Understanding calculators is not based on one's knowledge of technology, but on one's knowledge of mathematics. We will
illustrate it by two encounters, unfortunately fictitious, with mathematicians
of the past, Archimedes (of Syracuse) (287-212 BC) and Carl Friedrich Gauss
(1777-1855).

Archimedes

He was a scientist, engineer, and mathematician. So he wouldn't be too surprised to see a TI-83/84 calculator. He wouldn't think that there is any "magic" in a computing device. But as an engineer, he would probably ask so many technical questions that you would have to lend him your cell phone so he could talk directly to his colleagues at Texas Instruments. He wouldn't be impressed by our scientific notation for numbers. In his "Sand Reckoner", he invented his own notation, and he carried out complex computations in base 10,000. He would be interested in how we compute approximations of π, and modern methods based on calculus and not on geometry would be new to him. He would be critical about our way of computing integrals. He always gave two numbers, approximations from above and from below. He would say that giving just one value is "sloppy math". He would be very interested in the TI-83/84's Solver. Newton's method, which Solver uses, would be quite new for him. He would know a lot about rate of change, but he would always write it as a ratio of two numbers; and the concept of a derivative would seem to him rather strange and "unnatural".

Gauss

A talk with Gauss would be very different. He would know all the mathematics that is implemented on the TI-83/84 calculator. He could even explain the small details of the methods that are used. He would be pleased what his method of computing integrals, and his method of solving systems of linear equations, are still the best, more than 150 years after their invention. But one thing would puzzle him. He would say that he and his colleagues were sure that in the future, computations would be carried out automatically be computing devices. But he would think that different computations would be carried out by different devices, namely, that we would have one device for integrals, another for derivatives, and still another for solving equations. He would say that he simply doesn't understand how such a "Swiss army knife" of computing, which is small and has so many functions, could have been created.(Oops, sorry! The Swiss army knife was invented in 1891.) To satisfy his curiosity, you would have to tell him about the theoretical underpinnings of computer science that form the basis for computer technology. So you would tell him about the work of three mathematicians, David Hilbert (1862-1943), Kurt Gödel (1906-1978), and Allan Turing (1912-1954).

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