Abstract
This chapter discusses the foundations of computing, including the binary number system and the step reckoner calculating machine, which were invented by Leibniz. The difference engine was designed by Babbage to evaluate polynomials and to produce accurate mathematical tables. Babbage’s design of the analytic engine provided the vision of a modern computer, and he envisaged that it would be analogous to the operation of the Jacquard loom, which is designed to weave (i.e. execute on the loom) a design pattern represented by a set of cards. Boole’s symbolic logic provides the foundation for digital computing.
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- 1.
The sexagesimal (or base-60) system was employed by the Babylonians c. 2000 BC. Indian and Arabic mathematicians developed the decimal system between 800 and 900 AD.
- 2.
The power series expansion of the Sine function is given by Sin(x) = x − x3/3! + x5/5! − x7/7! + ···. The power series expansion for the Cosine function is given by Cos(x) = 1 − x2/2! + x4/4! − x6/6! + ···. Functions may be approximated by interpolation and the approximation of a function by a polynomial of degree n requires n + 1 points on the curve for the interpolation. That is, the curve formed by the polynomial of degree n that passes through the n + 1 points of the function to be approximated is an approximation to the function. The error function also needs to be considered.
- 3.
The Jacquard loom was invented by Joseph Jacquard in 1801. It is a mechanical loom which used the holes in punch cards to control the weaving of patterns in a fabric. The use of punched cards allowed complex designs to be woven from the pattern defined on the punched cards. Each punched card corresponds to one row of the design, and the cards were appropriately ordered. It was very easy to change the pattern of the fabric being weaved on the loom, as this simply involved changing cards.
- 4.
Lady Ada Lovelace was the daughter of the poet, Lord Byron.
- 5.
There is some controversy as to whether this was entirely her own work or a joint effort by Lovelace and Babbage.
- 6.
De Morgan was a nineteenth-century British mathematician based at University College London. De Morgan’s laws in Set Theory and Logic state that (A ∪ B)c = Ac ∩ Bc and \( \neg \,{\kern 1pt} ({\text{A}} \vee {\text{B}}){\text{ }} \equiv ~\neg {\kern 1pt} \,{\text{A}} ~\wedge \, ~\neg\,{\text{B}} \).
- 7.
Finite Differences are a numerical method used in solving differential equations.
- 8.
Victor Shestakov at Moscow State University also proposed a theory of electric switches based on Boolean algebra (published in Russian in 1941 whereas Shannon’s were published in 1937).
References
Babbage C, Menabrea LF (1842) Sketch of the analytic engine invented by charles babbage. Bibliothèque Universelle de Genève, October, No. 82 Translated by Ada, Augusta, Countess of Lovelace
Boole G (1848) The calculus of logic. Camb Dublin Math J III:183–98
Boole G (1958) An investigation into the laws of thought. Dover Publications (First published in 1854)
Leibniz WG (1703) Explication de l’Arithmétique Binaire. Memoires de l’Academie Royale des Sciences
McHale D (1985) George Boole, His Life and Work. Cork University Press
O’Regan G (2013) Giants of computing. Springer
Shannon C (1937) A symbolic analysis of relay and switching circuits. Masters thesis, Massachusetts Institute of Technology
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O’Regan, G. (2020). Foundations of Computing. In: Mathematics in Computing. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-34209-8_2
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DOI: https://doi.org/10.1007/978-3-030-34209-8_2
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