John was the eldest son of John Warren and Elizabeth Crooke, born on 4 October 1796, at Bangor, Caernarvonshire, Wales.
He was educated at Westminster school and Jesus College, Cambridge, of which he was a fellow and tutor.
In 1818 he was fifth wrangler, and in 1825 and 1826 served the office of moderator and examiner.
(Wrangler is the name given in the University of Cambridge to those who have attained the first class in the public mathematical honours examination.)
In 1830 he was elected a fellow of the Royal Society.
In 1828 he published at Cambridge "A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities", a subject which had previously attracted the attention of Wallis, Professor Heinrich Kühn of Danzig, M. Buée, and M. Mourey, whose researches were, however, unknown to Warren.
The work bears evident marks of originality, and has received honourable mention as well from continental as from English Mathematicians.
The title hardly conveys an exact idea of the main object, which is to represent every kind of quantity geometrically by the intervention of Symbolical expressions, which involve the square roots of negative quantities, and designate lines in position as well as magnitude.
He was strongly convinced of the superiority of geometry as a means of demonstration to the use of mere symbols of quantity, and thought that the obscurity attaching to the proofs of some of the fundamental rules of
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algebraic and analytical operations might be removed by adopting a geometrical representation of quantity such as he proposed.
On 19 February 1829, Warren read a paper before the Royal Society entitled "Considerations of the Objections raised against the Geometrical Representations of the Square Roots of Negative Quantities", which was followed on the 4th of June by another "On the Geometrical Representation of the Powers of Quantities whose Indices involve the Square Roots of Negative Quantities", in which he came to the conclusion "that all algebraic quantity may be geometrically represented both in length and direction, by lines drawn in a given plane from a given point".
Warren was chancellor of the diocese of Bangor and Rector of Graveley in Cambridgeshire, and of Caldecott in Huntingdonshire.
He owned the advowson of the latter, which, as well as an adjoining parish, was without a resident clergyman.
To remedy this evil he proposed to unite two parishes.
He sold the advowson of Caldecott to the patron of the other parish, and gave the purchase-money to build a parsonage for the united parishes - an incident characteristic of the man.
He married his cousin, Caroline Elizabeth, daughter of Captain and Lieutenant-colonel Richard Warren of the 3rd foot guards.
He died at Bangor on 15 August 1852, without issue.
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