A paper by Hamilton entitled
Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Timeappeared in the Transactions of the Royal Irish Academy, volume 17 (1837), pp. 293-422.
This paper is available in the following formats:
There are three parts to this lengthy paper.
The first part, General Introductory Remarks, distinguishes between various approaches to algebra, namely `Practical', `Philological' and `Theoretical'. Hamilton then gives his reasons for contending that Algebra may be regarded as the Science of Pure Time.
The second part, On Algebra as the Science of Pure Time conceives of real numbers of ratios of steps between moments of time, and derives the basic properties of the algebra of real numbers from this conception. The existence of square roots of positive numbers, of $n$th roots, of exponentials, and of logarithms is discussed.
The third part, The Theory of Conjugate Functions, or Algebraic Couples, defines complex numbers as `algebraic couples', which are ordered pairs of real numbers, with appropriately defined operations of addition, subtraction, multiplication and division. Hamilton gives a careful proof of basic properties of the exponential function, and discusses the nature of logarithms, providing a natural framework to justify results obtained by his friend John T. Graves concerning complex logarithms which had been questioned by eminent British mathematicians such as Peacock and Herschel.