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Binary and Decimal Number Systems in Decimal Computations
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The accuracy of computer calculations is limited by the accumulation of rounding errors caused by the finite bit width of processor registers. A geometric interpretation of number representation is proposed, where numbers are considered as coordinates of a point on a number line, with the error determined by the minimal segment (resolution) dependent on the number system base and bit width. The impact of binary and decimal arithmetic on the accuracy of floating-point computations according to the IEEE 754 standard is analyzed. A comparison of cumulative errors in the addition of decimal numbers and their binary equivalents showed that for numbers with a large number of significant digits, the error in binary arithmetic is significantly smaller than in decimal arithmetic for a large number of operations. The results highlight the importance of choosing the number system for high-precision computations in scientific and financial applications.
Title: Binary and Decimal Number Systems in Decimal Computations
Description:
The accuracy of computer calculations is limited by the accumulation of rounding errors caused by the finite bit width of processor registers.
A geometric interpretation of number representation is proposed, where numbers are considered as coordinates of a point on a number line, with the error determined by the minimal segment (resolution) dependent on the number system base and bit width.
The impact of binary and decimal arithmetic on the accuracy of floating-point computations according to the IEEE 754 standard is analyzed.
A comparison of cumulative errors in the addition of decimal numbers and their binary equivalents showed that for numbers with a large number of significant digits, the error in binary arithmetic is significantly smaller than in decimal arithmetic for a large number of operations.
The results highlight the importance of choosing the number system for high-precision computations in scientific and financial applications.
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