CSC 110 –Intro to Computing

Lecture 14:

Midterm Review

Announcements

Reminder: Midterm is this ThursdayCovers all material discussed so farYou will be given 75 minutes, arithmetic tables,

logical propertiesEssay must be completed in class, but topic

given at end of class Service learning logs due March 23

Note slight change of date

Important Bases’ Digits

Binary (base-2) uses 2 digits: 0, 1

Octal (base-8) uses 8 digits: 0, 1, 2, 3, 4, 5, 6, 7

Decimal (base-10) uses 10 digits:0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Hexadecimal (base-16) uses 16 digits:0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Decimal Positional Notation

Formal equation for a number dn...d3d2d1d0

d0 is digit in ones place, d1 is in tens place, …

d0 * 100

d1 * 101

d2 * 102

d3 * 103

…

+ dn * 10n

Converting To Decimal

Use positional notation to any number dn...d3d2d1d0 from base-b to decimal:

+ dn * bn

…

d2 * b2

d3 * b3

d1 * b1

d0 * b0

Converting Binary to Decimal

1000112 = 1 *

1 *

0 *

0 *

0 *

+ 1 *

Converting Octal to Decimal

1078 =

Convert Hexadecimal To Decimal

AF16=

3716=

Converting From Decimal

Converting from decimal to any base-b

While the decimal number is not 0Divide the decimal number by bMove remainder to left end of

answerReplace decimal number with

quotient

33510 =

Converting Binary to Octal

Special relationship between these basesOctal uses 8 digits: 0, 1, 2, 3, 4, 5, 6, 73 binary digits (bits) represent 1 octal digit:

0002 = 08 1002 = 48

0012 = 18 1012 = 58

0102 = 28 1102 = 68

0112 = 38 1112 = 78

Every 3 bits is equal to one octal digitMakes conversion easy

Converting Binary to Octal

Starting from the right, split number into groupings of 3 bits

For each group from right to leftConvert each group from binary to decimalAdd digit to left end of result

Converting Binary to Octal

Convert from binary to octal:11110011002

Converting Binary to Octal

Convert each group to decimal:

1 111 001 100 41002 = 0 * 20 = 0 * 1 = 0

0 * 21 = 0 * 2 = 0

1 * 22 = 1 * 4 = + 4

4

Converting Binary to Octal

Convert each group to decimal:

1 111 001 100 1 40012 = 1 * 20 = 1 * 1 = 1

0 * 21 = 0 * 2 = 0

0 * 22 = 0 * 4 = + 0

1

Converting Binary to Octal

Convert each group to decimal:

1 111 001 100 7 1 41112 = 1 * 20 = 1 * 1 = 1

1 * 21 = 1 * 2 = 2

1 * 22 = 1 * 4 = + 4

7

Converting Binary to Octal

Convert each group to decimal:

1 111 001 100 1 7 1 4 12 = 1 * 20 = 1 * 1 = 1

Converting Binary to Octal

That’s all there us to it!

11110011002 = 17148

Converting Octal To Binary

For each digit (working from right to left)Convert the digit from decimal to binaryAdd 0s to left of the binary to fill 3 bitsAdd binary number to left-end of solution

Finally, remove any 0s padding leftmost digit

Converting Octal to Binary

Convert from octal to binary:3168

Binary to powers-of-two bases

Special relationship between binary and any power-of-two (2, 4, 8, 16, 32…) baseOctal is base-8

23 = 8 Use groupings of 3 bits converting to/from octal

Hexadecimal is base-16 24 = 16 Use groupings of 4 bits converting to/from octal

Conversion Quick Guide

When converting to decimal… …use positional notation

When converting from decimal… …divide by base being converted into

When converting between binary and power-of-two base……use groupings of bits of appropriate size

Addition Algorithm

To add 2 numbers in base-b:

For each position from right to leftCompute sum of the digitsIf sum > b then

sum = sum- bCarry the 1

EndRecord sum at position

Adding Octal Numbers

For each position from right to left

1 2 68

+ 7 2 28

Adding Octal Numbers

Record sum at position

1

1 2 68

+ 7 2 28

08

Adding Octal Numbers

For each position from right to left

1

1 2 68

+ 7 2 28

5 08

Adding Octal Numbers

For each position from right to left

1 1

1 2 68

+ 7 2 28

0 5 08

Adding Hex Numbers

For each position from right to left

B A 616

+ E 0 216

Boolean Negation: NOT

NOT is simplest operationReturns the negation of the input

1 when a is 0; 0 when a is 1

Written as a

0 1

1 0

aaa

Boolean Operation: OR

OR takes two values as inputsWritten as Returns 1 when either value is 1

1 when either a OR b are 1; 0 otherwise

0 00 11 01 1

a b a b

a b

a

b

Boolean Operation: AND

AND also takes two inputsWritten asReturns 1 when both values are 1

1 when a AND b are 1; 0 otherwise

0 00 11 01 1

a b a b

a b

a

b

Boolean Operation: XOR

First slightly unfamiliar idea: XORXOR computes “exclusive or” Written as 1 when a or b, but not both, are 1

0 00 11 01 1

a b a b

a b

a

b

Boolean Operation: NAND

NAND computes NEGATIVE ANDWritten as

Combines notation for AND & NOT

1 when either value is 0 Equivalent to negation of AND

0 0 10 1 11 0 11 1 0

a b a b

a b

a

b

Boolean Operation: NOR

Last operation is NEGATIVE ORWritten as 1 when both values are 0

Also equal to negation of OR

0 0 10 1 01 0 01 1 0

a b a b

a b

a

b

Creating Truth Tables

There is a secret to making the inputs1st input – Alternate one 0 and one 12nd input – Repeat two 0s then two 1s3rd input – Repeat four 0s then four 1s4th input – Repeat eight 0s then eight 1s

Start with row of all 0s Stop when you reach a row with all 1s

Create the Truth Table

( ) ( ) c a a b

Drawing Logic Diagrams

Often use results from one gate as input into another gate

Show by connecting wire from first gate’s output to second gate’s inputE.g., result from used in ( ) a b cb c

b

c

a

Computing Propagation Delay

Set initial delay for circuit inputs to 0For each gate whose input’s delays are known

Find the input with the longest delay Add 1 to this delay time

Set gate’s delay to this new time

Keep going until delay for entire circuit is known

Draw the Circuit

( ) ( ) c a a b

More About Boolean Properties

Properties identify equivalent circuitsE.g., circuits with identical truth table results

Many ways we use these propertiesReduce delays by doing more work in parallelSimplify circuits by removing useless gates

0 1 0

1 0 1

aa a

Using Boolean Properties

Circuit Property Used(c·d)+(d+c) Identity(c·d)+(d·c) DeMorgan’s(c·d)+(d·c) Double Negation(c·d)+(c·d) Commutative

c·(d+d) Distributivec·1 Complementc Identity

Reduce to c:

Adder Circuits

How to know a circuit is equivalent to half-adder or full-adderCircuits are equivalent to if both truth tables

have identical outputEquivalence can be found in two ways:

Memorize circuit and its truth Memorize circuit and its truth tabletable

Check that the carry out and result match the carry, result of adding inputs together

Full-Adders

Full-adder adds two input bits and the carry bit from adding previous bitOutputs result bit and another carry bit

Include new full-adder to add each bit except least significant bit

Adders

Half-adder Full adder

Result

Carry Out

Result

Combining Adders

Circuit adding two 2-bit numbers together

C0

Carry Out

C1

C2

A1

B1

A0

B0

Computer Components

Input Unit Device through which data are entered into the computer

Output Unit Device through which data stored within computer made available to outside world

Secondary Storage Device Device which reads and writes information into computer memory

von Neumann Computer Connections

Harvard Computer Connections

Program

Data

Palgo Loop Selection

Each of Palgo’s loops serves different purpose repeat

Executing the commands a specific number of times for

Remembering the number of times that we are executing the instructions in a loop

Very useful for lists! while

Executing a loop for as long as needed to meet a specific condition

Midterm Essay

Electronic computers have gotten easier to use since they were first developed. Discuss improvements to input, output, and secondary storage devices, and software that have made these machines easier to use.For full credit, you must discuss improvements

including at least 1 input device, at least 1 output device, and at least 1 secondary storage device or program.