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Most Indian students meet G. H. Hardy as a supporting character in Srinivasa Ramanujan's story: the Cambridge professor who received his letter in 1913 and brought him to England. Hardy's own career is worth a look too. He wrote a textbook that changed how calculus was taught in English, ran one of the longest partnerships in mathematics, and, almost by accident, wrote down a rule that every biology student now learns. This post covers all three, with a worked example from his genetics letter.
Who was G. H. Hardy?
The MacTutor biography from the University of St Andrews gives his dates as 7 February 1877, at Cranleigh in Surrey, to 1 December 1947, in Cambridge. He went to Winchester College and then Trinity College, Cambridge, and was placed fourth wrangler in the Mathematical Tripos of 1898, a result that continued to annoy him even though he thought the system silly. He was honest about his early motives. In his later book A Mathematician's Apology (1940), he wrote: "I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively."
What changed his mind was a teacher. His tutor A. E. H. Love told him to read Camille Jordan's Cours d'analyse, and Hardy wrote in the Apology that he learnt from it "for the first time as I read it what mathematics really meant".
Why did his textbook matter?
In 1908 Hardy published A Course of Pure Mathematics. The historian J. C. Burkill, quoted by MacTutor, called it "the first rigorous English exposition of number, function, limit, and so on, adapted to the undergraduate", and said it transformed university teaching. The third edition is free on Project Gutenberg. It carries forward the careful definitions of limits that began with Cauchy; our post on Augustin-Louis Cauchy covers that earlier step.
How did Hardy collaborate?
In 1911 he began working with J. E. Littlewood, a collaboration that lasted 35 years. MacTutor says Hardy acknowledged Littlewood's greater technical skill, while Hardy brought insight and a gift for writing their results up clearly. Then, in early 1913, came Ramanujan's letter from Madras. Hardy brought him to Cambridge, and they wrote five remarkable papers together. Our post on Ramanujan's notebooks tells that side of the story.
One of those papers, Asymptotic Formulae in Combinatory Analysis (1918), tackled partitions: the number of ways p(n) of writing n as a sum of positive whole numbers. For n = 4 there are five: 4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1, so p(4) = 5. For n = 5 there are seven, so p(5) = 7. The values grow very fast, and Hardy and Ramanujan found a formula that approximates p(n) for large n.
How did a pure mathematician end up in biology?
In 1908 Hardy wrote a one-page letter to the journal Science, "Mendelian Proportions in a Mixed Population". It begins: "I am reluctant to intrude in a discussion concerning matters of which I have no expert knowledge, and I should have expected the very simple point which I wish to make to have been familiar to biologists."
The statistician Udny Yule had suggested that if a trait such as brachydactyly (short fingers) is dominant, then over time about three people in four should show it. Hardy showed why that expectation was "quite groundless". Suppose the genotypes AA, Aa and aa occur in the ratio p : 2q : r, mating is random, and all types are equally fertile. Then the next generation is in the ratio (p + q)² : 2(p + q)(q + r) : (q + r)², and after that the proportions stay the same.
Worked example, using Hardy's own numbers. Start with pure brachydactylous people (AA) and normal people (aa) in the ratio 1 : 10,000, so p = 1, q = 0, r = 10,000.
p + q = 1 and q + r = 10,000.
Next generation: AA = 1² = 1; Aa = 2 × 1 × 10,000 = 20,000; aa = 10,000² = 100,000,000.
If brachydactyly is dominant, the people who show it are AA plus Aa: 1 + 20,000 = 20,001 out of a total of 100,020,001.
That is practically 2 in 10,000, twice the first generation, and Hardy showed it would "afterwards have no tendency whatever to increase". A dominant trait does not spread just because it is dominant.
The modern form makes NEET-style questions quick. If 1 person in 100 shows a recessive trait, then q² = 0.01, so q = 0.1 and p = 0.9. Carriers are 2pq = 2 × 0.9 × 0.1 = 0.18, about 18 people in 100. MacTutor notes that Hardy thought the result unimportant, though it proved important in work on blood group distribution. Our guides to evolution for NEET and genetics problems have more practice.
What else marked his career?
He moved to Oxford as Savilian professor of geometry in 1919 and returned to Cambridge as Sadleirian professor in 1931. MacTutor lists his honours from the Royal Society: Fellow in 1910, the Royal Medal in 1920, the Sylvester Medal in 1940 and the Copley Medal in 1947. In the Apology he wrote: "Beauty is the first test: there is no permanent place in the world for ugly mathematics."
What can a student take from Hardy's way of working?
Keep a fixed working block. MacTutor describes his routine: research from 9 o'clock until 1 o'clock, then the afternoon off for cricket. A protected block of focused work each day beats scattered hours.
Settle an argument with one clean example. His genetics letter needed only a ratio and a page of arithmetic.
Collaborate with people whose strengths differ from yours. He paired his clear writing with Littlewood's technique and Ramanujan's intuition.
Be honest about what you do not know. His letter opens by admitting he is not a biologist, and then makes its point anyway.
For genetics ratios that refuse to come out, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer, so you can check each line of a genotype-frequency calculation.
Try Hardy's example yourself with a starting ratio of 1 : 100 and see what share shows the dominant trait in the next generation.
