Hardy left the cab. He was visiting an ill Ramanujan at Putney. Ramanujan asked about his trip. Hardy remarked the ride was dull; even the taxi number, 1729, was dull to the number theorist. He hoped the number’s dullness wasn’t an omen predicting Ramanujan’s declining health. “No Hardy,” Ramanujan replied, “it is a very interesting number.” The integer 1729 is actually the smallest number expressible as the sum of two positive cubes in two different ways. Indeed 1729 = 13 + 123 = 93 + 103.
The mathematicians Srinivasa Iyengar Ramanujan and G.H. Hardy comprise the characters of this story. Ramanujan’s casual discovery of the smallest so-called taxicab number was no fluke. Ramanujan and Hardy’s most famous result was an asymptotic formula for the number of partitions of a positive integer. A partition of a number n is a way of writing n as the sum of positive integers. Reordering terms doesn’t change a partition. Thus the partitions of 5 are 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1; hence 5 has 7 seven partitions. Counting the partitions becomes difficult as n grows. Ramanujan had made several conjectures based on numerical evidence, and Hardy credits many of the needed insights to Ramanujan. The formula is complicated and counter-intuitive. It involves values of √3, π, and e, all very strange numbers for a counting formula. Near his death, Ramanujan discovered mock theta functions, which mathematicians are still “rediscovering” today.
These are only the achievements of later Ramanujan. The early life of Ramanujan is even more surprising. He spent his childhood in the poor south Indian town, Kumbakonam. In school, he scored top marks. He probed college students for mathematical knowledge by age 11. When he was 13, he comprehended S.L. Loney’s advanced trigonometry book. He could solve cubic and quartic equations—the latter method he found himself—and finished math tests in half the allotted time. If asked, he could recite the digits of π and e to any number of digits. He borrowed G.S. Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics by age 16 and worked through its many theorems. By 17, he independently investigated the Bernoulli numbers, a set of numbers intimately connected to number theory.
So How did Ramanujan do it? No brilliant teacher—excluding Hardy and J.E. Littlewood when Ramanujan was already an adult—taught Ramanujan. The odds were against Ramanujan from the start. Ramanujan often used slate instead of paper because paper was expensive; he even erased slate with his elbows, since finding a rag would take too long. He lost his first college scholarship by neglecting his every subject that wasn’t math. The first two English mathematicians he sent letters to request publication did not respond; was it luck that Hardy did?
Ramanujan credits Namagiri, a family deity, for his mathematics. He claimed the goddess would write mathematics on his tongue, that dreams and visions would reveal the secrets of math to him.
Ramanujan was undoubtedly religious. Before leaving India, he respected all the holy customs of his caste: he shaved his forehead, tied his hair into a knot, wore a red U with a white slash on his forehead, and refused to eat any meat. In the Sarangapani temple, Ramanujan would work on his math in his tattered notebook.
Hardy, the ardent atheist, didn’t believe that gods communicated with Ramanujan. He thought flashes of insight were just more common in Ramanujan than in most other mathematicians. Sure, Ramanujan had his religious quirks, but they were just quirks.
Looking at Ramanujan’s work, I find it hard to deny that some god aided the man. I cannot even imagine Ramanujan’s impact had he lived longer or had better teachers.
Sources and cool stuff:
biographical stuff: http://www-history.mcs.st-andrews.ac.uk/Biographies/Ramanujan.html, Christopher Syke’s documentary Letter’s from and Indian Clerk, Robert Kanigel’s The Man Who Knew Infinity
Ramanujan and Hardy’s original paper on partitions: http://plms.oxfordjournals.org/content/s2-17/1/75.full.pdf
Numberphile on taxicab numbers: https://www.youtube.com/watch?v=LzjaDKVC4iY
Wolfram on taxicab numbers: http://mathworld.wolfram.com/TaxicabNumber.html
Wolfram on Hardy-Ramanujan number: http://mathworld.wolfram.com/Hardy-RamanujanNumber.html
Wolfram on Bernoulli numbers: http://mathworld.wolfram.com/BernoulliNumber.html
Ramanujan’s papers and notebooks: http://www.imsc.res.in/~rao/ramanujan/contentindex.html
mock theta functions: http://www.ams.org/notices/201011/rtx101101410p.pdf
G.S. Carr’s book: http://books.google.com/books?id=JLmCAAAAIAAJ&hl=en
S.L. Loney’s book: https://archive.org/details/planetrigonomet00lonegoog
Numberphile on Ramanujan constant: https://www.youtube.com/watch?v=DRxAVA6gYMM