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 pr...

Data files often consist of numbers having only a few significant decimal digits, whose information content would allow storage in only 32 bits. However, we may require that arithmetic operations involving these numbers be done with 64-bit floating-point precision, which precludes simply representing the data as 32-bit floating-point values. Decimal floating point gives a compact and exact repr...

Irreducible polynomials play an important role till now, in construction of 8-bit S-Boxes in ciphers. The 8-bit S-Box of Advanced Encryption Standard is a list of decimal equivalents of Multiplicative Inverses (MI) of all the elemental polynomials of a monic irreducible polynomial over Galois Field GF(2) [1]. In this paper a new method to search monic Irreducible Polynomials (IPs) over Galois f...

Almost twenty years ago the IEEE 754 binary floating-point standard was adopted. Since then almost every microprocessor as well as many programming languages have defined the floating-point arithmetic to be IEEE 754 compliant. From the many years experience in implementing the standard in hardware and writing floating-point programs, there have been numerous suggestions for revisions. All IEEE ...

We report the discovery of a new factor for each of the Fermat numbers F13, F15, F16. These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are seen to be composite numbers with 2391, 9808 and 19694 decimal digits respectively.

Given an integer b > 1 and a string s of decimal digits, one may ask whether there exists an integer n such that n (in decimal form) ends in s. This paper answers that question for the case where the exponent b is relatively prime to 10. It extends the earlier work [2], where the question was answered for cubes.

Problem P01: Compute the first N decimal digits after the decimal point of sin(sin(sin 1)), rounded toward zero. We have sin(sin(sin 1)) ≈ 0.678: the first N decimal digits after the decimal point match the first N mantissa digits. We use a target decimal precision N1 > N , and a binary precision p. We compute x = ◦(sin 1), y = ◦(sinx), z = ◦(sin y), with all roundings to nearest. It is easy to...

Any system that is used for naming or representing numbers is a number system, also known as numeral system. The modern civilization is familiar with decimal number system using ten digits. However digital devices and computers use binary number system instead of decimal number system, using only two digits namely, 0 and 1 based on the fundamental concept of the decimal number system. Various o...

This study explored event-related potential (ERP) correlates of common fractions (1/5) and decimal fractions (0.2). Thirteen subjects performed a numerical magnitude matching task under two conditions. In the common fraction condition, a nonsymbolic fraction was asked to be judged whether its magnitude matched the magnitude of a common fraction; in the decimal fraction condition, a nonsymbolic ...