**Rafael Bombelli** was an Italian mathematician who wrote an influential algebra text and made free use of both negative numbers and complex numbers.

- Rafael received no university education.
- He was taught by an engineer- architect Pier Francesco Clementi so it is perhaps not too surprising that Bombelli himself should turn to that occupation.
- Bombelli found himself a patron in Alessandro Rufini who was a Roman noble, later to become the Bishop of Melfi.
- Scipione del Ferro, the first to solve the cubic equation was the professor at Bologna, Bombelli's home town, but del Ferro died the year that Bombelli was born.
- The contest between Fior and Tartaglia (see Tartaglia's biography) took place in 1535 when Bombelli was nine years old, and Cardan's major work on the topic Ars Magna Ⓣ(The great art) was published in 1545.
- Clearly Bombelli had studied Cardan's work and he also followed closely the very public arguments between Cardan, Ferrari and Tartaglia which culminated in the contest between Ferrari and Tartaglia in Milan in 1548 (see Ferrari's biography for details).
- From about 1548 Pier Francesco Clementi, Bombelli's teacher, worked for the Apostolic Camera, a specialised department of the papacy in Rome set up to deal with legal and financial matters.
- It is probable that Bombelli assisted his teacher Clementi with this project, but we have no direct evidence that this was the case.
- We certainly know that around 1549 Bombelli became interested in another reclamation project in a neighbouring region.
- It was in 1549 that Alessandro Rufini, Bombelli's patron, acquired the rights to reclaim that part of the marshes of the Val di Chiana which belonged to the Papal States.
- By 1551 Bombelli was in the Val di Chiana recording the boundaries to the land that was to be reclaimed.
- While Bombelli was waiting for the Val di Chiana project to recommence, he decided to write an algebra book.
- Only Cardan had, in Bombelli's opinion, explored the topic in depth and his great masterpiece was not accessible to people without a thorough grasp of mathematics.
- Bombelli felt that a self-contained text which could be read by those without a high level of mathematical training would be beneficial.
- By 1557, the work at Val di Chiana still being suspended, Bombelli had begun writing his algebra text.
- Suffice to say for the moment that, in 1560 when work at Val di Chiana recommenced, Bombelli had not completed his algebra book.
- The scheme was a great success and through the project Bombelli gained a high reputation as an hydraulic engineer.
- In 1561 Bombelli went to Rome but failed in an attempt to repair the Santa Maria bridge over the Tiber.
- However, with reputation still high, Bombelli was taken on as a consultant for a project to drain the Pontine Marshes.
- Several emperors and popes made unsuccessful attempts to reclaim the area but all, including the one which Bombelli acted as consultant on for Pope Pius IV, came to nothing.
- Antonio Maria Pazzi, who taught mathematics at the University of Rome, showed Bombelli a manuscript of Diophantus's Arithmetica and, after Bombelli had examined it, the two men decided to make a translation.
- Despite never completing the task, Bombelli began to revise his algebra text in the light of what he had discovered in Diophantus.
- Bombelli does not identify which problems are his own and which are due to Diophantus, but he does give full credit to Diophantus acknowledging that he has borrowed many of the problems given in his text from the Arithmetica.
- Bombelli's Algebra was intended to be in five books.
- Unfortunately Bombelli was never able to complete these last two volumes for he died shortly after the publication of the first three volumes.
- In 1923, however, Bombelli's manuscript was discovered in a library in Bologna by Bortolotti.
- Bortolotti published the incomplete geometrical part of Bombelli's work in 1929.
- Bombelli's Algebra gives a thorough account of the algebra then known and includes Bombelli's important contribution to complex numbers.
- Before looking at his remarkable contribution to complex numbers we should remark that Bombelli first wrote down how to calculate with negative numbers.
- In Bombelli's Algebra there is even a geometric proof that minus time minus makes plus; something which causes many people difficulty even today despite our mathematical sophistication.
- Bombelli was the first person to write down the rules for addition, subtraction and multiplication of complex numbers.
- Finally we should make some comments on Bombelli's notation.
- Bombelli, however, used quite sophisticated notation.
- Here are some examples of Bombelli's notation.
- Bombelli is the first to give a treatment of any complex numbers...
- It seems to be quite fair to describe Bombelli as the inventor of complex numbers.
- imaginaries had been used long before Bombelli's book, and it is therefore not quite justified to call him the "first discoverer" of complex numbers.

Born January 1526, Bologna, Papal States (now Italy). Died 1572, (probably) Rome, Papal States (now Italy).

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Algebra, Architecture, Origin Italy, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive