# Arithmetic of binary equivalents of decimal numbers

Within the framework of the concept of decimal calculations proposed in the article using binary arithmetic, a theory of binary equivalents of decimal floating-point numbers has been developed. According to this theory, basic decimal arithmetic operations on finite decimal numbers are performed with decimal precision by a binary processor according to the rules of binary arithmetic on the binary equivalents of decimal numbers. These calculation results are entirely consistent with the classical decimal finite number arithmetic and do not require the use of test programs. The identity of calculation results in decimal and binary equivalent arithmetics guarantees the repeatability of results on any platform. The article shows that implementing binary equivalents arithmetic with an acceptable decimal calculation error requires significantly fewer bits of binary processor registers than in modern computers. Because of the uniqueness of binary decimal equivalents, the difference between equal, properly rounded binary decimal equivalents is strictly zero. The presence of an explicit zero in the arithmetic of binary equivalents of decimal numbers makes it possible to implement a bitwise comparison of such numbers and introduce the concept of an infinitesimal number when the significand of a floating-point number is equal to zero.

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0000-0002-1260-8909## Submitting Author's Institution

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- Canada