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Unformatted text preview: HISTORY OF MATHEMATICS – LECTURE 7 – WEDNESDAY 22ND SEPTEMBER Negative Numbers It took a very long time for the mathematical community to agree on the role of negative numbers. While Chinese texts originally written around 200 B.C. were accepting of them, using them for commercial purposes, even Greek mathematicians were dismissive, with Diophantus saying that the equation 4x + 20 = 4 has “an absurd result”. The first systematic approach to the properties of negative numbers was given by Brahmagupta, who lived in India between 598 and 668 A.D., where mathematics was becoming more algebraic, rather than geometric. Referring to “fortunes” (positive amounts) and “debts” (negative amounts), Brahmagupta declared: A debt minus zero is a debt. A fortune minus zero is a fortune. Zero minus zero is a zero. A debt subtracted from zero is a fortune. A fortune subtracted from zero is a debt. The product of zero multiplied by a debt or fortune is zero. The product of zero multiplied by zero is zero. The product or quotient of two fortunes is one fortune. The product or quotient of two debts is one fortune. The product or quotient of a debt and a fortune is a debt. The product or quotient of a fortune and a debt is a debt. However, even though negative numbers could be understood in financial terms, they were still discarded for centuries when considering the solution of an algebraic equation. For example (as we will see below), the Islamic method of solving quadratic equations involved drawing a diagram to demonstrate the process of completing the square, but (geometrically) this will only lead to positive values. It was not until the 16th century that Europeans started to see the benefit of allowing negative numbers. A great deal of time was spent by Italian mathematicians trying to solve cubic equations (see Lecture 8). Given their aversion to negative numbers, they reduced it down to many different cases, depending on which coefficients were on which side of the equation (all having to be positive). By allowing the existence of negative numbers (and eventually complex numbers), these cases reduced down to one, and a general solution could be given. This led to a growing acceptance of negative numbers among mathematicians, with John Wallis’ creation of the number line in 1685 giving a visual interpretation. Quadratic Equations It was shown in Lecture 3 that the Babylonians were not only aware of quadratic equations, but knew how to solve them algebraically. While their study was always connected to a physical problem, the first person to look at quadratic equations purely from an algebraic standpoint was the Persian mathematician Mohammed ibn Mûsâ al‐Khowârizmî (c. 780‐850 A.D.), from whose name the word “algorithm” is derived, and from whose work on al‐jabr (which literally means “restoring” in the sense of moving terms around so that they all have positive coefficients) we get the word “algebra”. In dealing with quadratics, he created six categories in order to avoid the use of negative numbers: ax2 = bx ax2 = c bx = c ax2 + bx = c ax2 + c = bx bx + c = ax2 Note: These equations were written out in words by al‐Khowârizmî, using none of the notation we see above. Ex. Solve x2 + 10x = 39. The methods of al‐Khowârizmî were studied and expanded upon by Abû Kâmil (c. 850‐930). He was the first to consider quadratic equations with irrational coefficients and his book containing 69 problems was an inspiration for Fibonacci three hundred years later. We will look at two of them (problems 8 and 61) below: Ex. (Problem 8) Divide 10 into two parts in such a way that when each part is divided by the other, their sum is 17/4. Ex. (Problem 61) Divide 10 into three parts in such a way that if the small part multiplied by itself is added to the middle part multiplied by itself, the result is the large part multiplied by itself, and when the small part is multiplied by the large part it equals the middle part multiplied by itself. Of course, the derivation of the quadratic formula is well‐known today, and presented in every algebra class using the method of completing the square. While we will not show it here, an interesting alternative formula is derived below. One final item of note is the method of François Viète for solving quadratic equations of the form x2 + ax = b. Viète showed that if we make the substitution x = y – a/2 that a quadratic equation in y arises where the first‐degree term is zero. Ex. ...
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This note was uploaded on 09/22/2011 for the course MAC 2311 taught by Professor Evinson during the Spring '08 term at University of Central Florida.

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