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Mathematics by Experiment and Experimentation in Mathematics

Is mathematics an experimental science? And is the computer the microscope (or the telescope) of mathematics? As a young mathematician I was certainly convinced this was so. Influenced by Kenneth Iverson (1920–2004), the inventor of the computer language APL, I saw the computer as an experimental tool that would reveal new mathematical worlds. As Iverson and others wrote in 1970, with the use of a computer, “mathematics becomes a laboratory science, open to experiment, conjecture, and discovery” [1]. Inspired by a 1972 paper of Lang and Trotter [3], as a teenager I wrote programs to compute the continued fraction expansion of various algebraic numbers. After verifying their results, I was naturally led to compute the expansions of other real numbers. One of the numbers I tried was