
In mathematics, the complex numbers are the final stage in the search to extend the number system to obtain all the numbers that might be needed. The process began with the natural numbers, then the integers, then the rational numbers, then the algebraic numbers and then the real numbers. (This is the order in which they were discovered and used; today textbooks usually skip the algebraic numbers.) The real numbers are a field; the problem is that they are not algebraically closed (â€˜algebraically closedâ€™ means that every polynomial equation has a solution; in the real numbers, the polynomial equation x2 + 1 = 0 has none). The complex numbers are formed by adding in an extra number i, which is defined to be a solution of x2 + 1 = 0 (i is also a solution) it is the square root of 1. This number can be multiplied by any real number; this gives rise to numbers known (somewhat unfortunately) as imaginary. Complex numbers are those which are sums of real and imaginary ones, such as 3 + 4i.
It turns out that the complex numbers are algebraically closed; any polynomial equation has a complex number solution. This fact is known as the fundamental theorem of algebra, and shows that the complex numbers are as far as we need to go. The name this result was given shows its perceived importance; it ranks it with the fundamental theorem of the calculus discovered by Newton and Leibniz, which precipitated some of the most important developments leading to modern mathematics. However, the theorem has failed to live up to this promise; it is a direct consequence of some of the deeper theorems of complex analysis. Although it is often used, it has not really expanded the horizons of mathematics.
Complex analysis, mentioned above, was developed by AugustinLouis Cauchy (1789  1857). He sought to extend the methods of analysis of functions of real numbers, recently made rigorous by himself, , Karl Weierstrass (1815  1897) and , Bernard Bolzano (1781  1848). The definitions they came up with for such concepts as the limit were translated into the terms of complex numbers with much ease. The results of complex analysis are, however, very different from those of real analysis; they all stemmed from the theorem which is named after Cauchy, and some of them were very different from what was expected. The concepts of differentiation and integration are even more closely linked for complex numbers than they are for real numbers. Complex analysis is difficult, partly because the functions are hard to conceptualize.
It certainly seemed as though complex numbers would never be used for anything concrete they did, after all, involve imaginary quantities. Electrical engineers discovered that alternating currents and the properties of circuits associated with them could most easily be represented by the use of complex numbers. This is another example where mathematical research produced something apparently totally abstract that eventually turned out to be extremely practical. SMcL 
