Number of Digits in Fractional Part of Arithmetic Operations of Two Terminating Decimals

By defining the valid digits and the standard fraction form of a terminating decimal, this paper makes an investigation on the number of digits in the fractional part of addition, subtraction, and multiplication of two terminating decimals. It is proved that the number of the valid digits in the fractional part of addition or subtraction of two terminating decimals does not exceed the maximal number of the valid digits of the two operands while that of multiplication does not exceed the total of the two. Several theorems are proved in detail mathematical deductions and a potential application is exhibited in integer factorization. The results in the paper are helpful to understand the change of the number of digits in the related arithmetic operations.