History of complex numbers: Closeness

   Another interesting mathematical problem is associated with the algebraic closeness of the set of complex numbers (C).

    Let us recall that the complex numbers first arose from the efforts to make possible the operation of extracting the square root of negative numbers. But the introduction of complex numbers made it possible to solve any algebraic equation of the 1st, 2nd, 3rd and 4th degrees. The impossibility, according to the Abel-Ruffini theorem, of solving general algebraic equations of n-th degree with n > 5 by radicals might suggest that for solving such equations, it is necessary to expand the set C. But as it turns out, however, no extensions are longer necessary, that is, that all the roots of any algebraic equation with complex (or, in particular, real) coefficients belong to the set C. Thus, for solving such equations, no other numbers, besides the complex ones, are required. This property is called the algebraic closeness of the set C. It was first noticed by the Dutch mathematician Albert Girard (1595- 632) in 1629. But the first rigorous proof was obtained more than half a century later by Gauss.

   Further mathematical research of problems associated with complex numbers concerned summarizing these very numbers as algebraic objects no longer in direct connection with the theory of algebraic equations. In particular, the Irish mathematician William Rowan Hamilton (1805-1865), engaged in this line of study, introduced quaternions, which, in turn, gave life to vector calculus (we can solve vector calculus assignment).

Unfortunately, this is the last article in this cycle. Hope, you've enjoyed it.

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