." Our modern term "algebra" was derived from the arabic word al-jabr, meaning "restoring." Algebra in turn came to be the branch of mathematics involving the solution of equations by means of transpositions and cancellations. Ancient algebra was very different from contemporary algebra. All early algebraic problems and numbers were written out using Latin, Arabic or Greek abbreviations for quantities and operations. Most of our familiar numbers, symbols and notations were not developed until much later.
The eleventh and twelfth centuries were a time of great advances in algebra and mathematical theory. During this time, Italy led the way in breakthroughs, producing several prominent mathematicians. The use of Hindu-Arabic numerals was encouraged by Italian mathematician Leonardo Fibonacci in 1202. Many years later, fellow Italians Girolamo Cardano and Niccolò Tartaglia produced noteworthy accomplishments. In 1539 Tartaglia was the first to work out a method for solving special types of third degree equations; however, Cardano published the general solution six years later after promising Tartaglia he would keep his method secret. Although Tartaglia was credited with the earlier work, the general method is still known as Cardano's rule.
Another important contributor was the English mathematician Thomas Harriot, who did much to simplify algebraic notation and attempted, although unsuccessfully, to create a uniform treatment for all algebraic equations. In the late 1500s, François Viète, known as the greatest French mathematician of the sixteenth century, developed algebraic symbolism, using letters to represent unknown quantities in equations, and he also advanced algebraic theory and notation. In the 1600s, French mathematician and philosopher René Descartes played a major role in the algebraic evolution. While in the French army, Descartes was unable to participate in active duty because of ill health. He found his inactivity gave him the chance to study mathematics, just one of his many interests. Many years later, he recommended that Viète's symbolism be modified to use letters near the beginning of the alphabet for constants and those near the end for variables. The change was accepted without hesitation; however Descartes' concept of combining algebra and geometry was of much greater importance. In 1637 he published Discours de le méthode, in which he described this combined system, an approach which allowed mathematicians to solve problems more easily than by using algebra or geometry separately. This new branch of mathematics became known as analytic geometry and paved the way for the formation of calculus.
By the 1700s, the use of exponential, negative, fractional and complex number equations, as well as the development of new algebraic symbols extended the study of algebra into new areas. Among the leaders was John Wallis, an English mathematician who wrote extensively and had the reputation of being a "calculating prodigy"; he worked out the square root of a fifty-three digit number in his head, correct to 17 places. Wallis also introduced the systematic use of algebraic formulae.
During the early nineteenth century, the French mathematician, Evariste Galois (1811-1832) concentrated on improving the solvability of equations in general, creating his own mathematical techniques toward problem-solving, an important concept which came to be called group theory. Unfortunately, Galois died in a duel in 1832 at the age of 21. In 1844 George Boole (1815-1864) introduced the theory of logical or Boolean algebra. Although his ideas were not immediately accepted by the mathematicians of the day, the usefulness of symbolic logic was proven years later in designing the digital computer. Max Noether (1844-1921) was considered to be one of the guiding spirits of nineteenth century algebraic geometry, accomplishing notable work on algebraic curves in 1873. Taking up the work of Boole in logical mathematics was the English mathematician Alfred North Whitehead. From 1910 to 1913 he published a three-volume work Principia Mathematica, which served to build mathematics up from symbolic logic. Irish mathematician William Rowan Hamilton was an amazing child prodigy; while Hamilton never attended school, he taught himself fourteen languages as well as a comprehensive study of mathematics. He is most noted for his work with quarternions and the use of quarternion algebra.
With the turn of the century in 1900, algebra became more abstract, focusing on the study of structures known as groups, rings, fields, lattices and vector spaces. Termed modern abstract algebra, the numbers as well as the calculations became more generalized in scope. American mathematician Joseph Wedderburn (1882-1948) independently studied abstract algebra. In 1905 he published A Theorem on Finite Algebras, which introduced the use of hypercomplex numbers and the creation of finite algebras. Another early pioneer in this area was Ernst Steinitz (1871-1928), known for his 1910 work with the algebra of fields. German mathematician Amalie (Emmy) Noether, eldest daughter of Max Noether, took up the work begun by her father. In 1921 she published her ideals in rings" theory, which became an important basis for future mathematical studies in noncommutative algebra.
The importance of abstract algebra has been confirmed in modern use. It is utilized in specific geometry, number theory, data encryption and cryptology problems with much success. Algebra has been widely used by scientists and engineers throughout the world and has also played a major role in uncovering the secrets of nuclear energy.
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