Rafael Bombelli

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Rafael Bombelli (1526–1573) 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 $\sqrt{n}$ sets $n=(a\pm r)^2=a^2\pm 2ar+r^2\$ with $0<r<1\$ from which it can be shown that $r=\frac{|n-a^2|}{2a\pm r}$ . Repeated substitution of the expression on the right hand side for $r$ into itself yields a continued fraction

$a\pm \frac{|n-a^2|}{2a\pm \frac{|n-a^2|}{2a\pm \frac{|n-a^2|}{2a\pm \cdots }}}$

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... :

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

The last convergent equals 3.605550883... . Bombelli's method should be compared with formulas and results used by Hero and Archimedes. The result $\frac{265}{153}<\sqrt{3}<\frac{1351}{780}$ used by Archimedes in his determination of the value of $\pi \$ can be found by using 1 and 0 for the initial values of $r$ .

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.

[edit] External link

Template:MacTutor Biofasf

[edit] 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