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Journal of Inforamtion Science and Engineering, Vol.18 No.3, pp.425-443 (May 2002)

Why is 1 + 1 = 2?*

Wuu Yang
Department of Computer and Information Science
National Chiao Tung University
Hsinchu, 300 Taiwan
E-mail: wuuyang@cis.nctu.edu.tw

Too often we take the method of decimal addition to be the definition of addition. We clarify this misunderstanding by first giving a definition of addition and then proving that the decimal addition method is both sound and complete with respect to the given definition. By showing the soundness and completeness of decimal addition with respect to the addition of natural numbers, we actually propose a new approach to algorithm correctness. We also discuss the differences between our logic approach and the traditional axiomatic approach.

Keywords: addition, algorithm correctness, arithmetic, decimal addition, decimal notation, logic, foundation of mathematics

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Received April 12, 2001; revised August 23, 2001; accepted October 5, 2001.
Communicated by Hsu-Chun Yen.
*This work was partially supported by the Republic of China National Science Council under Contract No. NSC 89-2213-E-009-014, NSC 89-2213-E-009-068 and NSC 89-2213-e-0009-146.