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Rafael BombelliItalian mathematician, hydraulic engineer.
Country:
 Italy |
Content:
- Biography of Raffaele Bombelli
- Meeting Antonio Maria Paci and Translation of "Arithmetic"
- Contributions to Algebra
- Contributions to Notation and Calculation
- Other Scientific Achievements
- The last fraction equals ..., while .
Biography of Raffaele Bombelli
Raffaele Bombelli was an Italian mathematician and hydraulic engineer. He was born in Bologna into a family of a wool merchant, Antonio Mazzoli, and a tailor's daughter, Diamante Scudieri. He was the oldest of their six children and studied architecture. During this time, the discoveries of del Ferro and Tartaglia sparked widespread interest in mathematics, which also captivated Bombelli.
Meeting Antonio Maria Paci and Translation of "Arithmetic"
While in Rome, Bombelli met Professor Antonio Maria Paci, who had recently discovered a manuscript of Diophantus' "Arithmetic" in the Vatican Library. The two friends agreed to translate it into Latin. Simultaneously with the translation, Bombelli wrote his treatise "Algebra" in three books, which included not only his own developments but also numerous problems from Diophantus with his own comments. He planned to add two more books on geometry to the treatise but did not manage to complete them.
Contributions to Algebra
Bombelli's main work, "Algebra," written around 1560 and published in 1572, is notable in many respects. He was the first in Europe to freely operate with negative numbers, providing rules for their manipulation, including the rule of signs for multiplication. He was also the first to recognize the usefulness of complex numbers, especially in solving cubic equations using Cardano's formulas. For example, an equation with a real root x = 4 can be transformed using Cardano's formulas to obtain complex roots. Bombelli discovered that , from which the desired real root is obtained. He emphasized that in such (irreducible) cases, complex roots are always conjugate, leading to the emergence of real roots. His explanations laid the foundation for the successful application of complex numbers in mathematics. However, a complete study of extracting roots from complex numbers was accomplished by de Moivre in the 18th century.
Contributions to Notation and Calculation
Bombelli also invented the first parentheses, which had the shape of a straight and inverted letter L. The familiar round parentheses that we use today appeared in the same 16th century, but they were only introduced into common usage by Leibniz and Euler. Bombelli was the first to use numerical (rather than verbal, as before) notation for exponents, marked by a special subscript. The modern notation for exponents was introduced into wide usage by Descartes.
Other Scientific Achievements
Among Bombelli's other scientific achievements, it is worth noting his application of continued fractions to calculate square roots of natural numbers. To find the value of , we first determine its integer approximation: , where . Then . From here, it is not difficult to derive the continued fraction expansion:
To estimate the accuracy of the obtained approximations, one can use one of the properties of continued fractions: consecutive values of convergents oscillate around the limit, alternating between overestimation and underestimation.
For example, for , we obtain the following consecutive approximations:
