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© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION There are 10 kinds of people in the world—those who understand binary and those who don’t. © Jones & Bartlett Learning, LLC © Jones—Anonymous & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION CHAPTER Data Representation © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE 2OR DISTRIBUTIONin ComputerNOT FOR Systems SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 2.1 INTRODUCTION he organization of any computer depends considerably on how it represents T numbers, characters, and control information. The converse is also true: Stan- © Jones & Bartlettdards Learning, and conventions LLC established over the years© Jones have determined& Bartlett certain Learning, aspects LLC of computer organization. This chapter describes the various ways in which com- NOT FOR SALE putersOR DISTRIBUTION can store and manipulate numbers andNOT characters. FOR SALE The ideas OR presented DISTRIBUTION in the following sections form the basis for understanding the organization and func- tion of all types of digital systems. The most basic unit of information in a digital computer is called a bit, which is a contraction of binary digit. In the concrete sense, a bit is nothing more than © Jones & Bartlett Learning, aLLC state of “on” or “off ” (or “high”© Jones and “low”)& Bartlett within Learning, a computer LLCcircuit. In 1964, NOT FOR SALE OR DISTRIBUTIONthe designers of the IBM System/360NOT FOR mainframe SALE OR computer DISTRIBUTION established a conven- tion of using groups of 8 bits as the basic unit of addressable computer storage. They called this collection of 8 bits a byte. Computer words consist of two or more adjacent bytes that are sometimes addressed and almost always are manipulated collectively. The word size repre- © Jonessents the & Bartlettdata size thatLearning, is handled LLC most efficiently by a particular© Jones architecture. & Bartlett Learning, LLC NOTWords FOR can SALE be 16 OR bits, DISTRIBUTION 32 bits, 64 bits, or any other size thatNOT makes FOR sense SALE in the OR DISTRIBUTION context of a computer’s organization (including sizes that are not multiples of eight). An 8-bit byte can be divided into two 4-bit halves called nibbles (or nyb- bles). Because each bit of a byte has a value within a positional numbering sys- © Jones & Bartletttem, Learning, the nibble LLCcontaining the least-valued© binary Jones digit & isBartlett called the Learning, low-order LLC nibble, and the other half the high-order nibble. NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 61 © Jones & Bartlett Learning, LLC© Jones & Bartlett Learning, LLC.© NOTJones FOR SALE& Bartlett OR DISTRIBUTION Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 9781284136852_CH02_Pass03.indd 61 03/02/18 2:25 PM © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 62 Chapter 2 / Data Representation in Computer Systems 2.2 © POSITIONALJones & Bartlett NUMBERING Learning, SYSTEMSLLC © Jones & Bartlett Learning, LLC NOTAt someFOR pointSALE during OR DISTRIBUTIONthe middle of the sixteenth century, NOTEurope FOR embraced SALE OR DISTRIBUTION the decimal (or base 10) numbering system that the Arabs and Hindus had been using for nearly a millennium. Today, we take for granted that the num- ber 243 means two hundreds, plus four tens, plus three units. Notwithstand- ing the fact that zero means “nothing,” virtually everyone knows that there © Jones & Bartlettis a substantial Learning, difference LLC between having 1© of Jones something & Bartlett and having Learning, 10 of LLC NOT FOR SALEsomething. OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION The general idea behind positional numbering systems is that a numeric value is represented through increasing powers of a radix (or base). This is often referred to as a weighted numbering system because each position is weighted by a power of the radix. © Jones & Bartlett Learning, LLCThe set of valid numerals© forJones a positional & Bartlett numbering Learning, system LLC is equal in NOT FOR SALE OR DISTRIBUTIONsize to the radix of that system.NOT For FOR example, SALE there OR are DISTRIBUTION 10 digits in the decimal system, 0 through 9, and 3 digits for the ternary (base 3) system, 0, 1, and 2. The largest valid number in a radix system is one smaller than the radix, so 8 is not a valid numeral in any radix system smaller than 9. To distinguish among numbers in different radices, we use the radix as a subscript, such as © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC in 3310 to represent the decimal number 33. (In this text, numbers written NOTwithout FOR a SALEsubscript OR should DISTRIBUTION be assumed to be decimal.) AnyNOT decimal FOR integer SALE OR DISTRIBUTION can be expressed exactly in any other integral base system (see Example 2.1). EXAMPLE 2.1 Three numbers are represented as powers of a radix. © Jones & Bartlett Learning, LLC210–© Jones1– & Bartlett2 Learning, LLC 243.5110 =×2104+×10 +×3105 +×10 +×110 NOT FOR SALE OR DISTRIBUTION21 0 NOT FOR SALE OR DISTRIBUTION 2123 =×23+×13+×23= 2310 4321 0 101102 =×12+×02+×12 +×12+×02= 2210 © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC The two most important radices in computer science are binary (base two) and NOT FOR SALE OR DISTRIBUTIONhexadecimal (base 16). AnotherNOT radix FOR of SALEinterest ORis octal DISTRIBUTION (base 8). The binary system uses only the digits 0 and 1; the octal system, 0 through 7. The hexa- decimal system allows the digits 0 through 9 with A, B, C, D, E, and F being used to represent the numbers 10 through 15. Table 2.1 shows some of the © radices.Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 2.3 CONVERTING BETWEEN BASES Gottfried Leibniz (1646−1716) was the first to generalize the idea of the (posi- tional) decimal system to other bases. Being a deeply spiritual person, Leibniz © Jones & Bartlettattributed Learning, divine qualities LLC to the binary system.© JonesHe correlated & Bartlett the fact Learning, that any LLC integer could be represented by a series of 1s and 0s with the idea that God (1) NOT FOR SALEcreated OR DISTRIBUTION the universe out of nothing (0). UntilNOT the FORfirst binary SALE digital OR DISTRIBUTION comput- ers were built in the late 1940s, this system remained nothing more than a © Jones & Bartlett Learning, LLC© Jones & Bartlett Learning, LLC.© NOTJones FOR SALE& Bartlett OR DISTRIBUTION Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 9781284136852_CH02_Pass03.indd 62 03/02/18 2:25 PM © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 2.3 / Converting Between Bases 63 © Jones & BartlettPowers of 2Learning, LLCDecimal 4-Bit Binary© Hexadecimal Jones & Bartlett Learning, LLC −2 1 0 0 NOT FOR SALE20==4 OR.2 5DISTRIBUTION 0000 NOT FOR SALE OR DISTRIBUTION −1 1 20==2 .5 1 0001 1 210 = 2 0010 2 221 = 3 0011 3 2 24= 4 0100 4 283 = © Jones & Bartlett Learning, LLC 5 © Jones0101 & Bartlett5 Learning, LLC 214 = 6 NOT FOR SALE OR DISTRIBUTION5 6 NOT 0110FOR SALE OR6 DISTRIBUTION 23= 2 7 0111 7 266 = 4 8 1000 8 27 = 128 9 1001 9 28 = 256 10 1010 A © Jones & Bartlett Learning, LLC 29 = 512 © Jones & Bartlett Learning, LLC 11 1011 B NOT FOR SALE OR DISTRIBUTION 210 = 1024 NOT FOR SALE OR DISTRIBUTION 12 1100 C 2315 = 2,768 13 1101 D 2616 = 5, 536 14 1110 E 15 1111 F © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR TABLEDISTRIBUTION 2.1 Some Numbers to RememberNOT FOR SALE OR DISTRIBUTION mathematical curiosity. Today, it lies at the heart of virtually every electronic device that relies on digital controls. © Jones & Bartlett Learning,Because of itsLLC simplicity, the binary numbering© Jones system & Bartlett translates Learning, easily into LLC NOT FOR SALE electronicOR DISTRIBUTION circuitry. It is also easy for humansNOT to understand.FOR SALE Experienced OR DISTRIBUTION com- puter professionals can recognize smaller binary numbers (such as those shown in Table 2.1) at a glance. Converting larger values and fractions, however, usually requires a calculator or pencil and paper. Fortunately, the conversion techniques are easy to master with a little practice. We show a few of the simpler techniques © Jones & Bartlett Learning, inLLC the sections that follow. © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 2.3.1 Converting Unsigned Whole Numbers We begin with the base conversion of unsigned numbers. Conversion of signed numbers (numbers that can be positive or negative) is more complex, and it is © Jonesimportant & Bartlettthat you first Learning, understand LLC the basic technique for conversion© Jones before & Bartlett con- Learning, LLC tinuing with signed numbers. NOT FORConversion SALE betweenOR DISTRIBUTION base systems can be done by using NOTeither FORrepeated SALE sub- OR DISTRIBUTION traction or a division-remainder method. The subtraction method is cumbersome and requires a familiarity with the powers of the radix being used. Because it is the more intuitive of the two methods, however, we will explain it first. As an example, let’s say we want to convert 538 to base 8. We know that © Jones & Bartlett Learning, LLC © Jones10 & Bartlett Learning, LLC 83 = 512 is the highest power of 8 that is less than 538, so our base 8 number NOT FOR SALE willOR beDISTRIBUTION 4 digits wide (one for each power ofNOT the FORradix: SALE0 through OR 3).
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