Rafael Bombelli
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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.
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[edit] Life
Rafael Bombelli was born in January of 1526 in Bologna, Italy. He was born to Antonio Mazzoli, a wool merchant, and Diamante Scudieri, a tailor's daughter. Rafael's father changed his name to Bombelli because of the bad reputation of the Mazzoli family. Once, the Mazzoli family was quite powerful in Bologna, but then Pope Julius II came to power and exiled the ruling family, the Bentivoglios, in 1506. There was then an attempt in 1508 to take back control of Bologna, but it failed. Among the participants of the attempted coup was Rafael's grandfather, who was captured and executed. Some time later, Antonio Bombelli was able to come back to Bologna as a wool merchant, and without the Mazzoli surname, he wasn't bothered. Besides Rafael, there were five other children in the family, of which Rafael was the oldest. Rafael received no college education, but was instead taught by an engineer-architect by the name of Pier Francesco Clementi.
Throughout his early life, Bombelli felt that many of the arguments amongst the leading mathematicians of the day was a lack of a careful and thorough exposition of the subject. Therefore, Rafael took it upon himself to write a book about algebra that would clear things up. In fact, he decided to not just write another convoluted mathematical book that only those on the highest echelons of the subject could comprehend, but a book that could be understood by anyone. This text would be self-contained and easily understandable by those without higher education.
Rafael Bombelli died in 1572 in Rome, Italy.
[edit] Bombelli's Algebra
In the book that he wrote in 1579, entitled Algebra, Bombelli gave a comprehensive account of the algebra known at the time. He was the first to write down how to perform computations with negative numbers. The following is an excerpt from the text:
"Plus times plus makes plus
Minus times minus makes plus
Plus times minus makes minus
Minus times plus makes minus
Plus 8 times plus 8 makes plus 64
Minus 5 times minus 6 makes plus 30
Minus 4 times plus 5 makes minus 20
Plus 5 times minus 4 makes minus 20"
As was intended, Bombelli used simple language as can be seen above so that anybody could understand it. But at the same time, he was thorough.
Perhaps more importantly than his work with algebra, however, the book also includes Bombelli's monumental contributions to complex number theory. Before he writes about complex numbers, he points out that they occur in solutions of equations of the form x^3 = ax + b, given that (a/3)^3 > (b/2)^2, which is another way of stating that the discriminant of the cubic is negative. The solution of this kind of equation requires taking the cube root of some number and adding the square root of some negative number.
Before Bombelli delves into using imaginary numbers practically, he goes into a detailed explanation of the properties of complex numbers. Right away, he makes it clear that the rules of arithmetic for imaginary numbers are not the same as for real numbers. This was a big accomplishment, as even numerous subsequent mathematicians were extremely confused on the topic.
Bombelli avoided confusion by giving a special name to square roots of negative numbers, instead of just trying to deal with them as regular radicals like other mathematicians did. This made it clear that these numbers were neither positive nor negative. This kind of system avoids the confusion that Euler encountered. Bombelli called the imaginary number i “plus of minus” or “minus of minus” for -i.
Bombelli had the foresight to see that imaginary numbers were crucial and necessary to solving quartic and cubic equations. At the time, people cared about complex numbers only as tools to solve practical equations, not because of how amazing complex numbers really were. As such, Bombelli was able to get solutions using Scipione del Ferro's rule, even in the irreducible case, where other mathematicians such as Cardano had given up.
In his book, Bombelli explains complex arithmetic as follows:
"Plus by plus of minus, makes plus of minus.
Minus by plus of minus, makes minus of minus.
Plus by minus of minus, makes minus of minus.
Minus by minus of minus, makes plus of minus.
Plus of minus by plus of minus, makes minus.
Plus of minus by minus of minus, makes plus.
Minus of minus by plus of minus, makes plus.
Minus of minus by minus of minus makes minus."
After dealing with the multiplication of real and imaginary numbers, Bombelli goes on to talk about the rules of addition and subtraction. He is careful to point out that real parts add to real parts, and imaginary parts add to imaginary parts.
[edit] Accomplishments
In honor of his accomplishments, a moon crater was named after Bombelli.
Bombelli is generally regarded as the inventor of complex numbers, as no one before him had made rules for dealing with such numbers, and no one believed that working with imaginary numbers would have useful results. Upon reading Bombelli's Algebra, Leibniz praised Bombelli as an ". . . outstanding master of the analytical art." Crossley writes in his book, "Thus we have an engineer, Bombelli, making practical use of complex numbers perhaps because they gave him useful results, while Cardan found the square roots of negative numbers useless. Bombelli is the first to give a treatment of any complex numbers. . . It is remarkable how thorough he is in his presentation of the laws of calculation of complex numbers. . ."
[edit] Bombelli method
Bombelli used a method related to continued fractions to calculate square roots. His method for finding 
begins with 
with 
, from which it can be shown that 
. Repeated substitution of the expression on the right hand side for r into itself yields a continued fraction
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 
while the actual value is 3.605551275... :
The last convergent equals 3.605550883... . Bombelli's method should be compared with formulas and results used by Heros and Archimedes. The result 
used by Archimedes in his determination of the value of 
can be found by using 1 and 0 for the initial values of r.
[edit] External links
- L'Algebra, original Italian text.
- O'Connor, John J.; Robertson, Edmund F., "Rafael Bombelli", MacTutor History of Mathematics archive.
- Background
[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 0-486-64690-4Template:Italy-scintist-stub
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