Conversion to binary number system of some types of ordinary fractions

Informatics in school ◽

10.32517/2221-1993-2020-19-10-42-44 ◽

2020 ◽

pp. 42-44

Author(s):

D. M. Zlatopolski

Keyword(s):

Natural Number ◽

Number System ◽

Binary Number ◽

Original Technique ◽

Independent Work

The article describes a technique developed by the author for converting a number of ordinary fractions into a binary number system without intermediate conversion to decimal. The technique is generally applicable to two types of fractions: the denominator of which is 2n – 1 or 2n + 1, where n is a natural number, and also for denominator values 2, 4, 8, ... times greater than the indicated values. For fractions with a denominator of the form 2n + 1 and numerators equal to powers of two, an original technique is implemented during translation, which consists in using a conditional negative digit 1 (its designation: 1). The analysis showed that the developed method is more effective than the traditional one. A large number of examples are given and assignments for independent work of students are proposed.

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Non-standard methods for converting numbers from one number system to another

Informatics in school ◽

10.32517/2221-1993-2020-19-9-28-30 ◽

2020 ◽

Vol 1 (9) ◽

pp. 28-30

Author(s):

D. M. Zlatopolski

Keyword(s):

Binary System ◽

Maximum Degree ◽

Number System ◽

Binary Form ◽

Binary Number ◽

Standard Methods ◽

Optimal Variant ◽

Decimal System ◽

Large Numbers ◽

Independent Work

The article describes a number of little-known methods for translating natural numbers from one number system to another. The first is a method for converting large numbers from the decimal system to the binary system, based on multiple divisions of a given number and all intermediate quotients by 64 (or another number equal to 2n ), followed by writing the last quotient and the resulting remainders in binary form. Then two methods of mutual translation of decimal and binary numbers are described, based on the so-called «Horner scheme». An optimal variant of converting numbers into the binary number system by the method of division by 2 is also given. In conclusion, a fragment of a manuscript from the beginning of the late 16th — early 17th centuries is published with translation into the binary system by the method of highlighting the maximum degree of number 2. Assignments for independent work of students are offered.

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Method for extracting square and cube roots in the binary number system

Informatics in school ◽

10.32517/2221-1993-2021-20-1-42-45 ◽

2021 ◽

Vol 1 (1) ◽

pp. 42-45

Author(s):

D. M. Zlatopolski

Keyword(s):

Number System ◽

Cube Root ◽

Binary Number ◽

Decimal System ◽

Column Method ◽

Standard Values ◽

Current Value ◽

Decimal Numbers ◽

Independent Work ◽

Selection Of

The article describes in detail the methods of extracting square and cube roots in the binary number system. The method for extracting the square root of a binary number is similar to the corresponding method for decimal numbers, which is called the "column method". As for decimal numbers, when choosing the next digit of the root, twice the current value of the root, represented in the binary system, is used. When extracting the cube root (also "column"), there are two differences from the decimal system. The first is that instead of 300 (the product of 3 and 100), the binary number 1100 is used (that is, the product of the binary equivalents of the numbers 3 and 4). The second difference is that instead of the number 30 (the product of 3 and 10), the binary number 110 is used (that is, the product of binary analogs numbers 3 and 2). To facilitate the selection of the next root digit (0 or 1), a number of standard values have been calculated, depending on the current root value. Assignments for independent work of students are offered.

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Complex Binary Number System-based Co-Processor Design for Signal Processing Applications

10.1109/iementech53263.2021.9614893 ◽

2021 ◽

Author(s):

Sudia Sai Santosh ◽

Tandyala Sai Swaroop ◽

Tangudu Kavya ◽

Ramesh Chinthala

Keyword(s):

Signal Processing ◽

Number System ◽

Binary Number ◽

Processor Design

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Decimal Hardware Multiplier

Encyclopedia of Information Science and Technology, Fourth Edition ◽

10.4018/978-1-5225-2255-3.ch400 ◽

2018 ◽

pp. 4607-4618

Author(s):

Mário Pereira Vestias

Keyword(s):

Number System ◽

Floating Point ◽

Binary Number ◽

Correct Result ◽

Fundamental Operation ◽

Floating Point Arithmetic ◽

Decimal Arithmetic ◽

Decimal Numbers ◽

Decimal Multiplication ◽

Point Arithmetic

IEEE-754 2008 has extended the standard with decimal floating point arithmetic. Human-centric applications, like financial and commercial, depend on decimal arithmetic since the results must match exactly those obtained by human calculations without being subject to errors caused by decimal to binary conversions. Decimal Multiplication is a fundamental operation utilized in many algorithms and it is referred in the standard IEEE-754 2008. Decimal multiplication has an inherent difficulty associated with the representation of decimal numbers using a binary number system. Both bit and digit carries, as well as invalid results, must be considered in decimal multiplication in order to produce the correct result. This article focuses on algorithms for hardware implementation of decimal multiplication. Both decimal fixed-point and floating-point multiplication are described, including iterative and parallel solutions.

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Decimal Hardware Multiplier

Advanced Methodologies and Technologies in Artificial Intelligence, Computer Simulation, and Human-Computer Interaction - Advances in Computer and Electrical Engineering ◽

10.4018/978-1-5225-7368-5.ch054 ◽

2019 ◽

pp. 722-736

Author(s):

Mário Pereira Vestias

Keyword(s):

Number System ◽

Floating Point ◽

Binary Number ◽

Correct Result ◽

Fundamental Operation ◽

Floating Point Arithmetic ◽

Decimal Arithmetic ◽

Decimal Numbers ◽

Decimal Multiplication ◽

Point Arithmetic

IEEE-754 2008 has extended the standard with decimal floating-point arithmetic. Human-centric applications, like financial and commercial, depend on decimal arithmetic since the results must match exactly those obtained by human calculations without being subject to errors caused by decimal to binary conversions. Decimal multiplication is a fundamental operation utilized in many algorithms, and it is referred in the standard IEEE-754 2008. Decimal multiplication has an inherent difficulty associated with the representation of decimal numbers using a binary number system. Both bit and digit carries, as well as invalid results, must be considered in decimal multiplication in order to produce the correct result. This chapter focuses on algorithms for hardware implementation of decimal multiplication. Both decimal fixed-point and floating-point multiplication are described, including iterative and parallel solutions.

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