William R. Hamilton developed a theory of conjugate functions, or algebraic couples, in which complex numbers are constructed as ordered pairs of real numbers.

Hamilton's friend from college days, John T. Graves, had developed a theory of imaginary logarithms, which was published in the Philosophical Transactions of the Royal Society of London for the year 1829. This theory was not generally accepted, and George Peacock (1791-1858) had criticised it in his Report on Algebra to the British Association for the Advancement of Science in 1833.

According to Hamilton, it was on meditation on those results of Graves that he was led to his theory of conjugate functions. He presented his theory at the meeting of the British Association for the Advancement of Science at Edinburgh in 1834 in a report entitled On conjugate functions, or algebraic couples, as tending to illustrate generally the doctrine of imaginary quantities, and as confirming the results of Mr Graves respecting the existence of two independent Integers in the complete expression of an imaginary logarithm.

Hamilton presented his theory more fully in a lengthy paper entitled Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time printed in the Transactions of Royal Irish Academy, Vol 17 (1837), pp. 293-422.

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