Al-Khwarizmi (780-850) Arab
Al-Khwarizmi’s contribution to Mathematics and Geography established the basis for innovation in Algebra and Trigonometry. He presented the first systematic solution of linear and quadratic equations. He is considered the founder of algebra. His work on arithmetic was responsible for introducing the Arabic numerals, based on the Hindu-Arabic numeral system developed in Indian Mathematics, to the Western world.
Algebra is an important and a very old branch of mathematics which deals with solving algebraic equations. In the third century, the Greek mathematician Diaphanous wrote a book “Arithmetic” which contained many of practical problems. In the sixth and seventh centuries, Indian mathematicians like Aryabhatta and Brahmagupta have worked on linear equations and quadratic equations and developed general methods of solving them. The next major development in algebra took place in ninth century by Arab mathematicians. In particular, Al-Khwarizmi’s book entitled “Compendium on calculation by completion and balancing” was an important milestone. There he used the word aljabra - which was latinized into algebra - translates as competition or restoration. In the 13th century, Leonardo Fibonacci’s books on algebra was important and influential. Other highly influential works on algebra were those of the Italian mathematician Luca Pacioli (1445-1517), and of the English mathematician Robert Recorde (1510-1558). In later centuries Algebra blossomed into more abstract and in 19th century British mathematicians took the lead in this effort. Peacock (Britain, 1791-1858) was the founder of axiomatic thinking in arithmetic and algebra. For this reason he is sometimes called the “Euclid of Algebra”. DeMorgan (Britain, 1806-1871) extended Peacock’s work toconsider operations defined on abstract symbols. In this chapter, we shall focus on learning techniques of solving linear system of equations and quadratic equations.
An ordered pair ( , ) x y 0 0 is called a solution to a linear system in two variables if the values x x y y , = = 0 0 satisfy all the equations in the system.
(i) An equation of the form ax by c + = is called linear because the variables are only
to the first power, and there are no products of variables in the equation.
(ii) It is also possible to consider linear systems in more than two variables. You will
learn this in higher classes.
The basic relationships between the zeros and the coefficients of p x ax bx c ( ) 2
= + + are
sum of zeros : a b + =
a
b - = coefficient of
coefficient of
x
x - 2 .
product of zeros : ab =
a
c = coefficient of
constant term
x2 .
A quadratic polynomial p x ax bx c ( ) 2
= + + may have atmost two zeros.
Now, for any a ! 0, a x x
2 ^ - + + ^a b ab h h is a polynomial with zeros a and b. Since
we can choose any non zero a, there are infinitely many quadratic polynomials with
zeros a and b.
Division algorithm :
If p x( ) is the dividend and q^ hx is the divisor, then by division
algorithm we write, p x( ) = s^ ^ ^h h h x q x r x + .
Now, we have the following results.
(i) If q(x) is linear , then r^ hx = r is a constant.
(ii) If q x( )deg = 1 (i.e., q(x) is linear), then p x( )deg = 1 + s x( )deg
(iii) If p x( ) is divided by x a + , then the remainder is p a ( ) - .
(iv) If r = 0, we say q(x) divides p(x) or equivalently q(x) is a factor of p(x).
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