**Richard Hamming**'s parents were Richard J Hamming and Mabel G Redfield. He was brought up in Chicago where he attended school and soon realised that he was a more able mathematician than his teacher. He decided to study engineering at university but the only offer of a scholarship came from the University of Chicago which had no engineering department. He entered the University of Chicago receiving his B.S. in 1937 after majoring in mathematics. Following his undergraduate studies, he went to the University of Nebraska where he was awarded his M.A. in 1939. He received his Ph.D. in mathematics in 1942 from the University of Illinois at Urbana-Champaign. His doctoral dissertation *Some Problems in the Boundary Value Theory of Linear Differential Equation* was supervised by Waldemar Trjitzinsky (1901-1973). Trjitzinsky, a Russian by birth, emigrated to the United States in the 1920s and was a professor of Mathematics at the University of Illinois from 1934 to 1969. His interests were in analysis, particularly measure theory, integration and differential equations. In a paper published in *Acta Mathematica* in 1936, Trjitzinsky studied linear differential systems with two point boundary conditions of a particular type and Hamming extended this work in his thesis. In particular, Hamming investigated the Green's function and also the characteristic solutions for which he obtained asymptotic expressions. He further developed methods introduced by Jacob D Tamarkin to investigate the characteristic numbers and to show that the series used converged uniformly. Hamming did, however, develop interests in ideas that were quite far removed from his study of differential equations [15]:-

As a graduate student I found, and studied, Boole's 'Laws of Thought', and I found it interesting, relevant, and believable.

After the award of his doctorate, Hamming married Wanda Little on 5 September 1942. He taught, for a short time, at the University of Illinois and then at the University of Louisville. In 1945, encouraged by a friend, he joined the Manhattan Project, a U.S. government research project to produce an atomic bomb. It was called the Manhattan Project because the first research had been done at Columbia University in Manhattan. However by the time that Hamming was recruited to the project it was being undertaken at Los Alamos. A month after he arrived at Los Alamos, he was joined by his wife Wanda who was also employed on the Manhattan Project. He was given the task of running the IBM computers which played a vital role in the project [9]:-

At Los Alamos I was brought in to run the computing machines which other people had got going, so those scientists and physicists could get back to business. ... I saw Feynman up close. I saw Fermi and Teller. I saw Oppenheimer. I saw Hans Bethe: he was my boss.

Wanda began working with desk calculators and, later, worked for Enrico Fermi and Edward Teller. Hamming relates an interesting episode in [15]:-

... at Los Alamos ... we were designing atomic bombs. Shortly before the first field test(you realize that no small scale experiment can be done - either you have a critical mass or you do not), a man asked me to check some arithmetic he had done, and I agreed, thinking to fob it off on some subordinate. When I asked what it was, he said, "It is the probability that the test bomb will ignite the whole atmosphere." I decided I would check it myself! The next day when he came for the answers I remarked to him, "The arithmetic was apparently correct but I do not know about the formulas for the capture cross sections for oxygen and nitrogen - after all, there could be no experiments at the needed energy levels." He replied, like a physicist talking to a mathematician, that he wanted me to check the arithmetic not the physics, and left. I said to myself, "What have you done, Hamming, you are involved in risking all of life that is known in the Universe, and you do not know much of an essential part?" I was pacing up and down the corridor when a friend asked me what was bothering me. I told him. His reply was, "Never mind, Hamming, no one will ever blame you."

After the Manhattan Project ended, Hamming remained at Los Alamos for six months, writing up details of the calculations that had been made. He felt that it was important to try to understand exactly what had been achieved there and why it had been so successful. It was at this time that he realised that he had done the right thing by not studying engineering - the engineers did much of the routine work but mathematicians like himself were more involved in the cutting edge innovations. He formed a view of mathematics, arising from this Los Alamos experience, that computation was of major importance but it made him doubt the significance of the standard approach through abstract mathematical theories (see, for example, [19]). He had already the offer of a position at the Bell Telephone Laboratories in New Jersey and, in 1946, he began working in the mathematics department there. However, he did not break his link with Los Alamos, continuing to make two week visits each summer to Los Alamos Scientific Laboratories as a consultant. At Bell Labs he was able to work with both Claude Shannon and John Tukey [15]:-

[

At]Bell Labs I came into a very productive department. Bode was the department head at the time; Shannon was there ... I shared an office for a while with Shannon. At the same time he was doing information theory, I was doing coding theory. It is suspicious that the two of us did it at the same place and at the same time - it was in the atmosphere.

The head of the Mathematical Research Department at Bell Labs was Hendrik Wade Bode (1905-1982) who first joined the Department at Bell Labs in 1929 and became head in 1944. In addition to Shannon and Tukey, some other young mathematicians had joined the Mathematical Research Department at Bell Labs just before Hamming. These included Donald Percy Ling (1912-1981) and Brockway McMillan (1915-) who had been at Los Alamos at the same time as Hamming. Shannon, Ling, McMillan and Hamming called themselves the Young Turks and Hamming spoke about the group (see [8]):-

We grew up in the great depression, so we believed we owed the world a living. During the war, we all had to learn things we didn't want to learn to get the war won, so we were all cross-fertilized. We were impatient with conventions and had often had responsible jobs very early. We were first-class troublemakers. We did unconventional things in unconventional ways and still got valuable results. Thus, management had to tolerate us and let us alone a lot of the time.

His unconventional approach was emphasised by Alan Chynoweth who worked for the Physics Research Department at Bell Labs [9]:-

... we were in the habit of lunching together as a physics group, and for some reason this strange fellow from mathematics was always pleased to join us. We were always happy to have him with us because he brought so many unorthodox ideas and views. Those lunches were stimulating, I can assure you.

Not all of Hamming's colleagues were as happy to tolerate his 'unconventional ways', however [8]:-

Some former colleagues from Bell Labs recall Hamming as egotistical and comment that he occasionally went off "half-cocked, after some half-baked idea," and he was slow to pick up on his misdirection. "He is very hard to work with," one former Bell scientist said, "because he does a lot of broadcasting and not a lot of listening."

We will discuss below some of Hamming's highly significant work on error-correcting codes, but here we note some of the many varied problems he worked on in Bell Labs [11]:-

... working as I did for the Bell System, I did many telephone computations and other mathematical work on such varied things as traveling wave tubes, the equalization of television lines, the stability of complex communication systems, the blocking of calls through a telephone central office, to name but a few.

He was to continue to work for Bell Telephones until 1976 although he became increasingly interested in teaching and held visiting or adjunct professorships at Stanford University, the City College of New York, the University of California at Irvine, and Princeton University between 1960 and 1976. After retiring from Bell Labs in 1976, he accepted a professorship of computer science at the Naval Postgraduate School at Monterey, California. At this point he gave up his research career, deciding to concentrate on teaching and writing books. He formed the belief that the way mathematics is being taught is wrong, but the only way to change this was to write textbooks that can be used for a new approach. Here we quote two examples from [17] illustrating his views on mathematics teaching:-

*We live in an age of exponential growth in knowledge, and it is increasingly futile to teach only polished theorems and proofs. We must abandon the guided tour through the art gallery of mathematics, and instead teach how to create the mathematics we need. In my opinion, there is no long-term practical alternative*.*The way mathematics is currently taught it is exceedingly dull. In the calculus book we are currently using on my campus, I found no single problem whose answer I felt the student would care about! The problems in the text have the dignity of solving a crossword puzzle - hard to be sure, but the result is of no significance in life. Probability and statistics have easily understood problems whose results are surprising, important and interesting in themselves*.

His attempt to move to a new way of teaching calculus is exhibited in his book

*Methods of mathematics applied to calculus, probability, and statistics*(1985). He wrote about this book in [17]:-

This book ... is very different from the standard texts and its success or failure will tell us something about the prospects for change and innovation. If we are to move to any sort of new curriculum ... this book is a first start.

Other texts he wrote, all attempting to change conventional approaches to the areas they covered, include *Numerical Methods for Scientists and Engineers* (1962), *Introduction to applied numerical analysis* (1971), *Computers and Society* (1972), *Digital filters* (1977),* Coding and information theory* (1980), *The Art of Probability for Scientists and Engineers* (1991), and *The Art of Doing Science and Engineering : Learning to Learn* (1997).

Extracts from reviews of most of these books are given at THIS LINK

Hamming is best known for his work on error-detecting and error-correcting codes. His fundamental paper on this topic *Error detecting and error correcting codes* appeared in April 1950 in the *Bell System Technical Journal*. With this paper, he started a new subject within information theory. Hamming codes, Hamming distance and Hamming metric are standard terms used today in coding theory but they are also used in many other areas of mathematics. These ideas, of fundamental importance in coding theory, all originated in this classic paper and are of practical use in computer design. How did Hamming come to do this work? He tells us in the paper that he:-

... was led to the study ... from a consideration of large scale computing machines in which a large number of operations must be performed without a single error in the end result.

In fact it was in 1947 when Hamming set computers at Bell Labs to work on a particular problem over the weekend. The result was needed by his colleagues but, come Monday, he discovered that an error had occurred early in the calculations and he had nothing to report. He decided that if the computer could detect when an error had occurred then it must be able to detect where it had occurred. Since the machines worked in binary, every entry was either 0 or 1 so knowing that a particular entry was wrong meant that the computer could correct it - if 0 was wrong the correct entry must be 1 and vice-versa. Error detecting was done by a parity check on each block of symbols - an extra digit was added (either 0 or 1) so that the sum of the digits in the block was even. One incorrect entry could be detected since the parity check would fail. Hamming thought out a way not only to determine if a single error had occurred but also, by adding extra parity checks, to detect where it had occurred. He then devised a way to both correct a single error in a block and detect a second error. These codes were described in the first part of the paper while the second part used a geometrical model to show that the codes he had described were best possible.

Work in codes is related to packing problems and the error-correcting codes discovered by Hamming led to the solution of a packing problem for matrices over finite fields. In 1956 Hamming worked on the early computer, the IBM 650. His work here led to the development of a programming language which has evolved into the high-level computer languages used to program computers today. Hamming also worked on numerical analysis, integrating differential equations, and the Hamming spectral window which is much used in computation for smoothing data before Fourier analysing it.

Hamming has received many awards for his pioneering work. In 1968 he was made a fellow of the Institute of Electrical and Electronics Engineers. The IEEE awarded Hamming the Emanuel R Piore Award in 1979:-

For introduction of error correcting codes, pioneering work in operating systems and programming languages, and the advancement of numerical computation.

The IEEE named a medal "The Richard W. Hamming Medal" in his honour and he was the first recipient of this $10,000 prize medal in 1988:-

For exceptional contributions to information sciences and systems.

Also in 1968 he was presented with the Turing Award from the Association for Computing Machinery:-

For his work on numerical methods, automatic coding systems, and error-detecting and error-correcting codes.

He was made a fellow of the Association for Computing Machinery in 1994.

He has received further honours included being elected a member of the National Academy of Engineering in 1980 and receiving the Harold Pender Award from the University of Pennsylvania in 1981. In 1996, in Munich, Hamming received the prestigious $130,000 Eduard Rheim Award for Achievement in Technology for his work on error correcting codes.

In 1997 Hamming retired from teaching at the Naval Postgraduate School and was made Distinguished Professor Emeritus. Shortly before he retired he said (see [8]):-

A friend told me recently, "Hamming, the day you quit teaching, you are going to fall apart." He's probably right. When I left Bell Labs, I knew that that was the end of my scientific career. When I retire from here, in another sense, it's really the end.

Indeed he was right for having taught up to December 1997, he died of a heart attack in the following month. On Hamming's death Richard Franke of the Naval Postgraduate School at Monterey wrote:-

He will be long remembered for his keen insights into many facets of science and computation. I'll also long long remember him for his red plaid sport coat and his bad jokes.

James F Kaiser, in a brief obituary of Hamming, writes:-

We will all miss his engaging mind and his penetrating insight into matters scientific, engineering, and of everyday living.

**Article by:** *J J O'Connor* and *E F Robertson*

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