# G. H. Hardy

Hardy was an important British mathematician who produced some highly original work on prime number theory and partitions, among many other areas of research. He is known for the discovery of a remarkable young Indian mathematician, Srinivasa Ramanujan,with whom he createdmany important papers in a short period of time. Hardy is also famous for his collaboration with fellow Britishmathematician*John Edensor* Littlewood.

**Early life and family**

Godfrey Harold Hardy was born on February 7, 1877, in Cranleigh, Surrey, England. His father, Isaac Hardy, worked at the local Cranleigh school as a bursar and an art master. Isaac was a gentle, kind individual but had great ambitions for his son. The highly disciplined environment of his childhood was also shaped by his mother Sophia. She was very strict and had worked as a teacher at Lincoln Teacher’s Training School. His parents were both very bright individuals with strong interests in mathematics and music. Indeed, G. H. Hardy showed mathematical promise from the age of two when he was already able to write numbers beyond one million. His mother was very religious and Hardy had to attend church services. However, he was uninterested in religion and often spent his time calculating the various factors of hymn numbers.

Later in 1890, Hardy left Cranleigh School after he had won a scholarship to Winchester College a public school with an outstanding reputation for teaching mathematics. Hardy ensured rapid development of his mathematical abilities by focussing purely on his studies. While he was fiercely competitive in an academic sense, he was a quiet, reserved boy who was uninterested in social activities of any kind. Winchester provided the ideal preparation for his entry into Trinity College, Cambridge in 1896. At Cambridge, Hardy was coached by Robert Rumsey Webb who had a strong reputation for preparing students for the Cambridge Mathematical Tripos examination. However, while Hardy admitted that in his youth he was less interested in mathematics and more in the competition of exams, with maturity, he soon lost interest in strategies for simply obtaining excellent marks for which Webb was well known. He wanted a more meaningful engagement with mathematics that went beyond complex problem solving and was assigned a new coach, Augustus Edward Hough Love, who introduced him to some of the most innovative mathematics of the day. Hardy came fourth in the Cambridge Mathematical Tripos of 1898. He was unhappy with the result as he believed that he was the best Mathematician among his contemporaries. This did not affect the development of his academic career though as he became a fellow of Trinity College in 1900.

In 1908 Hardy published his first book *A Course of Pure Mathematics* which became a standard textbook. An entire generation of mathematicians was inspired by Hardy’s work as they pursued their undergraduate studies. The book not only had a substantial influence on students; it was respected by many academics as an outstanding introduction to the analysis of number and calculus. Although this was an important first step in his published output, Hardy did not want to be a textbook academic and was more interested in developing more advanced research in pure mathematics.

**Srinivasa Ramanujan and****J. E.**** Littlewood**

G.H. Hardy is also known for his close association with Indian mathematician Srinivasa Ramanujan who had written a letter to Hardy in 1913 which included more than 100 theorems on number infinity and theory. He had already written to a number of other mathematicians in the hope that they would recognise the originality of his work and that he might be published. Ramanujan was working as a clerk in Madras when he contacted Hardy who was initially sceptical about Ramanujan’s work. Due to the fact that Ramanujan had a limited level of conventional mathematical instruction, the way he expressed his theorems was highly unorthodox and Hardy even thought it might be a sophisticated hoax. Nonetheless, Hardy became fascinated by one particular aspect of Ramanujan’s mathematical communication. Hardy was fascinated by prime numbers as he knew that they were at the root of all numbers. Ramanujan claimed to have discovered a solution to a famous prime number problem. As he put it: “I have found a function of *x* that exactly represents the number of prime numbers less than *x*”. In essence, this was a problem of how the prime numbers are distributed.Hardy was intrigued by Ramanujan’s claims to havediscovered a formula whereby it was possible to calculate the number of primes up to one hundred million. Hardy discussed Ramanujan’s mathematics with his long-term collaborator *J. E.*Littlewood who agreed that the young Indian had produced ingenious work. Hardy then asked Ramanujan for proof of his theory and, while the reply was unconvincing, Hardy knew that he had still discovered a mathematician of exceptional ability. Later, in a 1921 lecture, Hardy conceded that “no elementary proof of the prime number theorem is known and one may ask whether it is reasonable to expect one”.

Hardy later suggested that none of his discoveries had led to any particular practical benefit for humanity. Ironically, the inability to fully crack the mystery of prime numbers has meant that they have proved to be of great value for other purposes. Results of number theories developed by Hardy and his cohorts have been developed and used to create encryption algorithms. This has been of particular importance for the development of security on the Internet where valuable information from personal bank details to secret military information is ubiquitous. Codes that led to encryption on the Internet are based on our understanding of prime numbers.Websites have code numbers for security. To break codes, you have to work out combinations of prime numbers that form the code. For simple prime numbers this would be relatively easy, but these website codes are for numbers with hundreds of digits and it is very difficult in practice to crack these codes. So while Hardy did not see his work as practical in any way, in reality, it has been applied to the real world.

Hardy went to visit Ramanujan in a London hospital by taxi one day. Something made him aware of the taxi number which was 1729. He thought about the number considering whether or not it was significant and decided that it was not. He informed Ramanujan that he had come across the number but that he did not see it as meaningful in any way. Ramanujan is said to have challenged Hardy’s assertion, suggesting that the number was highly significant due to the fact that it is “the smallest number expressible as the sum of two cubes in two different ways”. In mathematical form Ramanujan’s idea is expressed as follows: 1729 = 1^{3} + 12^{3} = 9^{3} + 10^{3}. These numbers are occasionally referred to as “taxi cab” numbers.

The two mathematicians worked together on a number of theorems and proofs. One of their most fruitful collaborations was in the area of partitions. Partitioning involves the splitting of larger numbers into smaller, more manageable units. Partitioning a number consists of the different ways in which it can be considered as a sum of smaller numbers. For instance, the number 4 can be expressed as the sum of the following: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. This process is expressed mathematically as p(*n*) and in the latter case, p(4)=5. Hardy and Ramanujan took this fundamental idea further and were able to devise more complex formulas to work with larger values.

The other vital collaboration of G.H. Hardy’s career was with fellow British mathematician *J. E.* Littlewood. The two men worked together to produce nearly one hundred papers. These papers consisted of a wealth of research covering prime number theory, Diophantine approximations, inequalities and the Riemann zeta function. Hardy and Littlewood complemented each other very well. Littlewood was seen as a very technical mathematician who could work through extremely complex problems. On the other hand, Hardy was more of a mathematical artist seeing the aesthetic beauty of equations and possessing a more engaging style of writing than Littlewood.

**Genetics**

Hardy is usually classified as a pure mathematician (generally theoretical and abstract in nature) and not a scholar of applied mathematics, which usually has a practical use. While Hardy often talked about the beauty of pure mathematics, he had a range of interests beyond his main focus of research. In 1908 Hardy and a German doctor, Wilhelm Weinberg, discovered the so-called Hardy-Weinberg law. The purpose of this law is to establishwhether or not evolution has occurred. It does this by detecting changes to gene frequencies in the population during a certain period of time.The law suggests that if there is no evolution allele (a variant of a gene) frequencies will remain in equilibrium for each succeeding generation**. **For this equilibrium to remain constant and for there to be no evolution, a series of conditions need to be satisfied: There can be no mutations (no new genes) entering the population, no movement of individuals in or out of the population, individuals must mate randomly, there must be no genetic drift and there must be no selection. As these disruptive factors commonly occur, the law does not usually apply to reality. However, it allows us to identify some allele frequencies that may change from one generation to the next. This is a simplified way of establishing that evolution is in fact occurring. This law proved to be an important contribution to population genetics. It also shows that Hardy did not limit himself to one discipline; in this case, his interest unifies mathematics and biology.

**Politics**

Moreover, Hardy’s interests extended to other aspects of academic and public life. Hardy was an ardent follower of cricket and would often go to watch games in Cambridge. He was also a pacifist and had distinctly left-wing political sympathies. He actively took part in the activities of the National Union of Scientific Workers and during one of his speeches he pointed out that, although the work of scientists was very different to that of coalminers, both had a skill and were not “exploiters of other people’s work”. This sympathy with the underdog suggests an anti-capitalist outlook, consistent with his anti-religious views, and his office also contained a photograph of Lenin among others. Hardy also supported his colleague Bertrand Russell in 1916 after he had been dismissed from his academic post at Trinity College. Russell had been involved in an anti-conscription movement which tried to persuade men to refuse the call to military service. Hardy (and many others) opposed the decision and Russell was eventually reinstated a few years later.

Towards the end of his life, Hardy wrote an eccentric autobiography *A mathematician’s apology* in 1940. The book has a rather sad tone of resignation in which he suggests that, with age, he has lost his ability to be a mathematican: “I write about mathematics because like any other mathematician who has passed sixty I have no longer the freshness of mind, the energy or the patience to carry on effectively with my proper job”. The book is also a reflection on his life in mathematics and he discusses his biggest achievements in the discipline. Hardy lived through World War II, but,towards the end of the war, he developed health problems. He was no longer an active mathematician in the true sense and he sank into depression. He yearned to engage in his work once more, but he was simply too ill and had to rely on taxis to get out of his house. Hardy attempted to commit suicide in 1947 by taking an overdose of barbiturates; this attempt was unsuccessful although he only lived until December of that year.

**Legacy**

G.H. Hardy was an important figure in the history of British mathematics. The status of mathematics in the United Kingdom was not very high when Hardy came to prominence in the early years of the twentieth century. The work of mathematicians from France, Germany and Switzerland was regarded as being the best and Hardy’s work went some way to redressing that balance. In truth, G.H. Hardy is better known through his association with Ramanujan. In his autobiography Hardy himself makes the telling comment: “I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, ‘Well, I have done one the thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.’”