Math

Technology in Mathematics Education

Spring 2007, Volume 7, Issue 1

**Delights from Ramanujan**

Srinivasa Aiyangar Ramanujan (1887 – 1920) was an Indian mathematician who is widely considered among the most talented mathematicians in the heuristic aspects of number theory and insight into modular functions because of his originality and insight. He also made significant contributions to the development of partition functions and summation formulas involving constants such as p. A child prodigy, he was largely self-taught in mathematics and had compiled over 3,000 theorems between 1914 and 1918 at the University of Cambridge. However, Ramanujan was truly a self-taught person and never sought any degree from Cambridge. Often, his formulas were merely stated, without proof, and were only later proven to be true. His results were highly original and unconventional, and have inspired a large amount of research and many mathematical papers; however, some of his discoveries have been slow to enter the mathematical mainstream. Recently his formulae have started to be applied in the field of crystallography, and other applications in physics.

Ramanujan's series for p converges extraordinarily rapidly (exponentially) and forms the basis for the fastest algorithms currently used to calculate p.

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Editor: Scott S Albert

Meditating Mathematicians

M^{2}

Critical Thinking in Mathematics Education

where p is every prime

+ ... =

Ramanujan has been termed a "magical genius." In contrast, "ordinary geniuses" are merely an order of magnitude of two smarter than you and me. In Ramanujan's case, no one knows where his voluminous results came from. They appeared as if by magic, in a manner transcending ordinary human mental activity.

Ramanujan did complete high school, but his entire mathematical education seems to have come from the reading of just two books. Nevertheless, he was invited to Cambridge on the basis of a letter he wrote to G.H. Hardy in 1913. The letter contained about 60 theorems and formulas stated without proof. After some study, Hardy concluded that Ramanujan's results must be true be cause, "if they were not true, no one would have had the imagination to invent them."

Ramanujan lived for mathematics. He would work 24-36 hours and then collapse. He died in 1919, leaving behind three notebooks crammed with some 4000 "results," again stated without proof and again seeming to come from no where. Step by step, his results are being proved. Ramanujan evidently saw their truth without going through laborious proofs.