invitation to computer science 6th edition chapter 4 the building blocks: binary numbers, boolean...

Invitation to Computer Science 6th Edition

Chapter 4

The Building Blocks: Binary Numbers, Boolean

Logic, and Gates

Invitation to Computer Science, 6th Edition 2

Objectives

In this chapter, you will learn about:

• The binary numbering system

• Boolean logic and gates

• Building computer circuits

• Control circuits


Introduction

• Computing agent– Abstract concept representing any object capable of

understanding and executing our instructions

• Fundamental building blocks of all computer systems – Binary representation– Boolean logic– Gates– Circuits


The Binary Numbering System

• Binary representation of numeric and textual information– Two types of information representation

• External representation

• Internal representation

– Binary is a base-2 positional numbering system


Figure 4.1 Distinction Between External Memory and Internal Representation of Information


Binary Representation of Numeric and Textual Information

• Binary numbering system (Computer)– Base-2– Built from ones and zeros– Each position is a power of 2– 1101 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20

• Decimal numbering system (Daily Life)– Base-10– Each position is a power of 10– 3052 = 3 x 103 + 0 x 102 + 5 x 101 + 2 x 100


Binary Representation of Numeric and Textual Information

• Binary-to-decimal algorithm– Whenever there is a 1 in a column, add the

positional value of that column to a running sum– Whenever there is a 0 in a column, add nothing– The final sum is the decimal value of this binary

number


Figure 4.2 Binary-to-Decimal Conversion Table


Binary Representation of Numeric and Textual Information (continued)

• To convert a decimal value into its binary equivalent– Use the decimal-to-binary algorithm

• Maximum number of binary digits that can be used to store an integer: 16, 32, or 64 bits

• Given k bits, the largest unsigned integer is 2k-1

• Arithmetic overflow– Operation that produces an unsigned value greater

than 65,535


Signed Numbers

• Sign/magnitude notation(old computer)– One of a number of different techniques for

representing positive and negative whole numbers– Not used often in real computer systems

• Two’s complement representation(current computer)– Total number of values that can be represented with

n bits is 2n


Signed Numbers

• 1(sign bit) 110001 (total: 7 bits, computer A)– When it is a signed value, it is -49– When it is a unsigned value, it is 113

• 1(sign bit) 0110001 (total: 8 bits, computer B)– When it is a signed value, it is -49– When it is a unsigned value, it is 177

• You must tell the computer if it is signed or unsigned integer. The number of bit depends on computer (8bit, 16bit, 32bit, 64bit)


Two's complement notation

• Why we have this solution?– Because of two zeros problem in Sign/Magnitude

notation : 10000, 00000– if (a = b)– do operation 1– else– do operation 2

Invitation to Computer Science, Java Version, Third Edition 13


If A > 0 then do nothing else get complement value of each bit a <- a+1 Eg: -3 (3bit): 3 -> 011 -> 100 -> 101 -3 (4bit): 3 -> 0011 -> 1100 -> 1101 The number of bit depends on computer (8bit, 16bit, 32bit, 64bit)

Easier method: Get complement value of each bit until before right most 1 Eg: -3 (3bit): 3-> 011 -> 101 -3 (4bit) : 3 -> 0011 -> 1101

No substraction in two’s complement notation. Convert to complement value. Eg 5 – 3 = 5 + (-3)



• Bit pattern Decimal Value

• 001 +1

• 010 +2

• 011 +3

• 100 -4

• 101 -3

• 110 -2

• 111 -1


Compare value range

• Suppose we have k bit.• Sign/Magnitude notation: -(2k-1 – 1) to (2k-1 – 1)

– Eg. 3bit: -3 ~ +3

• Two’s complement notation:-(2k-1) to (2k-1 – 1) – Eg 3bit: -4 ~ +3

• Why different?– Eg. 3bit: -3 ~ +3

– Because Sign/Magnitude notation has two zeros while two’s complement notation has only one zero.

– Though two’s complement notation is difficult to human, it much clearer to computer.


Fractional Numbers

• Fractional numbers (12.34 and –0.001275)– Can be represented in binary by using signed-

integer techniques

• Scientific notation– ±M x B±E

– M is the mantissa, B is the exponent base (usually 2), and E is the exponent

• Normalize the number– First significant digit is immediately to the right of the

binary point

Binary <-> Decimal (Fractional number)

• Binary -> Decimal 0.1101 = 1*2-1 + 1*2-2 + 0*2-3 + 1*2-4 = 0.5 + 0.25 + 0 + 0.0625 = 0.8125

• Decimal -> Binary 0.8125

• 0.8125 * 2 = 1.625 ------ Get 1

• 0.625 * 2 = 1.25 ------ Get 1

• 0.25 * 2 = 0.5 ------ Get 0

• 0.5 * 2 = 1 ------ Get 1

• Final: 0.1101 (Attention: compare to integer conversion)


Textual Information

• Code mapping– Assigning each printable letter or symbol in our

alphabet a unique number

• ASCII– International standard for representing textual

information in the majority of computers– Uses 8 bits to represent each character (256characters)

• UNICODE – Uses a 16-bit representation for characters rather

than the 8-bit format of ASCII(655,36characters)


Figure 4.3 ASCII Conversion Table


Binary Representation of Sound and Images

• Digital representation– Values for a given object are drawn from a finite set

• Analog representation– Objects can take on any value

• Figure 4.4– Amplitude of the wave: measure of its loudness– Period of the wave (T): time it takes for the wave to

make one complete cycle– Frequency f: total number of cycles per unit time


Figure 4.4 Example of Sound Represented as a Waveform


Binary Representation of Sound and Images (continued)

• Sampling rate– Measures how many times per second we sample

the amplitude of the sound wave• Bit depth

– Number of bits used to encode each sample• MP3

– Most popular and widely used digital audio format• Scanning

– Measuring the intensity values of distinct points located at regular intervals across the image’s surface


Figure 4.5 Digitization of an Analog Signal(a) Sampling the Original Signal(b) Re-creating the Signal from the Sampled Values


• Raster graphics– Each pixel is encoded as an unsigned binary value

representing its gray scale intensity

• RGB encoding scheme– Most common format for storing color images

• True Color– 24-bit color-encoding scheme

• Data compression algorithms – Attempt to represent information in ways that preserve

accuracy while using significantly less space



Data Compression

• Why we need data compression?– Because the original data need too much space.– Eg. 3,000,000 pixels/photograph * 24 bits/pixel = 72million bits.

•Simple compression method – “run – length encoding” for image compression

– Replaces a sequence of identical values v1, v2, . . ., vn by a pair of values (v, n)

– Eg. (255 255 255), (255, 0, 0), (255, 255, 255), (255, 0, 0), (255, 255, 255) → (255, 4), (0, 2), (255, 4), (0, 2), (255, 3)


Data Compression

• Simple compression method - “variable length code sets” - for text compress

Letter 4 bit encoding Variable length encoding

A 0000 00

I 0001 10

H 0010 010

W 0100 110

H A W A I I → 0010, 0000, 0100, 0000, 0001, 0001

→ 001, 00, 110, 00, 10, 10


Figure 4.8 Using Variable Length Code Sets(a) Fixed Length(b) Variable Length


Data Compression

• Compression rate = size of the uncompressed data / size of the compressed data

– Measures how much compression schemes reduce storage requirements of data



• Lossless compression schemes– No information is lost in the compression– It is possible to exactly reproduce the original data

• Lossy compression schemes – Do not guarantee that all of the information in the

original data can be fully and completely recreated


The Reliability of Binary Representation

• Computers use binary representation for reasons of reliability

• Building a base-10 “decimal computer”– Requires finding a device with 10 distinct and stable

energy states that can be used to represent the 10 unique digits (0, 1, . . . , 9) of the decimal system

• Bistable environment– Only two (rather than 10) stable states separated by

a huge energy barrier


Binary Storage Devices

• Magnetic cores – Used to construct computer memories

• Core– Small, magnetizable, iron oxide-coated “doughnut,”

about 1/50 of an inch in inner diameter, with wires strung through its center hole


Figure 4.9 Using Magnetic Cores to Represent Binary Values


Binary Storage Devices (continued)

• Transistor – Solid-state device that has no mechanical or moving

parts– Constructed from semiconductors– Can be printed photographically on a wafer of silicon

to produce a device known as an integrated circuit

• Circuit board– Interconnects all the different chips needed to run a

computer system


Figure 4.10 Relationships Among Transistors, Chips, and Circuit Boards


Binary Storage Devices (continued)

• Mask – Can be used to produce a virtually unlimited number

of copies of a chip

• Figure 4.11– Control (base): used to open or close the switch

inside the transistor– ON state: current coming from the In line

(Collector) can flow directly to the Out line (Emitter), and the associated voltage can be detected by a measuring device


Figure 4.11 Simplified Model of a Transistor


Boolean Logic

• Boolean logic– Construction of computer circuits is based on this– Boolean expression

• Constructed by combining together Boolean operations

• Example: (a AND b) OR ((NOT b) AND (NOT a))

– Truth table• capture the output/value of a Boolean expression

• A column for each input plus the output

• A row for each combination of input values


Figure 4.12 Truth Table for the AND Operation


Boolean Logic (continued)

• Boolean operations– AND, OR, NOT

• Binary operators– Require two operands

• Unary operator– Requires only one operand

• NOT operation – Reverses, or complements, the value of a Boolean

expression


Figure 4.13 Truth Table for the OR Operation


Figure 4.14 Truth Table for the NOT Operation

Gates• Gate

– Hardware devices built from transistors to mimic Boolean logic

• AND gate– Two input lines, one output line– Outputs a 1 when both inputs are 1


Gates(continued)• OR gate

– Two input lines, one output line– Outputs a 1 when either input is 1

• NOT gate– One input line, one output line– Outputs a 1 when input is 0 and vice versa



Figure 4.15 The Three Basic Gates and Their Symbols

Gates(continued)• Abstraction in hardware design

– Map hardware devices to Boolean logic– Design more complex devices in terms of logic, not

electronics– Conversion from logic to hardware design can be

automated



Building Computer Circuits

• Introduction– Circuit: collection of logic gates that transforms a

set of binary inputs into a set of binary outputs

• Every Boolean expression: – Can be represented pictorially as a circuit diagram

• Every output value in a circuit diagram: – Can be written as a Boolean expression


Figure 4.19 Diagram of a Typical Computer Circuit


A Circuit Construction Algorithm

• Step 1: Truth Table Construction– Determine how the circuit should behave under all

possible circumstances– If a circuit has N input lines and if each input line can

be either a 0 or a 1, then:• There are 2N combinations of input values, and the

truth table has 2N rows


A Truth Table for a Circuit with 8 Input Combinations


A Circuit Construction Algorithm (continued)

• Step 2: Subexpression Construction Using AND and NOT Gates– Choose any one output column of the truth table built

in step 1, and scan down that column– Every place that you find a 1 in that output column,

you build a Boolean subexpression that produces the value 1 for exactly that combination of input values and no other


Output Column Labeled Output-1 from the Previous Truth Table


Taking Snapshots

• Step 3: Subexpression Combination Using OR Gates– Take each of the subexpressions produced in step 2

and combine them, two at a time, using OR gates

• Step 4: Circuit Diagram Production– Construct the final circuit diagram

• Algorithms for circuit optimization– Reduce the number of gates needed to implement a

circuit


Figure 4.20 Circuit Diagram for the Output Labeled Output-1


Figure 4.21 The Sum-of-Products Circuit Construction Algorithm


Examples of Circuit Design and Construction

• A Compare-For-Equality Circuit– Tests two unsigned binary numbers for exact

equality– Produces the value 1 (true) if the two numbers are

equal and the value 0 ( false) if they are not


Figure 4.22 One-Bit Compare for Equality Circuit


Figure 4.23 N-Bit Compare for Equality Circuit


An Addition Circuit

• Full adder– Performs binary addition on two unsigned N-bit

integers

• Figure 4.27– Shows the complete full adder circuit called ADD

• Addition circuits – Found in every computer, workstation, and handheld

calculator in the marketplace


Figure 4.24 The 1-ADD Circuit and Truth Table


Figure 4.25 Sum Output for the 1-ADD Circuit


Figure 4.26 Complete 1-ADD Circuit for 1-Bit Binary Addition


Figure 4.27 The Complete Full Adder ADD Circuit


Control Circuits

• Used to: – Determine the order in which operations are carried

out – Select the correct data values to be processed

• Multiplexor– Circuit that has 2N input lines and 1 output line– Function: to select exactly one of its 2N input lines

and copy the binary value on that input line onto its single output line


Figure 4.28 A Two-Input Multiplexor Circuit


Control Circuits (continued)

• Decoder – Has N input lines numbered 0, 1, 2, . . . , N – 1 and

2N output lines numbered 0, 1, 2, 3, . . . , 2N – 1– Determines the value represented on its N input

lines and then sends a signal (1) on the single output line that has that identification number


Figure 4.29 A 2-to-4 Decoder Circuit


Figure 4.30 Example of the Use of a Decoder Circuit


Figure 4.31 Example of the Use of a Multiplexor Circuit


Summary

• Digital computers– Use binary representations of data: numbers, text,

multimedia

• Binary values – Create a bistable environment, making computers reliable

• Boolean logic – Maps easily onto electronic hardware


Summary (continued)

• Circuits – Constructed using Boolean expressions as an abstraction

• Computational and control circuits – Can be built from Boolean gates

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