Chapter A.7, Problem 93AYU

To determine

To explain: The origin of complex numbers.

Explanation of Solution

Given information:

The origin of complex numbers.

Calculation:

Consider the search for the need of complex number, starts with finding the square root of a negative number while solving the equation of form $x^2+bx+c=0$ , when $b^2<4c$ . All efforts to assert were disregarded, while finding a meaningful solution to this situation. The situation changed towards the end of 15th century when two Italian mathematicians Del Ferro and Nicolo Fontana made progress towards solving cubic equations. Nicolo shared his achievement with his friend Cardano that further developed, with his student Lodovico Ferrari, this method and published both work in "Artismagnatesave de regulisalgebraicis liber unus" in 1545. This is considered the first published record on origin of complex numbers. Cardeno presented the formulae solving certain cubic and quadratic equations. Both involved the expressions involving square roots of negative numbers that led through the very alien territory of square roots of negative numbers. In year 1572, Rafael Bombelli published his work "L'algerbra", in which he suggested the rules for systematic computations for evaluate in a number of cases the expressions arising from Cardano's formulae for a root of cubic polynomials. Further several generations of mathematicians performed formula computations involving complex numbers. Surprising relations between real and complex numbers were discovered such as

  $\begin{array}{l}{i}^{-i}=e^{-\frac{\pi }{2}}\\ \cos x=\frac{1}{2}\left(e^{ix}+e^{-ix}\right)\end{array}$

However the discomfort about the nature of complex numbers persisted. Precursors to the interpretation of complex numbers points in plane are implicit in the work of Euler’s, John Wallis, Caspar Wessel and Jean Robert Argand. However real credit of clarifying the whole concept of complex numbers goes to Carl Friedrich Gau's 1831 publication in which he gave the idea of giving imaginary quantities a real existence. In 1835, the formal interpretation of complex numbers as an ordered pair of real numbers appeared for the first time in the work of Sir William Rowan Hamilton. To Hamilton, we also owe the introduction of the symbol i as the square root of -1.