*Bryan's list of mathematical theorems, lemmas, conjectures, inequalities and proofs

An attempt to extend "A Synopsis of Elementary Results in Pure and Applied Mathematics: Containing Propositions, Formulae, and Methods of Analysis with Abridged Demonstrations." (G. S. Carr, 1880)

Srinivasa Ramanujan Iyengar (22 December 1887 – 26 April 1920) was an Indian mathematician who is widely regarded as one of the greatest mathematical minds in recent history.

His father was an accountant to a cloth merchant at Kumbakonam, while his mother, a woman of "strong common sense," was the daughter of a Brahman petty official in the Munsiff's (or legal judge's) Court at Erode. For some time after her marriage she had no children, "but her father prayed to the famous goddess Namagiri, in the neighboring town of Namakkal, to bless his daughter with offspring. Shortly afterwards, her eldest child, the mathematician Ramanujan, was born on 22nd December 1887."

When he was 15 and in the sixth form at school, a friend of his secured for him the loan of Carr's Synopsis of Pure Mathematics from the library of the local Government College. Through the new world thus opened to him Ramanujan ranged with delight. It was this book that awakened his genius. He set himself at once to establishing its formulae. As he was without the aid of other books, each solution was for him a piece of original research. He first devised methods for constructing magic squares. Then he branched off to geometry, where he took up the squaring of the circle and went so far as to get a result for the length of the equatorial circumference of the earth which differed from the true length by only a few feet. Finding the scope of geometry limited, he turned his attention to algebra. Ramanujan used to say that the goddess of Namakkal inspired him with the formulae in dreams. It is a remarkable fact that, on rising from bed, he would frequently note down results and verify them, though he was not always able to supply a rigorous proof. The pattern repeated itself throughout his life.

His work was the work from which most of us would shrink. There's admiration there, but maybe a wisp of derision, too--as if in wonder that Ramanujan, of all people, could stoop so willingly to the realm of the merely arithmetical. And yet, Ramanujan was doing what great artists always do--diving into his material. He was building an intimacy with numbers, for the same reason that the painter lingers over the mixing of his paints, or the musician endlessly practices his scales. And his insight profited. For him, it wasn't what his equation stood for that mattered, but the equation itself, as pattern and form. And his pleasure lay not in finding in it a numerical answer, but from turning it upside down and inside out, seeing in it new possibilities, playing with it as the poet does words and images, the artist color and line, the philosopher ideas. Ramanujan's world was one in which numbers had properties built into them. Chemistry students learn the properties of the various elements, the positions in the periodic table they occupy, the classes to which they belong, and just how their chemical properties arise from their atomic structure. Numbers, too, have properties which place them in distinct classes and categories. Ramanujan was an artist. And numbers--and the mathematical language expressing their relationship--were his medium ... Ramanujan's was no cool, steady Intelligence, solemnly applied to the problem at hand; he was all energy, animation, force. He had a determination to succeed and to sacrifice everything in the attempt. That could be a prescription for an unhappy life; certainly for a life out of balance, sneering at timidity and restraint. Sometimes, as Ramanujan sat or squatted on the pial, he'd look up to watch the children playing in the street with what one neighbor remembered as 'a blank and vacant look.' But inside, he was on fire.

Algebra

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