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Wednesday, June 29, 2011

History of Mathematics - Algebra

Algebra is a branch of mathematics concerning the study of structure, relation and quantity. Elementary algebra is the branch that deals with solving for the operands of arithmetic equations. Modern or abstract algebra has its origins as an abstraction of elementary algebra. Some historians believe that the earliest mathematical research was done by the priest classes of ancient civilizations, such as the Babylonians, to go along with religious rituals. The origins of algebra can thus be traced back to ancient Babylonian mathematicians roughly four thousand years ago.

The term Algebra comes from Arabic word ‘Al Jabr’ that was found in Al-Kitab Al Mukhtasar fi hisab al jabr wa’l muqabala or in Arabic authored by Abu Jafar Muhammad ibn Musa al-Khawarizmi, also known as Al-Khwarizmi which can be translated as The Compendious Book on Calculation by Completion and Balancing. His book eventually helped him become the ‘Father Of Algebra’. The exact meaning of the word al-jabr is still unknown, most historians agree that the word meant something like "restoration", "completion", "reuniter of broken bones" or "bonesetter." The term is used to describe the operations that he introduced, "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.

Development Of Algebra

The development of symbolic algebra can be divided into 3 stages.
Rhetorical algebra
Syncopated algebra
Simbolic algebra

Conceptual Stages

In addition to the three stages of expressing algebraic ideas, there were four conceptual stages in the development of algebra which occurred alongside the changes in expression. These four stages were as follows.
Static Equation-Solving
Dynamic Function

Babylonian Algebra

Babylions were more concerned with quadratic and cubic equations.They were familiar with many simple forms of factoring, three-term quadratic equations with positive roots and many qubic equations although it is not known if they were able to reduce the general cubic equations. Tthe Babylions were not interested in exact solutions but there interest about approximations, so they would commonly use linear interpolations to approximate intermediate values. The Babylonians had developed flexible algebraic operations with which they were able to add equals to equals and multiply both sides of an equation by like quantities so as to eliminate fractions and factors. They also dealt with the equivalent of system of two equations in two unknown. They considered some problems involving more than two unknowns and a few equivalent to solve equations of higher degree.

Egyptian Algebra

The Egyptians were mainly concerned with linear equations.
Example :
x + ax = b
x + ax + bx =c
can solved when a, b and c are known.

The similarity between the Babylonians and Egyptians are :
Their algebra essentially rhetorical, that is, it is use no symbols. Problems were stated and solved verbally. Only recognize positive rational numbers

Greek Geometrical Algebra

It is sometimes that the Greeks had no algebra, but this is inaccurate. But the time of Plato, Greek mathematicians had undergone a drastic change. For example, the geometric algebra can be solve by the linear equations ax = bc. The ancient Greeks solve this equation by looking at it as an equality of areas rather than as an equality between the ratio a:b and c:x. The rectangle can be construct length b and c and the side will extend of, finally they would complete the extended rectangle as to find the side of rectangle, that is the solution. The significant achievement was in applying deductive reasoning and describing general procedures. The Greeks of classical period, who did not recognize the existence of irrational numbers, avoided the problem thus created by representing quantities as geometrical magnitudes. In content, there was little beyond what the Babylion had done.

Chinese Algebra

The Chinese introduced magic square in Chiu-chang Suan-Shu or The Nine Chapters on the Mathematical Art, written around 250 BCE, to solve determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns

Indian Algebra

The method known as "Modus Indorum" or the method of the Indians have become our algebra
today. Brahmagupta solves the general quadratic equation for both positive and negative roots. He was the first to give a general solution to the linear Diophantine equation ax + by = c, where
a, b, and c are integers. In writing Brahmagupta wrote addition by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.

Islamic algebra

Al-jabr wa'l muqabalah

Al-Jabr is divided into six chapters, each of which deals with a different type of formula. The first chapter of Al-Jabr deals with equations whose squares equal its roots (ax2 = bx), the second chapter deals with squares equal to number (ax2 = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax2 + bx = c), the fifth chapter deals with squares and number equal roots (ax2 + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax2). In Al-Jabr, al-Khwarizmi uses geometric proofs, he only deals with positive roots. He also recognizes that the discriminant must be positive and described the method of completing the square. Al-Jabr is fully rhetorical with the numbers even being spelled out in words. For example, what we would write as
x^2+ 10x = 39
And al-Khawarizmi would have written as One square and ten roots of the same amount to thirty-nine dirhems.

European Algebra

In 13th century, the Italian mathematicians Leonardo Fibonacci achieved a closed approximation to the solution of the cubic equation x3 + 2x2 + cx = d. He used Arabic method of successive approximations. The Italian mathematicians Scipione del Ferro, Nicollo Tartaglia and Gerolamo Cardano solved the general cubic equations in terms of the constants appearing in the equation in 16th century.

From Gauss, algebra had entered its modern phase. Attention shifted from solving polynomial equations to studying the structure of abstract mathematical systems whose axioms were based on the behavior of mathematical objects, such as complex numbers, that mathematicians encountered when studying polynomial equations. Two examples of such systems are algebraic groups and quaternions which share some of the properties of number systems but also depart from them in important ways. Groups began as systems of permutations and combinations of roots of polynomials, but they became one of the chief unifying concepts of 19th-century mathematics.

Immediately after Hamilton's discovery, the German mathematician Hermann Grassmann began investigating vectors. Despite its abstract character, American physicist J. W. Gibbs recognized in vector algebra a system of great utility for physicists, just as Hamilton had recognized the usefulness of quaternions. The widespread influence of this abstract approach led George Boole to write The Laws of Thought (1854), an algebraic treatment of basic logic. Since that time, modern algebra also called abstract algebra has continued to develop. Important new results have been discovered, and the subject has found applications in all branches of mathematics and in many of the sciences as well.

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