Test #1 rescheduled to Test #1 rescheduled to next Tuesday, 09/20/05next Tuesday, 09/20/05

The contents will cover chapter 1, 2, The contents will cover chapter 1, 2, and part of Chapter 4.and part of Chapter 4.

Read the related chapters and do the Read the related chapters and do the exercises in the text book.exercises in the text book.

I will give you a brief review on I will give you a brief review on Thursday. Thursday.

"Computers in the future may "Computers in the future may weigh no more weigh no more

than 1.5 tons." than 1.5 tons." Popular Mechanics, forecasting Popular Mechanics, forecasting

the relentlessthe relentless march of science, 1949 march of science, 1949

"I think there is a world market for may be five computers."

Thomas Watson, chairman of IBM, 1943

"There is no reason anyone would want a computer in their home."

Ken Olson, president, chairman and founder of Digital Equipment Corp., 1977

Chapter 4: ObjectivesChapter 4: Objectives

Understand purpose of binary numbersUnderstand purpose of binary numbers Be able to work with binary numbers.Be able to work with binary numbers. Know conditions of bistable Know conditions of bistable

environment and how they are met.environment and how they are met. Determine truth value of Booleans.Determine truth value of Booleans. Identify transistor diagrams for AND, Identify transistor diagrams for AND,

OR, and NOT gates.OR, and NOT gates. Construct & interpret circuit diagrams. Construct & interpret circuit diagrams.

Chapter 4, Sections 1 & 2Chapter 4, Sections 1 & 2

Introduction; Binary Numbering Introduction; Binary Numbering SystemSystem

Objectives:Objectives: Understand the reason binary numbers are used.Understand the reason binary numbers are used. Translate binary and other bases to decimal.Translate binary and other bases to decimal. Do simple binary calculations.Do simple binary calculations. Know conditions of bistable environment and the Know conditions of bistable environment and the

way in which light switches, magnetic cores and way in which light switches, magnetic cores and transistors meet them.transistors meet them.

4.1 Introduction4.1 Introduction

Our computing agent: Electronic Our computing agent: Electronic Digital Computer, from the bottom up.Digital Computer, from the bottom up.

Ch. 4: At the bottom: digital logic Ch. 4: At the bottom: digital logic components (transistors, gates, components (transistors, gates, circuits)circuits)

Ch. 5: Four basic subsystems Ch. 5: Four basic subsystems (memory, ALU, CU, I/O) built from (memory, ALU, CU, I/O) built from digital logic components, and digital logic components, and organized into a complete computer organized into a complete computer system. system.

4.2 Binary Numbering 4.2 Binary Numbering SystemSystem

What is it?What is it? What are the digits in decimal What are the digits in decimal system?system? In the binary system? In the binary system?

Why we use binary in computers:Why we use binary in computers: More reliable More reliable Can be built from bi-stable Can be built from bi-stable devicesdevices

Bases commonly used in Bases commonly used in Computer ScienceComputer Science

DecimalDecimal (base 10) Digits: 0, 1, 2, ..., 9 (base 10) Digits: 0, 1, 2, ..., 9 Place Value: 10 Place Value: 10n n ... 10... 103 3 10 1022 10 1011 10 1000

BinaryBinary (base 2) Digits: 0, 1 (base 2) Digits: 0, 1 Place Value: 2 Place Value: 2n n ... 2... 23 3 2 222 2 211 2 200

OctalOctal (base 8) Digits: 0, 1, 2, ..., 7 (base 8) Digits: 0, 1, 2, ..., 7 Place Value: 8 Place Value: 8n n ... 8... 83 3 8 822 8 811 8 800

HexidecimalHexidecimal (base 16) (base 16) Digits: 0, 1, ..., 9, A, B, C, D, E, F Digits: 0, 1, ..., 9, A, B, C, D, E, F Place Value: 16 Place Value: 16n n ... 16... 163 3 16 1622 16 1611 16 1600

Converting from other Converting from other bases to Decimalbases to Decimal

Binary to decimal: 0101 1011Binary to decimal: 0101 10110*20*277 + 1*2 + 1*266 + 0*2 + 0*25 5 + 1* 2+ 1* 244

+ 1*2+ 1*233 + 0*2 + 0*222 + 1*2 + 1*21 1 + 1* 2+ 1* 200 = 91= 91 Octal to decimal: 2537Octal to decimal: 2537

2*82*833 + 5*8 + 5*822 + 3*8 + 3*81 1 + 7* 8+ 7* 800 = 1375= 1375 Hexadecimal to decimal: A3FHexadecimal to decimal: A3F

10*1610*1622 + 3*16 + 3*161 1 + 15* 16+ 15* 1600 = 2623= 2623

Representing Data in Representing Data in BinaryBinary

using 8 bits (one byte)using 8 bits (one byte) unsigned numbers:unsigned numbers:

(same as previous slide) (same as previous slide) signed numbers:signed numbers:

Use leftmost bit for sign: Use leftmost bit for sign: 1 for - 0 for + 1 for - 0 for + Represent: -56 +56 Represent: -56 +56

characters:characters: ASCII code for each. (See figure ASCII code for each. (See figure 4.3) 4.3)

Bistable DevicesBistable Devices

Two stable energy statesTwo stable energy states Two states widely separatedTwo states widely separated Can sense which state without Can sense which state without

destroying itdestroying it Can switch from one state to Can switch from one state to

anotheranother

ExamplesExamples

Two balls at different potential Two balls at different potential energy stateenergy state

. .

. .

ExamplesExamples

A capacitor in charged state and A capacitor in charged state and discharged state.discharged state.

+ + + +

_ _ _ _

It is similar to the a charged battery and discharged battery.

ExamplesExamples

Different magnetized state.Different magnetized state.

N

S

S

N

Binary Storage DevicesBinary Storage Devices

Devices which have been used for Devices which have been used for

primary computer memory--primary computer memory-- vacuum tubes vacuum tubes magnetic coremagnetic core transistortransistor integrated circuitintegrated circuit

4.1, 4.2 Homework4.1, 4.2 Homework

Read Sections 4.1 and 4.2.Read Sections 4.1 and 4.2. Count to 32 in binary (Use 6 bits)Count to 32 in binary (Use 6 bits) Exercises, p. 179 # 1, 3, 5, 8Exercises, p. 179 # 1, 3, 5, 8

4.3.1 Boolean 4.3.1 Boolean LogicLogic

A branch of mathematics A branch of mathematics which deals with two logical which deals with two logical

values:values:

True and FalseTrue and False

Boolean ExpressionsBoolean Expressions

A mathematical expression that has a A mathematical expression that has a value of True or False.value of True or False.

e.g. For which integer values of A is e.g. For which integer values of A is each of the following true?each of the following true?

A < 0A < 0 A + 5 A + 5 >> 7 7

Boolean OperationsBoolean Operations

Operations on real numbers: Operations on real numbers: + - * /+ - * /

Operations on Booleans: Operations on Booleans: NOT AND NOT AND OROR

NOT OperationNOT Operation

Similar to unary minus for numbers -- Similar to unary minus for numbers -- for an integer A. for an integer A.

-A gives the opposite of A-A gives the opposite of A e.g. if A = -3, -A = ? e.g. if A = -3, -A = ?

for a Boolean P, for a Boolean P, NOT P gives the opposite of PNOT P gives the opposite of P e.g. if P = False, NOT P = ? e.g. if P = False, NOT P = ?

Either a zero “0” or a “1”Either a zero “0” or a “1”

In Boolean Algebra, there are only In Boolean Algebra, there are only two values can be taken by any two values can be taken by any variables:variables:

““00” or “” or “11””

AND OperationAND OperationBinary operation like integer Binary operation like integer

multiplication multiplication

Partial Partial multiplication multiplication

table for A * Btable for A * B

Complete truth table Complete truth table for Boolean ANDfor Boolean AND

A B A*B-1 -1 10 -1 01 -1 -1

P Q P & QF F FF T FT F FT T T

OR OperationOR Operation

P OR Q is True P OR Q is True when either one is when either one is true or both are true or both are true.true.

Complete the truthComplete the truth

table for Boolean table for Boolean OROR

P Q P O R Q F FF TT FT T

Evaluating Boolean Evaluating Boolean ExpressionsExpressions

Evaluate each to True or False for the Evaluate each to True or False for the given values. given values.

For A = -1, B = 5, and C = 0For A = -1, B = 5, and C = 0 (A < 0) AND (B > 10)(A < 0) AND (B > 10) (A * C = C) OR NOT(C = 0) (A * C = C) OR NOT(C = 0) NOT [ (A < C) AND (B >= C) ]NOT [ (A < C) AND (B >= C) ] (B = 5) OR ([A + C] = A) AND ([A + B] = (B = 5) OR ([A + C] = A) AND ([A + B] =

3) (Precedence rule: Do AND before OR)3) (Precedence rule: Do AND before OR)

4.3.2 Electronic Gates 4.3.2 Electronic Gates for for

AND, OR, NOTAND, OR, NOTBuilt from transistors, the gates Built from transistors, the gates

produce the correct output for any produce the correct output for any given inputs.given inputs.

Symbols for the gates: Figure 4.15Symbols for the gates: Figure 4.15

How gates are built from transistors:How gates are built from transistors:

Figures 4.16, 4.17, 4.18Figures 4.16, 4.17, 4.18

Homework for 4.3Homework for 4.3

Read Section 4.3Read Section 4.3 Exercises pp. 179-180 #17, and #18Exercises pp. 179-180 #17, and #18

4.4 Circuits4.4 Circuits

CircuitCircuit: a set of logic gates that take : a set of logic gates that take binary inputs and transform them binary inputs and transform them into binary outputs.into binary outputs.

Some things computer circuits do:Some things computer circuits do: Math operations like additionMath operations like addition Comparisons (e.g. Are two inputs Comparisons (e.g. Are two inputs

equal?) equal?)

Three Ways to Represent Three Ways to Represent a Circuita Circuit

Circuit diagramCircuit diagram

Boolean expressionBoolean expression

Truth tableTruth table

Circuit Diagram: Circuit Diagram: Building BlockBuilding Block

AND, OR, and NOT gatesAND, OR, and NOT gates

+

A

B

C

C = A AND BC = A B.

A

B

C

C = A OR BC = A + B

A C

C = A NOTC = A

Construct a Circuit from Construct a Circuit from aa

Boolean ExpressionBoolean ExpressionConstruct a circuit diagram from the Construct a circuit diagram from the

following Boolean expression:following Boolean expression:

(a or b) and (c or d)(a or b) and (c or d)

Construct a Boolean Construct a Boolean Expression from a Truth Expression from a Truth

TableTable Example: to write the Example: to write the

Boolean expression forBoolean expression for

the following truth the following truth table table

a b a b c(output)c(output)

0 0 00 0 0

0 1 10 1 1

1 0 11 0 1

1 1 01 1 0

Sum of products Sum of products methodmethod

for each 1 outputfor each 1 output construct an construct an and and

sub-expressionsub-expressionusing using notnot for 0 for 0 inputsinputs

construct an construct an oror expression from expression from the the and and sub-sub-expressionsexpressions

Circuit DesignCircuit Design

Design a circuit to compare two bits. Design a circuit to compare two bits. The output should be 1 if the bits are The output should be 1 if the bits are equal and 0 otherwise.equal and 0 otherwise.

To design the circuit:To design the circuit: Construct the truth tableConstruct the truth table Write the Boolean expressionWrite the Boolean expression Construct the circuit diagramConstruct the circuit diagram

(Check on simulator.) (Check on simulator.)

Circuit to do AdditionCircuit to do Addition

A full adder circuit must be able to do A full adder circuit must be able to do additions such as the following:additions such as the following:

1 0 1 11 0 1 1

+ 0 1 1 1+ 0 1 1 1

What is the correct answer?What is the correct answer?

Diagram of a Full 4-bit Diagram of a Full 4-bit AdderAdder

a4 b4 c4=0c3

b3a3

s4s3s2s1

c1 c2b1 b2a1 a2

c0

one-bitadder

Construct a One-Bit Construct a One-Bit AdderAdder

Once we construct a one-bit adder, Once we construct a one-bit adder, we may connect as many together we may connect as many together as we like, as shown on the as we like, as shown on the previous slide.previous slide.

Construct the truth tableConstruct the truth table Write the Boolean expressionWrite the Boolean expression Construct the circuit diagramConstruct the circuit diagram

HomeworkHomework

Read Section 4.4Read Section 4.4 Do #19, p.180 and give:Do #19, p.180 and give:

truth tabletruth table boolean expressionboolean expression circuitcircuit