Richard W. Hamming

One Man's View of Computer Science

A number of observations and comments are directed toward suggesting that more than the usual engineering flavor be given to computer science. The engineering aspect is important because most present difficulties in this field do not involve the theoretical question of whether certain things can be done, but rather the practical question of how can they be accomplished well and simply. The teaching of computer science could be made more effective by various alterations, for example, the inclusion of a laboratory course in programming, the requirement for a strong minor in something other than mathematics, and more practical coding and less abstract theory, as well as more seriousness and less game playing.

The art of probability: for scientists and engineers

Probability * Introduction * Models in General * The Frequency Approach Rejected * The Single Event Model * Symmetry as the Measure of Probability * Independence * Subsets of a Sample Space * Conditional Probability * Randomness * Critique of the Model Some Mathematical Tools * Permutations * Combinations * The Binomial DistributionBernoulli Trials * Random Variables, Mean and the Expected Value * The Variance * The Generating Function * The Weak Law of Large Numbers * The Statistical Assignment of Probability * The Representation of Information Methods for Solving Problems * The Five Methods * The Total Sample Space and Fair Games * Enumeration * Historical Approach * Recursive Approach * Recursive Approach * The Method of Random Variables * Critique of the Notion of a Fair Game * Bernoulli Evaluation * Robustness * InclusionExclusion Principle Countably Infinite Sample Spaces * Introduction * Bernoulli Trials * On the Strategy to be Adopted * State Diagrams * Generating Functions of State Diagrams * Expanding a Rational Generating Function * Checking the Solution * Paradoxes Continuous Sample Spaces * A Philosophy of the Real Number System * Some First Examples * Some Paradoxes * The Normal Distribution * The Distribution of Numbers * Convergence to the Reciprocal Distribution * Random Times * Dead Times * Poisson Distribution in Time * Queing Theorem * Birth and Death Systems * Summary Uniform Probability Assignments Maximum Entropy * What is Entropy? * Shannons Entropy * Some Mathematical Properties of the Entropy Function * Some Simple Applications * The Maximum Entropy Principle Models of Probability * General Remarks * Maximum Likelihood in a Binary Choice * Von Mises Probability * The Mathematical Approach * The Statistical Approach * When The Mean Does Not Exist * Probability as an Extension of Logic * Di Finetti * Subjective Probability * Fuzzy Probability * Probability in Science * Complex Probability Some Limit Theorems * The Biomial Approximation for the case p=1/2 * Approximation by the Normal Distribution * Another Derivation of the Normal Distribution * Random Times * The Zipf Distribution * Summary An Essay on Simulation

Impact of Computers

(1965). Impact of Computers. The American Mathematical Monthly: Vol. 72, No. sup2, pp. 1-7.

How do you know the simulation is relevant

are entitled to know a bit about what I have done in addition to what I preach. My first large-scale simulation experience was on the design of atomic bombs at Los Alamos during the Second World War. The experience made a great impression upon me, not only because of the accuracy achieved when compared with reality but also because of the more obvious fact that small-scale experiments are impossible in this field. This experience was followed by almost 30 years of doing simulations at Bell Laboratories, where I have been involved at various times in managing and using large-scale digital and analog computers. Thus I have had a great deal of experience with simulations in the areas of physics, engineering, aerodynamics, guidance and control, transistor research and design, chemistry, telephone research and design, the verification of mathematical theories--especially those involving probabilities-and even inventory control.

A Note on the Location of the Binary Point in a Computing Machine

Necessary conditions that a switching function be, in particular, a threshold function have been introduced by Paull and McCluskey [11]; named and studied at some length by Winder [13], [14], [15] and Muroga, et al. [8]; and generalized by Elgot [3], Winder [14], [15] and Gabelman [4]. The basic conditions are the k-monotonicities and have been used not only to recognize threshold functions but also to determine realizing weights and threshold.

How I was led to the frequency approach

Probably the first necessary step in adopting the frequency approach to numerical analysis is to realize that there are serious defects in the polynomial approach. I will give a few examples of these, more or less in the order that they came to my attention. In 1945 while in the computer group at Los Alamos I discovered that the classical Simpson’s iterated formula for numerical integration makes no statistical sense. If you think of the integration of a function y (1~) from 0 to 1 as computing the average value of the function in that interval, then the iterated formula is