Rafello Bombelli was the last of a long line of Italian algebraists who contributed to the theory of equations during the Renaissance. He was the first to develop a consistent theory of imaginary numbers which included the rules for the four operations on complex numbers. Gottfried Wilheim von Leibniz complimented Bombelli years later referring to him as an "outstanding master of the analytical art."
Bombelli was born in 1526 in Bologna, Italy. His father, Antonio Bombelli, was a wool merchant. Bombelli choose not his father's profession but instead became an engineer. He never attended any university but instead was trained by Pier Francesco Clementi of Corinaldo, who as an engineer-architect was responsible for draining swamps. For the major part of his working life, Bombelli was employed as an engineer under the patronage of Monsignor Alessandro Rufini, who later became the Bishop of Melfi. Bombelli's engineering career included two major projects; the reclaiming of the Val di Chiana marshes from 1551 to 1560, and the failed attempt to repair the Ponte Santa Maria Bridge in Rome in 1561.
Writes Algebra Text
Bombelli wrote his Algebra text in 1560, just a few years after Girolamo Cardano's Ars magnawas published. Bombelli's Algebra was not published the first time until 1572 (and only a partial edition), and it became a very important work for several reasons. It featured 143 problems found originally in Diophantus' Arithmetica. Bombelli had found a copy of Diophantus' book in the Vatican Library and until that time the ancient Greek's Diophantus' work was mainly ignored. In Algebra, Bombelli made notable contributions to improvements in algebraic notation. He introduced an index notation for denoting powers, which he referred to by the term "dignita." Bombelli also introduced in this work a new symbol to indicate the root of an entire expression, was the predecessor of a modern bracket.
What Bombelli is best known for is his justification of conjugate imaginary roots for the "irreducible case" of cubic equations. Starting with the Cardano- Tartaglia formula, he developed a very detailed theory of imaginary numbers by arguing by analogy with the known rules for operating on real numbers. The irony of Bombelli's discovery was that the irreducible case actually results in three real roots, but the Cardano-Tartaglia formula produces two conjugate imaginary roots involving the square root of a negative number. Bombelli knew that this type of cubic equation has three real roots and used the results of the Cardano-Tartaglia formula to demonstrate that real numbers can be the result of operations on complex numbers.
Bombelli never realized his great contribution, since he still referred to complex numbers as useless. Nonetheless, Bombelli's Algebra was widely read and respected. Leibniz used it to study cubic equations and Leonhard Euler quoted from it in his text, Algebra.
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