This thesis uses primary and secondary sources to study advances in complex number theory during the 17th and 18th Centuries. Some space is also given to the early 19th Century. Six questions concerning their rules of operation, usage, symbolism, nature, representation and attitudes to them are posed in the Introduction. The main part of the thesis quotes from the works of Descartes, Newton, Wallis,
Saunderson, Maclaurin, d'Alembert, Euler, Waring, Frend, Hutton, Arbogast, de Missery, Argand, Cauchy, Hamilton, de Morgan, Sylvester and others, mainly in chronological order, with comment and discussion. More attention has been given tp algebraists, the originators of most advances in complex numbers, than to writers in trigonometry, calculus and analysis, who tended to be users of them. The last chapter summarises the most important points and considers the extent to which the six questions have been resolved. The most important developments during the period are identified as follows:
(i) the advance in status of complex numbers from 'useless' to
'useful'.
(ii) their interpretation by Wallis, Argand and Gauss in arithmetic, geometric and algebraic ways.
(iii) the discovery that they are essential for understanding
polynomials and logarithmic, exponential and trigonometric
functions.
(iv) the extension of trigonometry, calculus and analysis into
the complex number field.
(v) the discovery that complex numbers are closed under exponentiation, and so under all algebraic operations.
(vi) partial reform of nomenclature and symbolism.
(vii) the eventual extension of complex number theory to n dimensions
