On Two Conversion Methods of Decimal-To-Binary
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Representing Information in English Until 1598
In the twenty-first century, digital computers are everywhere. You al- most certainly have one with you now, although you may call it a “phone.” You are more likely to be reading this book on a digital device than on paper. We do so many things with our digital devices that we may forget the original purpose: computation. When Dr. Howard Figure 1-1 Aiken was in charge of the huge, Howard Aiken, center, and Grace Murray Hopper, mechanical Harvard Mark I calcu- on Aiken’s left, with other members of the Bureau of Ordnance Computation Project, in front of the lator during World War II, he Harvard Mark I computer at Harvard University, wasn’t happy unless the machine 1944. was “making numbers,” that is, U.S. Department of Defense completing the numerical calculations needed for the war effort. We still use computers for purely computational tasks, to make numbers. It may be less ob- vious, but the other things for which we use computers today are, underneath, also computational tasks. To make that phone call, your phone, really a com- puter, converts an electrical signal that’s an analog of your voice to numbers, encodes and compresses them, and passes them onward to a cellular radio which it itself is controlled by numbers. Numbers are fundamental to our digital devices, and so to understand those digital devices, we have to understand the numbers that flow through them. 1.1 Numbers as Abstractions It is easy to suppose that as soon as humans began to accumulate possessions, they wanted to count them and make a record of the count. -
ACP Course07
Applied Computer Programming Representation of Numbers. Bitwise Operators Course 07 Lect.eng. Adriana ALBU, PhD Politehnica University Timisoara Internal representation • All data, of any type, processed by a computer must have the same fundamental representation • In order to work correctly with these data, a programmer should understand: – the representation of numbers and their memory storage – the fundamental manner a computer operates with numbers (bitwise operations) Representation of numbers Binary Octal Hexadecimal Representation of numbers • Most of our programs don’t need concern regarding operations at the bit level • We’re free to think in bytes, or integers and doubles, or even higher level data types composed of a combination of these • But there are times when we'd like to be able to go to the level of an individual bit Representation of numbers • Any value (variable or constant) that is processed by our programs needs a memory space and it has a memory representation (sequence of bits) • Bit – unit of data storage – may have one of two values: 0 or 1 – usually is not individually addressable • Byte – unit of addressable data – in all usual computer architectures it contains 8 bits (CHAR_BIT from limits.h specifies its dimension) Representation of numbers • The byte is the lowest level at which we can access data; there's no "bit" type, and we can't ask for an individual bit • In fact, we can't even perform operations on a single bit – every bitwise operator will be applied to, at a minimum, an entire byte at a time • This -
Binary Numbers Bob Brown Information Technology Department Southern Polytechnic State University
Binary Numbers Bob Brown Information Technology Department Southern Polytechnic State University Positional Number Systems The idea of “number” is a mathematical abstraction. To use numbers, we must represent them in some way, whether by piles of pebbles or in some other way. It is common to create a code for representing numbers. Such codes are called number systems. Every number system uses symbols to convey information about the value of a number. A positional (or radix) number system is one in which the value that a symbol contributes to a number is determined by the both symbol itself and the position of the symbol within the number. That's just a fancy way of saying that 300 is far different from 3. Compare the idea of a positional number system with Roman numbers, where X means 10 no matter where in a number it appears. The decimal number system we use every day is a positional number system. Decimal numbers are also called base 10 or radix 10 numbers. The symbols are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, the plus and minus signs, and the period or decimal point. The position of each digit within the number tells us the multiplier used with it. Consider the number 1037. We learned in elementary school to call the rightmost digit the ones place, the next digit the tens place, then the hundreds place, then the thousands place, 3 2 1 0 and so on. Mathematically, 1037 means 110 + 010 + 310 + 710 . Each digit in the number is multiplied by some power of ten, starting with 100 at the right and increasing by one for each position to the left. -
Multi–Precision Math
Multi–Precision Math Tom St Denis Algonquin College Mads Rasmussen Open Communications Security Greg Rose QUALCOMM Australia August 29, 2017 This text has been placed in the public domain. This text corresponds to the v0.39 release of the LibTomMath project. This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the LATEX book macro package and the Perl booker package. Contents 1 Introduction 1 1.1 MultiplePrecisionArithmetic . 1 1.1.1 WhatisMultiplePrecisionArithmetic? . 1 1.1.2 The Need for Multiple Precision Arithmetic . 1 1.1.3 BenefitsofMultiplePrecisionArithmetic . 3 1.2 PurposeofThisText......................... 4 1.3 DiscussionandNotation . 5 1.3.1 Notation............................ 5 1.3.2 PrecisionNotation . 5 1.3.3 AlgorithmInputsandOutputs . 6 1.3.4 MathematicalExpressions . 6 1.3.5 WorkEffort.......................... 7 1.4 Exercises ............................... 7 1.5 IntroductiontoLibTomMath . 9 1.5.1 WhatisLibTomMath?. 9 1.5.2 GoalsofLibTomMath . 9 1.6 ChoiceofLibTomMath . 10 1.6.1 CodeBase........................... 10 1.6.2 APISimplicity ........................ 11 1.6.3 Optimizations......................... 11 1.6.4 PortabilityandStability . 11 1.6.5 Choice............................. 12 2 Getting Started 13 2.1 LibraryBasics ............................ 13 2.2 WhatisaMultiplePrecisionInteger? . 14 2.2.1 The mp intStructure .................... 15 iii 2.3 ArgumentPassing .......................... 17 2.4 ReturnValues............................. 18 2.5 InitializationandClearing . 19 2.5.1 Initializing an mp int .................... 19 2.5.2 Clearing an mp int...................... 22 2.6 MaintenanceAlgorithms . 24 2.6.1 Augmenting an mp int’sPrecision . 24 2.6.2 Initializing Variable Precision mp ints ........... 27 2.6.3 Multiple Integer Initializations and Clearings . 29 2.6.4 ClampingExcessDigits . 31 3 Basic Operations 35 3.1 Introduction............................. -
Multi–Precision Math
Multi–Precision Math Tom St Denis Algonquin College Mads Rasmussen Open Communications Security Greg Rose QUALCOMM Australia March 10, 2007 This text has been placed in the public domain. This text corresponds to the v0.39 release of the LibTomMath project. Tom St Denis 111 Banning Rd Ottawa, Ontario K2L 1C3 Canada Phone: 1-613-836-3160 Email: [email protected] This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the LATEX book macro package and the Perl booker package. Contents 1 Introduction 1 1.1 Multiple Precision Arithmetic . 1 1.1.1 What is Multiple Precision Arithmetic? . 1 1.1.2 The Need for Multiple Precision Arithmetic . 1 1.1.3 Benefits of Multiple Precision Arithmetic . 3 1.2 PurposeofThisText......................... 4 1.3 DiscussionandNotation . 5 1.3.1 Notation............................ 5 1.3.2 PrecisionNotation . 5 1.3.3 Algorithm Inputs and Outputs . 6 1.3.4 MathematicalExpressions. 6 1.3.5 WorkEffort.......................... 7 1.4 Exercises ............................... 7 1.5 IntroductiontoLibTomMath . 9 1.5.1 WhatisLibTomMath?. 9 1.5.2 GoalsofLibTomMath . 9 1.6 ChoiceofLibTomMath .. .. .. .. .. .. .. .. .. .. 10 1.6.1 CodeBase........................... 10 1.6.2 APISimplicity ........................ 11 1.6.3 Optimizations.. .. .. .. .. .. .. .. .. .. .. 11 1.6.4 Portability and Stability . 11 1.6.5 Choice............................. 12 2 Getting Started 13 2.1 LibraryBasics ............................ 13 2.2 WhatisaMultiplePrecisionInteger? . 14 2.2.1 The mp intStructure .................... 15 iii 2.3 ArgumentPassing .......................... 17 2.4 ReturnValues............................. 18 2.5 InitializationandClearing . 19 2.5.1 Initializing an mp int .................... 19 2.5.2 Clearing an mp int...................... 21 2.6 MaintenanceAlgorithms .