The Theory of Equations
History shows the necessity for the invention of new numbers in the orderly progress of civilization and in the evolution of maths. We must review briefly the growth of the number system in the light of the theory of equations and see why the complex number system need not be enlarged further. Suppose we decide that we want all polynomial equations to have roots. Now let us imagine that we have no numbers in our possession except the natural numbers. Then a simple linear equation like However, a simple quadratic equation like G. Cardano (1545) is credited with some progress in introducing complex numbers in his solution of the cubic equation, even though he regarded them as "fictitious". He is credited also with the first use of the square root of a negative number in solving the now-famous problem, "Divide 10 into two parts such that the product... is 40", which Cardano first says is "manifestly impossible"; but then he goes on to say, in a properly adventurous spirit, "Nevertheless, we will operate." Thus he found
and so forth. Now, we may well expect that there may be some equation of degree 3 or higher which has no roots even in the entire system of complex numbers. That this is not the case was known to K. F. Gauss, who proved in 1799 the following theorem, the truth of which had long been expected: Every algebraic equation of degree n withcoefficient in the complex number system has a root (and hence n roots) among the complex numbers, later Gauss published three more proofs of the theorem. It was he who called it "fundamental theorem of algebra". Much of the work on complex number theory is Gauss'. He was one of the first to represent complex numbers as points in a plane. Actually, Gauss gave four proofs for the theorem, the last when he was seventy; in the first three proofs, he assumes, the coefficients of the polynomial equation are real, but in the fourth proof the coefficients are any complex numbers. We can be sure now that for the purpose of solving polynomial equations we do not need to extend the number system any further.
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