He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the underlying principles of which he formalized.

As his father and grandfather had both been born in India, De Morgan used to say that he was neither English nor Scottish nor Irish, but a Briton "unattached," using the technical term applied to an undergraduate of Oxford or Cambridge who was not a member of any one of the colleges.

[2] His mathematical talents went unnoticed until he was fourteen when a family friend discovered him making an elaborate drawing of a figure from one of Euclid's works with a ruler and compasses.

Following a series of squabbles between the faculty, including De Morgan, and the administration, in particular the Warden, Leonard Horner, a dispute arose over the handling of medical student protests calling for the removal of the Professor of Anatomy, Granville Sharp Pattison, on the grounds of incompetence.

[15][16] In 1826 Lord Brougham, one of the founders of London University, founded the Society for the Diffusion of Useful Knowledge (SDUK) with the goal of promoting self-education and improving the moral character of the middle- and working- classes through cheap and accessible publications.

[17] De Morgan became involved with the SDUK in March 1827; his unpublished manuscript Elements of Statics for the society may have played a role in his appointment to London University.

[10] One of its most voluminous and effective writers, De Morgan published several books with SDUK: On the Study and Difficulties of Mathematics (1831),[14] Elementary Illustrations of the Differential and Integral Calculus (1832), The Elements of Spherical Trigonometry (1834), Examples of the Processes of Arithmetic and Algebra (1835), An Explanation of the Gnomic projection of the sphere (1836),[18] The Differential and Integral Calculus (1842),[19] and The Globes Celestial and Terrestrial (1845),[20] as well as over 700 articles in the Penny Cyclopedia and contributions to the Quarterly Journal of Education, the Gallery of Portraits, and the Companion to the British Almanac.

[26] He published several articles on actuarial subjects as well as the book An Essay on Probabilities and Their Application to Life Contingencies and Insurance Offices.

Many of his non-mathematician students rated him highly; William Stanley Jevons described De Morgan as "unrivalled" as a teacher.

[13][32] Jevons, heavily influenced by De Morgan, would go on to do independent work in logic and become best known for the development of the theory of utility as part of the so-called Marginal Revolution.

[33][34] In 1866, the Chair of Mental Philosophy and Logic at University College fell vacant and James Martineau was recommended formally by the Senate to the Council.

While symbolical algebra could mechanically construct negative and imaginary numbers, as in the work of Adrien-Quentin Buée [fr], Jean-Robert Argand, and John Warren, it could not provide their interpretation; De Morgan observed that a similar problem troubled the classical Indian mathematician Bhāskara II in his work Bijaganita.

On the one hand, argued by William Whewell, logic, particularly syllogism as emphasized by Whately, could not arrive at "new truths" and was therefore inferior to and distinct from scientific reasoning; on the other hand, argued by the Scottish philosopher Sir William Hamilton, Whately's effort to equate logic to a "grammar for reasoning" was wrong and reductive.

[49][28] Recruited by Elizabeth Jesser Reid, in 1849 De Morgan taught mathematics for one year at the newly founded Ladies College in Bedford Square.

[50] In 1850 De Morgan received a book from John Elliot Drinkwater Bethune, A Treatise on Problems of Maxima and Minima, written and self-published by the self-taught Indian mathematician Ramchundra.

De Morgan was so struck by the work that he entered into correspondence with Ramchundra and arranged for the book's re-publication in London in 1859, targeting a European audience; De Morgan's preface surveyed classical Indian mathematical thought and urged a contemporary return of Indian mathematics:[51][52][28] On examining this work I saw in it, not merely merit worthy of encouragement, but merit of a peculiar kind, the encouragement of which, as it appeared to me, was likely to promote native effort towards the restoration of the native mind in India.

On the occasion of the installation of his friend, Lord Brougham, as Rector of the University of Edinburgh, the Senate offered to confer on him the honorary degree of LL.

He humorously described himself using the Latin phrase 'Homo paucarum literarum' (man of few letters), reflecting his modesty about his extensive contributions to mathematics and logic.

He disliked the provinces outside London, and while his family enjoyed the seaside and men of science were having a good time at a meeting of the British Association in the country, he remained in the hot and dusty libraries of the metropolis.

[58] Despite a strict Church of England upbringing[59] De Morgan was publicly a non-conformist, at some personal cost: His refusal to conform debarred him from further advancement at Cambridge; his marriage was without Church ceremony;[60] and on several occasions he fought with the University College administration to maintain religious neutrality,[61] eventually resigning over the issue.

[62] In private De Morgan was a dissenter: He married into a Unitarian family, where his essentially Christian deist interpretations of scripture were welcome.

De Morgan's first original paper on logic, "On the structure of the syllogism",[45] appeared in the Transactions of the Cambridge Philosophical Society in 1846.

[70] The subsequent dispute with the philosopher Sir William Stirling Hamilton over the "quantification of the predicate" referred to in De Morgan's paper would lead George Boole to write the pamphlet Mathematical Analysis of Logic (1847).

De Morgan elaborated upon his initial paper in the book Formal Logic, or the Calculus of Inference, Necessary and Probable (1847),[71] published the same week as Boole's pamphlet and was immediately overshadowed by it.

[75] The calculus was described as the logic of relatives by Charles Sanders Peirce, who admired De Morgan and met him shortly before his death.

[76] In fact, a theorem articulated by De Morgan in 1860 was later expressed by Schrŏder in his textbook on binary relations, and is now commonly called Schröder rules.

While these papers are perhaps most notable for their influence on Sir William Rowan Hamilton and the development of quaternions,[13][77] they are also recognized to contain De Morgan's steps towards a fully abstract algebra:"Inventing a distinct system of unit-symbols, and investigating or assigning relations which define their mode of action on each other".

De Morgan was also a well known popularizer of science and mathematics; he contributed over 600 articles to the Penny Cyclopedia, ranging from Abacus to Young, Thomas.

One such angle-trisector was James Sabben, whose work received a one-line review from De Morgan: "The consequence of years of intense thought": very likely, and very sad.

De Morgan gives a favorable review of Godfrey Higgins' Anacalypsis[96] and provides several anecdotes about the views of great mathematicians on religion, notably Laplace[97] and Euler.