Rafael Bombelli

Rafael Bombelli (1526–1572) was an Italian mathematician.

Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers.

He was the one who finally managed to settle the problem with imaginary numbers. In Algebra 1569, Bombelli solved equations, using the method of del Ferro/Tartaglia, he introduced +i and -i and described how they both worked in Algebra.

The lunar crater Bombelli is named after him.

Bombelli method

Bombelli used a method related to continued fractions to calculate square roots. His method for finding $$sqrt{n} begins with $$n=(apm r)^2=a^2pm 2ar+r^2 with $$0, from which it can be shown that $$r=frac{2apm frac{2apm cdots }

for the root but Bombelli is more concerned with better approximations for $$r. The value chosen for $$a is either of the whole numbers whose squares $$n lies between. The method gives the following convergents for $$sqrt{13} while the actual value is 3.605551275... :

: $$3frac{2}{3}, 3frac{3}{5}, 3frac{20}{33}, 3frac{66}{109}, 3frac{109}{180}, 3frac{720}{1189}, cdots

The last convergent equals 3.605550883... . Bombelli's method should be compared with formulas and results used by Hero and Archimedes. The result $$frac{265}{153} used by Archimedes in his determination of the value of $$pi can be found by using 1 and 0 for the initial values of $$r.

External links

* [http://a2.lib.uchicago.edu/pip.php?/pres/2005/pres2005-188.pdf L'Algebra] , original Italian text.
*MacTutor Biography|id=Bombelli

References

* Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972, Oxford University Press, New York, ISBN 0-19-501496-0
* David Eugene Smith, "A Source Book in Mathematics", 1959, Dover Publications, New York, ISBN 0-486-64690-4