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. Bombelli used a method related to continued fractions to calculate square roots. His method for finding <math> \sqrt{n} </math> sets <math> n=(a\pm r)^2=a^2\pm 2ar+r^2\ </math> with <math> 0<r<1\ </math> from which it can be shown that <math> r=\frac{|n-a^2|}{2a\pm r}</math>. Repeated substitution of the expression on the right hand side for <math>r</math> into itself yields a continued fraction
- <math>a\pm \frac{|n-a^2|}{2a\pm \frac{|n-a^2|}{2a\pm \frac{|n-a^2|}{2a\pm \cdots }}}</math>
for the root but Bombelli is more concerned with better approximations for <math>r</math>. The value chosen for <math>a</math> is either of the whole numbers whose squares <math>n</math> lies between. The method gives the following convergents for <math>\sqrt{13}\ </math> while the actual value is 3.605551275... :
- <math> 3\frac{2}{3},\ 3\frac{3}{5},\ 3\frac{20}{33},\ 3\frac{66}{109},\ 3\frac{109}{180},\ 3\frac{720}{1189},\ \cdots</math>
The last convergent equals 3.605550883... . Bombelli's method should be compared with formulas and results used by Hero and Archimedes. The result <math>\frac{265}{153}<\sqrt{3}<\frac{1351}{780}</math> used by Archimedes in his determination of the value of <math>\pi \ </math> can be found by using 1 and 0 for the initial values of <math>r</math>. 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.
External links
- L'Algebra, original Italian text.
- O'Connor, John J; Edmund F. Robertson "Rafael Bombelli". MacTutor History of Mathematics archive.
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
