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

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.

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

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.

Conversion to binary number system of some types of ordinary fractions

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.

scholarly journals How the Brain might use Division

2020 ◽  
Vol 8 ◽  
pp. 126-137
Author(s):  
Kieran Greer
Keyword(s):  
Artificial Intelligence ◽  
Binary System ◽  
Problem Size ◽  
Symbol Manipulation ◽  
Mathematical Functions ◽  
Binary Operations ◽  
Neural Architecture ◽  
Large Numbers ◽  
The Brain

One of the most fundamental questions in Biology or Artificial Intelligence is how the human brainperforms mathematical functions. How does a neural architecture that may organise itself mostly throughstatistics, know what to do? One possibility is to extract the problem to something more abstract. This becomesclear when thinking about how the brain handles large numbers, for example to the power of something, whensimply summing to an answer is not feasible. In this paper, the author suggests that the maths question can beanswered more easily if the problem is changed into one of symbol manipulation and not just number counting.If symbols can be compared and manipulated, maybe without understanding completely what they are, then themathematical operations become relative and some of them might even be rote learned. The proposed systemmay also be suggested as an alternative to the traditional computer binary system. Any of the actual maths stillbreaks down into binary operations, while a more symbolic level above that can manipulate the numbers andreduce the problem size, thus making the binary operations simpler. An interesting result of looking at this is thepossibility of a new fractal equation resulting from division, that can be used as a measure of good fit and wouldhelp the brain decide how to solve something through self-replacement and a comparison with this good fit.

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

2021 ◽  
Author(s):  
Sudia Sai Santosh ◽  
Tandyala Sai Swaroop ◽  
Tangudu Kavya ◽  
Ramesh Chinthala
Keyword(s):  
Signal Processing ◽  
Number System ◽  
Binary Number ◽  
Processor Design

Decimal Hardware Multiplier

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.

Decimal Hardware Multiplier

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.

scholarly journals Algorithm for obtaining palindromes of the binary number system

2020 ◽  
Vol 1679 ◽  
pp. 032069
Author(s):  
V V Lyubimov ◽  
R V Melikdzhanyan
Keyword(s):  
Number System ◽  
Binary Number

In the classroom: Binary can be F-U-N

1963 ◽  
Vol 10 (6) ◽  
pp. 354-355
Author(s):  
Marion E. Ochsenhirt ◽  
Mary M. Wedemeyer
Keyword(s):  
Binary System ◽  
Place Value ◽  
Seventh Grade ◽  
Educational Value ◽  
Decimal System ◽  
The North ◽  
New Situation ◽  
Modern Mathematics

The seventh-grade students of the North Hills Joint Schools have found that there is fun as well as educational value in using the binary system. As all teachers of modern mathematics know, one of the main reasons for teaching the binary system is that the pattern for place value in this system is identical to that of the traditional decimal system. Developing the pattern in an entirely new situation gives the student a better understanding of the decimal system.

A Historical Reconstruction of Our Number System

1991 ◽  
Vol 38 (7) ◽  
pp. 46-48
Author(s):  
Kim Krusen
Keyword(s):  
Sixth Grade ◽  
Number System ◽  
General Knowledge ◽  
Mathematics Students ◽  
Historical Reconstruction ◽  
Number Systems ◽  
Large Numbers ◽  
Primitive Society ◽  
Humanistic Approach ◽  
Ancient Civilizations

Imagine your class as a “primitive society” just on the brink of civilization. Your society has been using tally sticks to represent numerical quantities. But now that your society is becoming more involved in commerce with other societies, you need an easier way to represent large numbers and some structure so that numbers can be manipulated. You need an organized number system. Creating a number system from scratch was the recent task of my sixth-grade class. My objective was to offer a more humanistic approach for my students further to understand and appreciate the structure of our number system. As the teacher. I was armed with a general knowledge of the number systems of the great ancient civilizations, and my students were armed with an enthusiasm to be cave dwellers for the day instead of mathematics students. With these resources, we began our project.

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