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Contents
Ancient Times
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Algebra in Ancient Times
 Al-jabr and Islamic Mathematics The word "algebra" is derived from the Arabic word al-jabr. This term is found in Mohammed ibn Musa al-Khwarizmi’s book The Comprehensive Book of Calculation by Balance and Opposition, written around the year 825. Balance is a translation of the word al-jabr, which eventually became algebra. In al-Khwarizmi’s book, he did not use the modern algebraic notation, neither did he use equations. Instead, everything was in words. For example, he used the Arabic word shay, or thing, in place of X. The text was a manual for solving equations. He mainly dealt with square (of the unknown), roots of the square, and absolute numbers (constants). He noted six different types of quadratic equations, such as squares equal to roots (ax2=bx) and squares equal to numbers (ax2=c). However, Islamic mathematics did not deal with negative numbers, hence the modern day general form of the quadratic equation ax2+bx+c did not make sense then, as the roots may be negative. They did not consider the negative roots.
Algebra of Polynomials Abu Bakr al-Karaji continued the work on algebra, focusing on the inclusion of techniques of arithmetic in algebra. He developed a method of naming the powers xn and their reciprocals 1/xn. Once this was established, he could work on the addition, subtraction, multiplication and division of polynomials. Al-Samaw’al furthered this work, introducing negative coefficients. To aid in the multiplication of polynomials, which requires the law of exponents, he developed a table:
He used this to explain the law of exponents, xnxm=xn+m. "The distance of the order of the product of the two factors is equal to the distance of the order of the other factor from the unit. If the factors are in different directions then we count from the order of the first factor towards the unit; but, if they are in the same direction, we count away from the unit." Hence, to multiply x2 by x3, count 3 orders to the left of x2, and you’ll get x5.
Euclid’s Elements—The Quadratic Equation Although the tern algebra occurred only during the Islamic times, many algebraic concepts were established before that. Book 2 of Greek mathematician Euclid’s Elements (about 300 B.C.) deals with quadratic equations. Algebraic symbolism was inadequate then, thus Euclid represented numbers by line segments. Algebraic identities were represented in geometric form , like the identity (a+b)2=a2+2ab=b2.Linear equations were solved by geometric construction. Quadratic equations were reduaced to the geometric equivalent of one of the forms, which were then solved by applying already established theorems on area. Although the method is similar to the Babylonian way, the Greek method could produce irrational numbers. The quadratic equation was established for solving problems, especially those involving the Pythagorean theorem, whereby the square of the unknown is used. The work of the Greeks was eventually continued by the Arabs.
China & the Nine Chapters of Mathematical Art The Nine Chapters of Mathematical Art was a recording of the development of early Chinese mathematics. It’s main purpose was, however, to present the knowledge acquired in the study of astronomy, so it wasn’t specific to mathematics. Systems of linear equations was dealt with in Chapter 8, Rectangular Arrays. This method, called "fang cheng", is actually solving simultaneous linear equations, using counting rods, which is quite similar to the modern method. The addition and subtraction involving negative numbers was also mentioned in the work. Extraction of square and cube roots is also dealt with in the book, using the same method as is used today, but with counting rods. The Pascal’s triangle was used to extract roots of higher degrees.
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