Lecture3 en Number Repr Boolean 2014(1)

8/9/2019 Lecture3 en Number Repr Boolean 2014(1)

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Representation of numbers incomputers. Logical

foundations. Boolean algebra,

Boolean variables, functions,

basic relationshipsBit, byte and machine word. Sign-and-magnitude, ones'

complement and two's complement. Integers and floatingpoint numbers. Symbol information in computer systems.

Introduction in Boolean algebra. Operations and laws.Boolean expressions, Conjunction, disjunction, negation,

implication, exclusive or, logical equivalence. De Morganslaws. Converting logical expressions


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Bit

A bitis the basic unit for information used in computersand communications. Bit can be defined as a variable orcomputed quantity that can have only twopossible valuesi.e. a binary digit which represents

either 1 or 0 (one or zero). The language of the computer is a set (sequence) of

binary digits (bits). The computers simply executecommands, called instructions, given by theprogrammers. An instruction is simply a sequence of

bits, which the computer interprets in a certain way. Forexample the bits 1000110010100000 instruct certaintypes of processors to add 2 numbers


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Byte 8 bits = 1 byte A byte is a unit of digital information in computing and

telecommunications that most commonly consists ofeight bits

From a historical prospective, a byte was the number of

bits used to encode a single character of text in acomputer. So it was and still is thebasic addressable element in most computerarchitectures

The size of the byte was hardware dependent, so therewas no standards to define its size. Eight bits were veryconvenient to code values between 0 and 255 in onebyte

The term octet was defined to explicitly denote asequence of 8 bits because of the ambiguity associatedin the past with the term byte


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Coding byte values

byte = 8 bits

binary 000000002 to111111112 decimal: 0

10

to25510 hexadecimal 0016 toFF16

- For hexadecimal numbers

- use digits 0 to 9 and A to F

- present as FA1D37B16 in C language as0xFA1D37B

>> or 0xfa1d37b


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Machine word

A machine word is the the natural unit of dataused by a particular processor design. It is apredefined group of bits that are handeled as aunit byt the instruction set. The number of bits in

a word (the word size, word width, or wordlength) is an important characteristic of a specificprocessor design or computer architecture

In modern computers, the typical group is 32 bits(the first computers had 4 or 8 bits, some of thenewest64 and 128 bits), and is the number ofbits in most registers of the architecture (e.g. theMIPS architecture)


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Byte Ordering

Issue - how should bytes be ordered inmemory within multi-byte word

Conventions Alphas, PCs are Little Endian machines

- Least significant byte has lowest address Suns Macs are Big Endian machines

- Least significant byte has highest address

Example

Variable x has 4-byte representation 0x01234567 Address given by &x is 0x100

0x100 0x101 0x102 0x103

4523 67

Big Endian

0x100 0x101 0x102 0x103

2345 01Little Endian

01

67


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Representing Integersint A = 15213;

int B = -15213;Long int C = 15213;

Decimal: 15213

Binary: 0011 1011 0110 1101Hex: 3 B 6 D

6D3B

00

00

0000

3B

6D

93

C4

FF

FF

FF

FF

C4

93

6D3B

00

00

6D3B

00

00

00

00

00

00

0000

3B

6D

Linux/Alpha A Sun A Linux C Alpha C Sun C

Linux/Alpha B Sun B

Twos complement representation

(Covered later in the lecture)


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Bi-endian architectures

Some architectures (incl. ARM, PowerPC, Alpha,SPARC V9, MIPS, PA-RISC and IA-64) can change theendianness in data segments, code segments or both.Such feature can improve the performance and simplifythe logic of network devices and software. The term bi-

endian, when used for hardware, denotes the possibilityof the machine to compute or transfer data in any of thetwo formats

Many of these architectures can switch through softwareto one or another endian format by default (usually this is

done upon the start of the computer) but for somesystems, the default format can be chosen by thehardware on the main board and can not be changed bythe software


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Bi-endian architectures

The term "bi-endian" refers to the way theprocessor treats access to data.Instruction accesses (fetches of instruction

words) on a given processor may stillassume a fixed endianness, evenif dataaccesses are fully bi-endian, though

this is not always the case, such as onIntel's IA-64-based Itanium CPU, whichallows both


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Endianness and operation systems

Operation systems with Big-endian AIX on POWER AmigaOS on PowerPC and 680x0 FreeBSD on PowerPC and SPARC HP-UX on Itanium and PA-RISC Linux on MIPS, SPARC, PA-RISC, POWER, PowerPC, 680x0,

ESA/390, z/Architecture, H8, FR-V, AVR32, Microblaze,ARMEB, M32R, SHEB and Xtensa.

Mac OS on PowerPC and 680x0 Mac OS X on PowerPC

NetBSD on PowerPC, SPARC, and etc. OpenBSD on PowerPC, SPARC, and etc. MVS and DOS/VSE on ESA/390, and z/VSE and z/OS on

z/Architecture Solaris on SPARC


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Representation of numbers

People use the decimal number system, whichfor the computers, the most convenient to use isthe binary system.

The hardware can be designed to add, subtract,

multiply and divide binary numbers The computer programs must be able to work

with both positive and negative numbers Computer programs must also recognize an

event, when an operation ends with a number,bigger than the biggest possible number in thecomputer


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Representation of numbers

In binary system/numbers we can not use aminus sign (-) to represent negative numbers We would like to represent binary numbers with

only two symbols, 0 and 1, are allowed fornumber representation

The easiest solution is to have a bit for positiveand negative numbers

Where should the bit reside? (left or right). Theadders will need one more operation for this bit,

as they can not know in advance what the resultwill be (whether positive or negative) and inadditionthere will be a positive 0 and anegative 0, which can be a problem


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Sign-and-magnitude, ones'complement and two's complement

Sign-and-magnitudepositive:

0 -1... 10 (0000 1010) = ?

negative

1 -1... 10 (1000 1011) = ?

Ones' complementpositive:

0 -1... 10 (0000 1100) = ?

negative

1 (1111 0101) = ?011

,..., xxxxkk


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Sign-and-magnitude, ones'complement and two's complement

Two's complementpositive: 0

-1... 10 (0000 1001) = ?

Negative

1 (1111 0100) = ?

00000000= 00000001=

00000010=

01111111=

10000000=

10000001= 10000010=

11111110=

11111111=

1,...,011

xxxxkk


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Two's complement

10000000 is not in use, as it has nocorresponding positive number

Modern computers use the twos compliment

representation as it requires least additionalinterventionsthe simplicity of this code is that itdoes not have two 0s, as do the sign andmagnitute and the ones complement

All negative numbers for the twos complementcode start with 1 in the most significant bitthehardware checks this bit to determine whetherthe number is positive or negative


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Two's complement

For the twos complement, an overflowcan occurwhen the sign bit is differentfrom the one that should be (0 for

operations with positive numbers and 1 foroperations with negative numbers)

When adding operands with differentsigns, overflow cannot occur. The reasonfor this is the sum must be no larger thanone of the operands


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Two's complement

The value of a two's-complement binarynumber can be calculated by adding upthe power-of-two weights of the "one" bits,

but with a negative weight for the mostsignificant (sign) bit

for example: 111110112= - 128 + 64 + 32

+ 16 + 8 + 0 + 2 + 1 = -5


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Two's complement

sign Integer part

+ 37 0 0000000 00100101

Two'scomplement

1 1111111 11011010

add +1 1

- 37 1 1111111 11011011

Operations are equivalent to:

1 00000000 00000000

0 00000000 00100101

11111111 11011011


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Two's complement

For operations in twos complement, additiondiffers from ordinary addition only by ignoringthe carry for the sign (watching for an overflow isdone separately)

Subtraction is an addition, changing the sign bitof the second operand If the left two carry bits are both 1s or both 0s,

the result is valid; if the left two carry bits are "1

0" or "0 1", a sign overflow has occurred Conveniently, an XOR operation on these twobits can quickly determine if an overflowcondition exists


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Two's complement

0 1001 + 0 0101 = 0 1110

0 10010 0101 =

0 1001 + 1 1011 = 10 0100 = 0 0100 1 0111 + 0 0101 = 1 1100

0 1100 + 0 0101 = 1 0001 (?)

The carry bit is ignored!


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Integers and floating point numbers

An integeris a datum of integral data type-a finite subset of themathematical integers. Integral data types may be of different sizes andmay or may not be allowed to contain negative values. Integers arerepresented in a computer as a group of binary digits. The place ofdecimal/binary point is fixed. The set of integer sizes available variesbetween different types of computers. The hardware for operations with

integers is simpler and less expensive than for floating point numbers. Floating point numbersignificant digits, or fraction (M1), base (p) and

exponent (E1)

The advantage of floating-point representation over fixed-point and integer representation is that it can support a much widerrange of values

The floating-point format needs slightly more storage (to encode theposition of the radix point), so when stored in the same space, floating-point numbers achieve their greater range at the expense of precision.

1*1

EpMN


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Floating point operations

A simple method to add floating-point numbersis to first represent them with the same exponent

The significand of the number with lower

exponent is shifted right by the absolute value ofthe difference of the exponents, the result willhave the bigger exponent, then the significantnumbers are added or subtracted as for integers

123456.7 + 101.7654 = (1.234567 10

5

) +(1.017654 102) = (1.234567 105) +(0.001017654 105) = (1.234567 +0.001017654) 105= 1.235584654 105


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Floating point operations

To multiply, the significands are multiplied whilethe exponents are added, and the result isrounded and normalized

Similarly, division is accomplished by subtractingthe divisor's exponent from the dividend'sexponent, and dividing the dividend's significandby the divisor's significand

e=3; s=4.734612 x e=5; s=5.417242 ; e=8; s=25.648538980104 (true product)

e=9; s=2.564854 (after normalization)


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Floating point standard

Prior to the IEEE-754 standard, computers usedmany different forms of floating-point, differing inthe word sizes, the format of therepresentations, and the rounding behavior of

operations. These differing systemsimplemented different parts of the arithmetic inhardware and software, with varying accuracy

The IEEE-754 standard was created in the early1980s after word sizes of 32 bits (or 16 or 64)had been generally settled upon. This wasbased on a proposal from Intel who weredesigning the i8087 numerical coprocessor


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Floating point arithmetic

Binary floating-point arithmetic is at its bestwhen it is simply being used to measure real-world quantities over a wide range of scales(such as the orbital period of a moon around

Saturn or the mass of a proton), and at its worstwhen it is expected to model the interactions ofquantities expressed as decimal strings that areexpected to be exact

An example of the latter case is financialcalculations. For this reason, financial softwaretends not to use a binary floating-pointnumber representation


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Strings In formal languages, which are used in mathematical

logic and theoretical computer science, a stringis afinite sequence of symbols that are chosen from a set oralphabet

In computer programming, a string is traditionally asequence of characters, either as a literal constant or as

some kind of variable. The latter may allow its elementsto be mutated and/or the length changed, or it may befixed (after creation)

A string is generally understood as a data type and isoften implemented as a byte (or word) array that stores a

sequence of elements, typically characters, using somecharacter encoding. A string may also denote moregeneral array data types and/or other sequential datatypes and structures; terms such as byte string, or moregeneral, string ofdatatype, or datatype-string, aresometimes used to denote strings in which the stored

data does not (necessarily) represent text


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Strings in C

Strings in C

Represented by array of characters

Each character encoded in ASCII format

- Standard 7-bit encoding of character set - Other encodings exist, but uncommon

- Character 0 has code 0x30

Digit ihas code 0x30+i String should be null-terminated

- Final character = 0


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String compatibility

Compatibility

Byte ordering not an issue

- Data are single byte quantities

Text files generally platform independent

- Except for different conventions ofline termination character!


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String operations

Two most often used operations areconcatenation and subset

S=love

T=you

ST = loveyou (not commutative operationST ^=TS)

Subset (loveyou), S=love


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Introduction in Boolean algebra

Boolean algebra, as developed in 1854 by GeorgeBoole It is a variant of ordinary elementary algebradiffering in its values, operations, and laws. Instead ofthe usual algebra of numbers, Boolean algebra is the

algebra of truth values 0 and 1, or equivalently ofsubsets of a given set. The operations are usually takento be conjunction, disjunction, and negation, withconstants 0 and 1. And the laws are definable as thoseequations that hold for all values of their variables, for

examplex

(y

x) =x. Applications include mathematicallogic, digital logic, computer programming, set theory,and statistics


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Whereas elementary algebra deals mainly withreal numbers, Boolean algebra deals with thevalues 0 and 1. These can be thought of as two

integers, or as the truth values falseand truerespectively. In either case they are called bits orbinary digits, in contrast to the decimal digits 0through 9

Boolean algebra also deals with other values onwhich Boolean operations can be defined, suchas sets or sequences of bits


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In the 1930s, Claude Shannon observed thatone could also apply the rules of Boole's algebrafor switching circuits and introduced switchingalgebraas a way to analyze and design circuits

by algebraic means in terms of logic gates Shannon already had at his disposal theabstract mathematical apparatus, thus he casthis switching algebra as the two-elementBoolean algebra. In circuit engineering settings

today, there is little need to consider otherBoolean algebras, thus "switching algebra" and"Boolean algebra" are often usedinterchangeably


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Efficient implementation of Booleanfunctions is a fundamental problem in thedesign of combinatorial logic circuits

Modern electronic design automation toolsfor VLSI circuits often rely on an efficientrepresentation of Boolean functions knownas (reduced ordered) binary decisiondiagrams (BDD) for logic synthesis andformal verification


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A Truth Table

Developed in 1854 by George Boole. Further developed by Claude Shannon (Bell Labs).

Outputs are computed for all possible input combinations(how many input combinations are there?).

Consider a room with two light switches. How must theywork?

SwitchA SwitchB

GND

Hot

LightZ


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Basic operations

Some operations of ordinary algebra likemultiplicationxy, additionx+y, and negation x,have their counterparts in Boolean algebra

The corresponsing Boolean operations are AND,OR, and NOT, also calledconjunction

xy, disjunctionxy, and negationorcomplement x, or sometimes !x

Some use instead the same arithmeticoperations as ordinary algebra reinterpreted forBoolean algebra, treatingxyas synonymouswithxyandx+ywithxy


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Boolean expressions

Duality principle:

the duality principleis when we substitute

AND with OR (or ORwith AND) and at thesame time substitute1s with 0s and 0s with1s

Positive AND logic

Negative OR logic

A B F

0

0

1

1

0

1

0

1

0

0

0

1

A B F

1

10

0

1

01

0

1

11

0


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Gate logicpositive vs negative

Positive logic: 1 is represented by high voltage;logic 0 is represented by low voltage.

Negative logic: logic 0 is represented by high

voltage; logic 1 is represented by low voltage. Normal Convention: Positive Logic/Active

High

Low Voltage = 0; High Voltage = 1

Alternative Convention sometimes used:

Negative Logic/Active Low

/


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Positive/Negative Logic Assignments

A B F

0 0 0

0 1 0

1 0 0

1 1 1

A B F

1 1 1

1 0 1

0 1 1

0 0 0

A B F

low low low

low high low

high low low

high high high

Voltage Levels Negative Logic LevelsPositive Logic Levels

Physical

AND gateAB

FA

B F = A B F = A + BA

B

A B F

0 0 1

0 1 1

1 0 1

1 1 0

A B F

1 1 0

1 0 0

0 1 0

0 0 1

A B F

low low high

low high high

high low high

high high low

Voltage Levels Negative Logic LevelsPositive Logic Levels

Physical

NAND gate

A

B

FA

BF = A B F = A + B

A

B


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Expressing duality with gates

F = A + BA

B

A

B

A + B

F = ABA

B

A

B

AB


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Properties of Boolean Algebra

Relationship Dual Property

A B = B

A (B + C) = A B + A C

1 A = A

A A = 0

postulates

A + B = B + A

A + B C = (A + B) (A + C)

0 + A = A

A + A = 1

commutative

Distributive

Identity

Complement

0 A = 0

A A = A

A (B C) = (A B) C

A = A

A B = A + B

AB + AC + BC = AB + AC

A (A + B) = A

Theorems

1 + A = 1

A + A = A

A + (B + C) = (A + B) + C

A + B = A B

(A + B)(A + C)(B + C) = (A + B)(A + C)

A + A B = A

0 and 1 theorems

Idempotence

Associative

Involution

DeMorgans Theorem

Consensus Theorem

Absorption Theorem


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Sum of products form: majority function

The sum of products form forthe 3-input majority functionis:

M = ABC + ABC + ABC + ABC= m3 + m5 + m6 + m7 = (3, 5,

6, 7) Each of the 2nterms are

called minterms, ranging from0 2n1.

Note relationship betweenminterm number and booleanvalue

minterm

indexA B C F

0

1

23

4

5

6

7

0

0

00

1

1

1

1

0

0

11

0

0

1

1

0

1

01

0

1

0

1

0

0

01

0

1

1

1


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AND-OR implementation of majority

A B C

A B C

A B C

A B C

F

CBA

Gate count is 8,gate input

count is 19


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OR-ANDimplementation of majority

A + B + C

A + B + C

A + B + C

A + B + C

F

CBA

Combinational Logic


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Combinational Logic Translates a set of inputs into a set of outputs according to

one or more mapping functions.

Inputs and outputs for a CLU normally have two distinct(binary) values: high and low, 1 and 0, 0 and 1, or 5 V and0 V for example.

The outputs of a CLU are strictly functions of the inputs,

and the outputs are updated immediately after the inputschange. A set of inputs i0in are presented to the CLU,which produces a set of outputs according to mappingfunctions f0fm.

Combinational

logic unit

i0

i1

in

f0(i0, i1)

f1(i1, i3, i4)

fm

(i9, in)


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Some Definitions

Combinational logic: a digital logic circuit in whichlogical decisions are made based only oncombinations of the inputs. e.g. an adder.

Sequential logic: a circuit in which decisions aremade based on combinations of the current inputsas well as the past history of inputs. e.g. a memoryunit.

Finite state machine: a circuit which has an internalstate, and whose outputs are functions of both

current inputs and its internal state. e.g. a vendingmachine controllerexamplecoffee vendingmachinemust remember how many and what kindof coins have been inserted!


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Finite state machine

An FSM iscomposed of acombinationallogic unit anddelay elements(called flip-flops) in afeedback path,whichmaintains stateinformation

Combinational

logic unit

Q0 D0

s0

Qn Dn

sn

Delay elements(memory)

Synchronization signal

State bits

OutputsInputs

f0

fm

i0

ik


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The mebi prefix

The mebi-prefix was defined by the InternationalElectrotechnical Commission (IEC) in December 1998.Its use (and related units) is presently endorsed bythe Institute of Electrical and Electronics

Engineers (IEEE) and the International Committee forWeights and Measures (CIPM) in contexts where use ofa binary prefix makes sense

Binary prefixes are increasingly used in scientific

literature and open source software. In productadvertising and other non-scientific publications, thekilobyte sometimes refers to a power of ten andsometimes a power of two


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Binary prefixes

Value IEC

1024 210 - Kilobyte KiB kibibyte

1048576 220 - MB Megabyte MiB mebibyte

1073741824 230 - GB Gigabyte GiB gibibyte

1099511627776 240- TB Terabyte TiB tebibyte

1125899906842624 250- PB Petabyte PiB pebibyte

1152921504606846976 260- EB Exabyte EiB exbibyte

1180591620717411303424 270- ZB Zettabyte ZiB zebibyte


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Some data

It is estimated that the human brain's ability to store memories isequivalent to about 2.5 petabytes of binary data

In January 2012, Cray began construction of the Blue WatersSupercomputer, which will have a capacity of 500 petabytes makingit the largest storage array ever if realized

As of March 2010 the global Internet traffic is estimated at 21exabytes per month

The 64-bit microprocessors found in many computerscan address 16 exabytes of memory

The total amount of global data was expected to be 2.7 zettabytesduring 2012

the new NSA Data Center south of Salt Lake City, UT, will be ableto contain up to 5 zettabytes of data, "according to some reports"


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Tasks Express the following numbers in sign-magnitude, 1`s complement, and

2`s complement notations, assuming 8-bit representation: -119 - 55

Show how the decimal integer -120 would be represented in 2`scomplement notation using:

8 bits = ? 16 bits = ?

The smallest (negative) number and the largest (positive) number thatcan be stored using the sign-magnitude notation.

Smallest in Binary = Smallest in Decimal = Largest in Binary = Largest in Decimal =

The smallest (negative) number and the largest (positive) number thatcan be stored using the 2`s complement notation. Smallest in Binary = Smallest in Decimal = Largest in Binary = Largest in Decimal =

Lecture3 en Number Repr Boolean 2014(1)

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