History of complex numbers: Introduction

   Europe’s return to high ideals of ancient art and science at the end of the Middle Ages was called the Renaissance, which literally means Rebirth. This process began in Italy in the early XIV century. The XV century gave the world Leonardo da Vinci, Michelangelo and Raphael.

   Almost every prominent figure of the Renaissance was a man of wide scientific and artistic interests – an encyclopedist, as we would say today. Mathematics had a special place among the sciences, which not only was the foundation of a scientific outlook, as in ancient times, but the primary means for the improvement of arts and crafts. Think at least of the use of geometry in search of the laws of perspective for painting! Another important area of mathematics’ applications was harmonious proportions in sculpture and architecture. Leonardo da Vinci asks one of his close friends – the Franciscan monk Luca Pacioli (c. 1445 - c. 1515), already known then for his broad mathematical knowledge – to write a work on this subject. Pacioli complied, and his book "On Divine Proportion" with illustrations by Leonardo da Vinci was completed in 1497, and was published in Venice in 1509.

   More admirable, however, was the fate of the other previously created work by Luca Pacioli, which became a kind of mathematical encyclopedia of his time. Its comprehensive nature is mirrored in its very title: "The sum of knowledge in arithmetic, geometry, ratios and proportionality" (written in 1477, and published in Venice in 1494). It was one of the first printed books on mathematics. It obtained a wide circulation in Europe, and later became generally accepted as a "reference point" for all scientists who set out to further develop science.

   In its part on algebra, Pacioli’s "Sum" brings the reader to the solution of linear and quadratic equations, i.e. material which until now completes the initial course of algebra (here is algebra homework help service). As for equations of higher degrees, Pacioli, taking cubic equations by way of example, simply says that for these equations "the art of algebra has not yet found a way of solving, the same way as it has not found a way for squaring the circle." (Squaring the circle was a name for a geometric construction problem: With ruler and a pair of compasses, construct a square of an area, equal to the area of a given circle.

   The fact that this problem could not be solved for many centuries has long led up to the idea that it may not be solved, or rather that it’s impossible to perform the demanded construction by such simple means. But it was rigorously proven only in the XIX century.

Here is the next article History of complex numbers: cubic equations formula of the cycle.

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