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Logic Circuits and Switching Theory

Digital Electronics : A Practical Approach by William KleitzDigital Design by Morris Mano

Prepared by Engr. Nerissa L. Serrano

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A prerequisite to

understandingmicroprocessor system is

understanding Logic

Design CombiningSimple Blocks intoComplicated Systems

Physics

Devices

Analog

Circuits

Digital

Circuits

Logic

Micro-

architecture

Architecture

Operating

Systems

Application

Software

electrons

transistors

diodes

amplifiers

filters

AND gates

NOT gates

adders

memories

datapaths

controllers

instructions

registers

device drivers

programs

focus

ofthis

cour

se

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Base 2, Base 3, Base 4, Base 5, Base 6,..

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Why Learn the Number System?

The computer does not understand the words andletters. Rather, those words and letters are translatedinto numbers. Computers talk and understand innumbers.

If you have an understanding of the number systems,you can understand how to calculate NetworkAddresses, Memory Address Locations in core dumps,Boolean Algebra, etc.

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Each system is characterized by the

number of independent digits or

symbols they use.

The number of symbols are often called

the base, radix, modulo, or mod.

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Binary (Base-2)0,1

Octal (Base-8)0,1,2,3,4,5,6,7

Decimal (Base-10)0,1,2,3,4,5,6,7,8,9

Hexadecimal (Base 16)

0,1,2,3,4,5,6,7,8,9,A,B

,C,D,E,F

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To indicate the type of number system used,the radix is used as a subscript.

e.g. 10101012

6548

A216

8910

NOTE: The absence of a subscript usually denotes Decimalnumber system

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GENERAL PRINCIPLES

1. Whenever it is desired to find the decimal equivalent ofa given number in another number system, it is given bythe sum of all the digits multiplied by their weights orplace values. The integer and fractional parts should be

handled separately. Starting from the radix point, theweights of different digits are r0, r1, r2 for the integer partand r1, r2, r3 for the fractional part, where r is the radix ofthe number system whose decimal equivalent needs to be

determined

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BASE-N to BASE-10 EXAMPLES

1. Convert 1011.18 to Decimal1011.1

x 0 = 1

x 1 =x = 0x 3 = 8

11.5

x -1 = 0.5

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BASE-N to BASE-10 EXAMPLES

2. Convert 27.158 to Decimal27.158

x 8-2 = 0.078125

x 8-1 = 0.125x 80 = 7x 81 = 16

23.203125

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BASE-N to BASE-10 EXAMPLES

3. Convert 4AC.816 to Decimal4AC.816x 16-1 = 0.5

x 160 = 12x 161 = 160x 162 = 1024

1196.5

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GENERAL PRINCIPLES

2. To convert a given mixed decimal number into an equivalentin another number system, the integer part is progressively

divided by r and the remainders noted until the result of

division yields a zero quotient. The remainders written in

reverse order constitute the equivalent. r is the radix of thetransformed number system. The fractional part is

progressively multiplied by r and the carry recorded until the

result of multiplication yields a zero or when the desired

number of bits has been obtained. The carrys written in forwardorder constitute the equivalent of the fractional part.

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BASE-10 toBASE-N Examples

1.Convert 235.5 to Binaryr Dividend R

2 235 1

2 117 1

2 58 0

2 29 12 14 0

2 7 1

2 3 1

2 1 1

0

0.5 x 2 = 1.0

Ans:

11101011.12

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BASE-10 toBASE-N Examples

1.Convert 99.8125 to Octal

r Dividend R

8 99 3

8 12 4

8 1 1

8

.8125 x 8 = 6.5

Ans: 143.648

.5 x 8 = 4.

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BASE-10 toBASE-N Examples

1.Convert 68.75 to Hex

r Dividend R

16 68 4

16 4 4

16 0

0.75 x 16 = 12.0

Ans: 44.C16

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GENERAL PRINCIPLES

3. For octalbinary conversion, replace each digit in the octal

number with its three-bit binary equivalent. For hexadecimalbinary

conversion, replace each hex digit with its four-bit binary equivalent.

For binaryoctal conversion, split the binary number into groups of

three bits, starting from the binary point, and, if needed, complete

the outside groups by adding Os, and then write the octalequivalent of these three-bit groups. For binaryhex conversion,

split the binary number into groups of four bits, starting from the

binary point, and, if needed, complete the outside groups by adding

Os, and then write the hex equivalent of the four-bit groups..

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Octal to Binary Example

1. Convert 4568 to Binary

4568

100 101 110

Ans: 1001011102

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Binary to Octal Example

Convert 1011101110112 to Octal

101101110112Ans: 26738

62 7 3

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Hex toBinary Conversion Example

Convert 5AF16 to Binary

5AF8

0101 1010 1111

Ans:

101101011112

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Binary to Hexadecimal Conversion

Convert 101110111011012 to Hex

101110111011012Ans:2EED16

E2 E D

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GENERAL PRINCIPLES

4. For octalhexadecimal conversion, we can gofrom the given octal number to its binaryequivalent and then from the binary equivalent toits hex counterpart. For hexadecimaloctal

conversion, we can go from the hex to its binaryequivalent and then from the binary number to itsoctal equivalent.

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Hex to Octal Conversion Example

Convert C3F16 to Octal

C3F16

1100 0011 1111

1100001111112

ANS:

60778

6 0 7 7

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Octal to Hex Conversion Example

Convert 1 to Hex

1

110 001 010

110001010

ANS: 1 A1

A1

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You have to learn these by heart!

Any Number System to Decimal

Multiply each digit by rn where ris the radix and n is the position

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You have to learn these by heart!

Decimal to Any Number System Whole number Successive Division.

The first digit of the answer is the last

remainder None-Whole Number Progressive

Multiplication.

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You have to learn these by heart!

Binary to Octal

Group the bits by 3 and convertdirectly

Binary to Hex Group the bits by 4 and convert

directly

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You have to learn these by heart!

Octal to Binary

Convert each digit to 3 bits

Hex to Binary Convert each digit to 4 bits

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You have to learn these by heart!

To convert Hex to Octal andOctal to Hex it is easier to

convert first to binary rather thanto decimal.

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Solve without using a calculator please

What is the Decimal Equivalent of

1234.315?

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What is the Decimal Equivalent of 1234.315?

1234.315x 5-2 = 0.04

x 5-1 = 0.6

x 50 = 4

x 52 = 50x 51 = 15

x 53 = 125

194.64

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Solve without using a calculator please

Convert 125.75 into its Base 4

equivalent.

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Convert 125.75 into its Base 4 equivalent.

r Dividend R

4 125 1

4 31 3

4 7 3

4 1 1

4 0

0.75 x 4 = 3.0

Ans: 1331.34

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Perform the indicated conversion:

1.3568 HEX

2.AE616 DEC

3.4108 BIN4.1203 OCT

5.101 OCT

6.13410 HEX

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Perform the indicated conversion:

1.3568 HEX = EE162.AE616 DEC = 2790

3.4108 BIN = 10000100024.1203 OCT = 1785.101 OCT = 14586.13410 HEX = 86