History of complex numbers: Interpretation

   After Bombelli, the new "sophistic" numbers for a long time appeared in mathematical research only sporadically. The attitude to them was distrustful until their geometric interpretation became known. It was similar to the interpretation of the real (including negative) numbers on the number axis. (It should be noted that negative numbers also gained full "citizenship" status in mathematics only after their interpretation on the number axis was found).

    Such interpretation of the "sophistic" numbers - using points or two-dimensional vectors - was first carried out by the Danish land surveyor and mathematician Kaspar Wessel (1745 - 1818). But the book which contained this result was published in Copenhagen in the Danish language, and therefore did not gain widespread in European scientific circles, their "official" language being Latin). Wessel’s idea was first rediscovered by the Swiss Jean Argan (1768-1822), and then by the great Gauss. Gauss introduced the i character, and called the sophistic numbers complex numbers, thus removing an aura of mystery from them. Successful use of complex numbers in Gauss’s studies, and before him – the Swiss Leonhard Euler’s studies (1707 - 1783), contributed to the full recognition of complex numbers as an effective and convenient tool for mathematical research.

   Despite the fact that the operations with complex numbers are no more complex than operations with conventional (real) numbers, before Gauss complex numbers were considered to be mysterious, almost mystical entities. Among scientists, an intense debate between supporters and opponents of "imaginary" numbers was underway. The main objections of opponents focused on the fact that the very expression was devoid of meaning, since i was not a number. To put the theory of complex numbers upon a stable foundation, a geometric interpretation of these numbers was needed, similar to the existing interpretation of real numbers on the number axis. This was the interpretation found by Gauss.

   Upon a closer look Gauss’s idea in modernized form is that an arbitrary complex number is represented by a point  on the coordinate plane, the abscissa of which is equal to the real part of a and the ordinate to the imaginary part of b. Thus each complex number corresponds to a single point on the plane, and every point of the plane can be associated with a single complex number. That is, such a correspondence between the complex numbers and the points on the coordinate plane is mutually identical. The real numbers are represented along the horizontal axis. Therefore, this axis is called the real axis. Purely imaginary numbers are represented by points on the vertical axis. Therefore, this axis is called the imaginary axis. The origin of coordinates corresponds to the number 0.

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