Digital Fundamentals

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Digital FundamentalsDigital Fundamentals

CHAPTER 1 (CONT…)CHAPTER 1 (CONT…)

Number Systems, Operations, and CodesNumber Systems, Operations, and Codes

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1. Number Systems1. Number Systems

1.1 Decimal Numbers 1.1 Decimal Numbers

• The decimal number system has ten digits: The decimal number system has ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, and 90, 1, 2, 3, 4, 5, 6, 7, 8, and 9

• The decimal numbering system has a base The decimal numbering system has a base

of 10 with each position weighted by a factor of 10 with each position weighted by a factor

of 10:of 10:

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1.2 Binary Numbers1.2 Binary Numbers

• The binary number system has two digits: The binary number system has two digits: 0 and 10 and 1

• The binary numbering system has a base of 2 The binary numbering system has a base of 2 with each position weighted by a factor of 2:with each position weighted by a factor of 2:

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2.2. ConversionConversion

2.1 Binary to Decimal Conversion2.1 Binary to Decimal Conversion

• Convert binary to decimal by summing the Convert binary to decimal by summing the positions that contain a 1.positions that contain a 1.

1 0 0 1 0 12

10012345 371432222222

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2.2 Decimal to Binary Conversion2.2 Decimal to Binary Conversion

• Two methods to convert decimal to Two methods to convert decimal to binary:binary:

– Reverse process described in 2.1Reverse process described in 2.1– Use repeated divisionUse repeated division

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Decimal to Binary Conversion (cont…)Decimal to Binary Conversion (cont…)

• Reverse process described in 2.1Reverse process described in 2.1– Note that all positions must be accounted forNote that all positions must be accounted for

02510 20200237

1 0 0 1 0 21

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• Repeated division steps:Repeated division steps:– Divide the decimal number by 2Divide the decimal number by 2– Write the remainder after each division until a Write the remainder after each division until a

quotient of zero is obtained.quotient of zero is obtained.– The first remainder is the LSB and the last is The first remainder is the LSB and the last is

the MSBthe MSB• Note, when done on a calculator, a fractional Note, when done on a calculator, a fractional

answer indicates a remainder of 1.answer indicates a remainder of 1.


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• Repeated division – Repeated division – This flowchart This flowchart describes the describes the process and can be process and can be used to convert from used to convert from decimal to any other decimal to any other number system.number system.


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3. Hexadecimal Number System3. Hexadecimal Number System

• Decimal, binary, and hexadecimal Decimal, binary, and hexadecimal numbersnumbers

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• Most digital systems deal with groups Most digital systems deal with groups of bits in even powers of 2 such as 8, of bits in even powers of 2 such as 8, 16, 32, and 64 bits.16, 32, and 64 bits.

• Hexadecimal uses groups of 4 bits.Hexadecimal uses groups of 4 bits.

• Base 16Base 16– 16 possible symbols16 possible symbols– 0-9 and A-F0-9 and A-F

Hexadecimal Number System (cont….)Hexadecimal Number System (cont….)

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• Convert from hex to decimal by Convert from hex to decimal by multiplying each hex digit by its multiplying each hex digit by its positional weight.positional weight.

Example:Example:

)16(3)16(6)16(1163 01216

131662561

10355


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• Convert from decimal to hex by using the Convert from decimal to hex by using the repeated division method used for decimal repeated division method used for decimal to binary and decimal to octal conversion.to binary and decimal to octal conversion.

• Divide the decimal number by 16Divide the decimal number by 16

• The first remainder is the LSB and the last The first remainder is the LSB and the last is the MSB.is the MSB.– Note, when done on a calculator a decimal remainder Note, when done on a calculator a decimal remainder

can be multiplied by 16 to get the result. If the can be multiplied by 16 to get the result. If the remainder is greater than 9, the letters A through F remainder is greater than 9, the letters A through F are used.are used.


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• Example of hex to binary conversion:Example of hex to binary conversion:

9F29F216 16 = 9= 9 F 2F 2

1001 1111 0010 = 1001111100101001 1111 0010 = 10011111001022

Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F

Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111


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• Convert from binary to hex by Convert from binary to hex by grouping bits in four starting with the grouping bits in four starting with the LSB.LSB.

• Each group is then converted to the Each group is then converted to the hex equivalenthex equivalent

• Leading zeros can be added to the Leading zeros can be added to the left of the MSB to fill out the last left of the MSB to fill out the last group.group.


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• Example of binary to hex conversion.Example of binary to hex conversion. (Note the addition of leading zeroes)(Note the addition of leading zeroes)

1110100110111010011022 = 0011 1010 0110 = 0011 1010 0110

= 3 A 6= 3 A 6

= 3A6= 3A61616

• Counting in hex requires a reset and carry after Counting in hex requires a reset and carry after reaching F.reaching F.

Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F

Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111


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• Hexadecimal is useful for Hexadecimal is useful for representing long strings of bits.representing long strings of bits.

• Understanding the conversion Understanding the conversion process and memorizing the 4 bit process and memorizing the 4 bit patterns for each hexadecimal digit patterns for each hexadecimal digit will prove valuable later.will prove valuable later.


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4. Binary Coded Decimal (BCD)4. Binary Coded Decimal (BCD)

• Binary Coded Decimal (BCD) is another Binary Coded Decimal (BCD) is another way to present decimal numbers in binary way to present decimal numbers in binary form.form.

• BCD is widely used and combines BCD is widely used and combines features of both decimal and binary features of both decimal and binary systems.systems.

• Each digit is converted to a binary Each digit is converted to a binary equivalent.equivalent.

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Binary Coded Decimal (cont…)Binary Coded Decimal (cont…)

Decimal and BCD digitsDecimal and BCD digits

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• To convert the number 874To convert the number 8741010 to BCD: to BCD:

88 77 44

0100 0111 0100 = 0100011101000100 0111 0100 = 010001110100BCDBCD

• Each decimal digit is represented using 4 Each decimal digit is represented using 4 bits.bits.

• Each 4-bit group can never be greater Each 4-bit group can never be greater than 9.than 9.

• Reverse the process to convert BCD to Reverse the process to convert BCD to decimal.decimal.


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• BCD is not a number system.BCD is not a number system.• BCD is a decimal number with each BCD is a decimal number with each

digit encoded to its binary equivalent.digit encoded to its binary equivalent.• A BCD number is not the same as a A BCD number is not the same as a

straight binary number.straight binary number.• The primary advantage of BCD is the The primary advantage of BCD is the

relative ease of converting to and from relative ease of converting to and from decimal.decimal.


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5. Digital Codes5. Digital Codes

• Gray code:Gray code: The gray code is The gray code is

used in applications used in applications where numbers where numbers change rapidly.change rapidly.

In the gray code, In the gray code, only one bit changes only one bit changes from each value to from each value to the next.the next.

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• ASCII – American Standard Code for ASCII – American Standard Code for Information Interchange.Information Interchange.– Seven bit code: 2Seven bit code: 277 = 128 possible code = 128 possible code

groupsgroups– Table 2-4 lists the standard ASCII codesTable 2-4 lists the standard ASCII codes– Examples of use are: to transfer information Examples of use are: to transfer information

between computers, between computers and between computers, between computers and printers, and for internal storage.printers, and for internal storage.

Digital Codes (cont…)Digital Codes (cont…)

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• ASCII code (control characters)ASCII code (control characters)

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• ASCII code (graphic symbols 20h – 3Fh)ASCII code (graphic symbols 20h – 3Fh)

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Extended ASCII code (80h – FFh)Extended ASCII code (80h – FFh)

• Non-English alphabetic charactersNon-English alphabetic characters

• Currency symbolsCurrency symbols

• Greek lettersGreek letters

• Math symbolsMath symbols

• Drawing charactersDrawing characters

• Bar graphing charactersBar graphing characters

• Shading charactersShading characters


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6. Putting It All Together6. Putting It All TogetherDecimal Binary Hexadecimal BCD Gray

0 0 0 0 01 1 1 0001 00012 10 2 0010 00113 11 3 0011 00104 100 4 0100 01105 101 5 0101 01116 110 6 0110 01017 111 7 0111 01008 1000 8 1000 11009 1001 9 1001 110110 1010 A 0001 0000 111111 1011 B 0001 0001 111012 1100 C 0001 0010 101013 1101 D 0001 0011 101114 1110 E 0001 0100 100115 1111 F 0001 0101 1000

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7. Complements of Binary Numbers7. Complements of Binary Numbers

• 1’s complements1’s complements

• 2’s complements2’s complements

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Complements of Binary Numbers (cont…)Complements of Binary Numbers (cont…)

• 1’s complement1’s complement

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Complements of Binary Numbers (cont…)Complements of Binary Numbers (cont…)

• 2’s complement2’s complement

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8. Signed Numbers8. Signed Numbers

• Signed-magnitude formSigned-magnitude form

• 1’s and 2’s complement form1’s and 2’s complement form

• Decimal value of signed numbersDecimal value of signed numbers

• Range of valuesRange of values

• Floating-point numbersFloating-point numbers

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Signed Numbers (cont…)Signed Numbers (cont…)

• Signed-magnitude formSigned-magnitude form

– The sign bit is the left-most bit in a The sign bit is the left-most bit in a signed binary numbersigned binary number

– A 0 sign bit indicates a positive A 0 sign bit indicates a positive magnitudemagnitude

– A 1 sign bit indicates a negative A 1 sign bit indicates a negative magnitudemagnitude

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• 1’s complement form1’s complement form– A negative value is the 1’s complement of A negative value is the 1’s complement of

the corresponding positive valuethe corresponding positive value

• 2’s complement form2’s complement form– A negative value is the 2’s complement of A negative value is the 2’s complement of

the corresponding positive valuethe corresponding positive value

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• Decimal value of signed numbersDecimal value of signed numbers– Sign-magnitudeSign-magnitude– 1’s complement1’s complement– 2’s complement2’s complement

• Range of ValuesRange of Values2’s complement form:2’s complement form:

– – (2(2n n – – 11) to + (2) to + (2n – 1 n – 1 – – 1)1)

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• Floating-point numbersFloating-point numbers– Single-precision (32 bits)Single-precision (32 bits)– Double-precision (64 bits)Double-precision (64 bits)– Extended-precision (80 bits)Extended-precision (80 bits)

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8.1 Arithmetic Operations with 8.1 Arithmetic Operations with Signed Numbers Signed Numbers

• AdditionAddition

• SubtractionSubtraction

• MultiplicationMultiplication

• DivisionDivision

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Arithmetic Operations with Signed Arithmetic Operations with Signed Numbers (cont…)Numbers (cont…)

Addition of Signed NumbersAddition of Signed Numbers

• The parts of an addition function are:The parts of an addition function are:– AddendAddend– AugendAugend– SumSum

Numbers are always added two at a timeNumbers are always added two at a time..

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Four conditions for adding numbers:Four conditions for adding numbers:

• Both numbers are positive.Both numbers are positive.

• A positive number that is larger than a A positive number that is larger than a negative number.negative number.

• A negative number that is larger than a A negative number that is larger than a positive number.positive number.

• Both numbers are negative.Both numbers are negative.

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Signs for AdditionSigns for Addition

• When both numbers are positive, the When both numbers are positive, the sum is positive.sum is positive.

• When the larger number is positive and When the larger number is positive and the smaller is negative, the sum is the smaller is negative, the sum is positive. The carry is discarded.positive. The carry is discarded.

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Signs for AdditionSigns for Addition

• When the larger number is negative and When the larger number is negative and the smaller is positive, the sum is the smaller is positive, the sum is negative (2’s complement form).negative (2’s complement form).

• When both numbers are negative, the When both numbers are negative, the sum is negative (2’s complement form). sum is negative (2’s complement form). The carry bit is discarded.The carry bit is discarded.

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Subtraction of Signed NumbersSubtraction of Signed Numbers

• The parts of a subtraction function are:The parts of a subtraction function are:– SubtrahendSubtrahend– MinuendMinuend– DifferenceDifference

Subtraction is addition with the sign of the Subtraction is addition with the sign of the subtrahend changed.subtrahend changed.


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SubtractionSubtraction

• The sign of a positive or negative binary The sign of a positive or negative binary number is changed by taking its 2’s number is changed by taking its 2’s complementcomplement

• To subtract two signed numbers, take To subtract two signed numbers, take the 2’s complement of the subtrahend the 2’s complement of the subtrahend and add. Discard any final carry bit.and add. Discard any final carry bit.


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Multiplication of Signed NumbersMultiplication of Signed Numbers

• The parts of a multiplication function are:The parts of a multiplication function are:– MultiplicandMultiplicand– MultiplierMultiplier– ProductProduct

Multiplication is equivalent to adding a Multiplication is equivalent to adding a number to itself a number of times equal to number to itself a number of times equal to the multiplier.the multiplier.


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There are two methods for multiplication:There are two methods for multiplication:

• Direct additionDirect addition

• Partial productsPartial products

The method of partial products is the most The method of partial products is the most commonly used.commonly used.


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Multiplication of Signed NumbersMultiplication of Signed Numbers

• If the signs are the same, the product is If the signs are the same, the product is positive.positive.

• If the signs are different, the product is If the signs are different, the product is negative.negative.


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Division of Signed NumbersDivision of Signed Numbers

• The parts of a division operation are:The parts of a division operation are:– DividendDividend– DivisorDivisor– QuotientQuotient

Division is equivalent to subtracting the Division is equivalent to subtracting the divisor from the dividend a number of divisor from the dividend a number of times equal to the quotient.times equal to the quotient.


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Division of Signed NumbersDivision of Signed Numbers

• If the signs are the same, the quotient is If the signs are the same, the quotient is positive.positive.

• If the signs are different, the quotient is If the signs are different, the quotient is negative.negative.


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• Binary data and codes are frequently moved Binary data and codes are frequently moved between locations. For example:between locations. For example:– Digitized voice over a microwave link.Digitized voice over a microwave link.– Storage and retrieval of data from magnetic and Storage and retrieval of data from magnetic and

optical disks.optical disks.– Communication between computer systems over Communication between computer systems over

telephone lines using a modem.telephone lines using a modem.

• Electrical noise can cause errors during Electrical noise can cause errors during transmission.transmission.

• Many digital systems employ methods for error Many digital systems employ methods for error detection (and sometimes correction).detection (and sometimes correction).

9. Parity Method for Error Detection9. Parity Method for Error Detection

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Parity Method for Error Detection Parity Method for Error Detection (cont…)(cont…)

• The parity method of error detection The parity method of error detection requires the addition of an extra bit to a requires the addition of an extra bit to a code group.code group.

• This extra bit is called the parity bit.This extra bit is called the parity bit.

• The bit can be either a 0 or 1, depending The bit can be either a 0 or 1, depending on the number of 1s in the code group.on the number of 1s in the code group.

• There are two methods, even and odd.There are two methods, even and odd.

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• Even parity method – the total Even parity method – the total number of bits in a group including number of bits in a group including the parity bit must add up to an even the parity bit must add up to an even number.number.– The binary group 1 0 1 1 would require The binary group 1 0 1 1 would require

the addition of a parity bit the addition of a parity bit 11 1 0 1 11 0 1 1• Note that the parity bit may be added at Note that the parity bit may be added at

either end of a group.either end of a group.


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• Odd parity method – the total number Odd parity method – the total number of bits in a group including the parity of bits in a group including the parity bit must add up to an odd number.bit must add up to an odd number.– The binary group 1 1 1 1 would require The binary group 1 1 1 1 would require

the addition of a parity bit the addition of a parity bit 11 1 1 1 1 1 1 1 1


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• The transmitter and receiver must The transmitter and receiver must “agree” on the type of parity checking “agree” on the type of parity checking used.used.

• Two bit errors would not indicate a Two bit errors would not indicate a parity error.parity error.

• Both odd and even parity methods are Both odd and even parity methods are used, but even seems to be used used, but even seems to be used more often.more often.


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• Parity error codesParity error codes

Parity Method for Error Detection (cont…)Parity Method for Error Detection (cont…)

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• Hamming error codesHamming error codes– Hamming code wordsHamming code words– Hex equivalent of the Hex equivalent of the

data bitsdata bits

00000000000000

00001110000111

0011011 0011011

00111100011110

01010100101010

01011010101101

01100110110011

01101000110100

10010111001011

10011001001100

10100101010010

10101011010101

11000011100001

11001101100110

11110001111000

11111111111111

00

11

22

33

44

55

66

77

88

99

AA

BB

CC

DD

EE

FF

Parity Method for Error Detection (cont…)Parity Method for Error Detection (cont…)

Digital Fundamentals

Documents

decimal conversion

binary number system

decimal remainder

decimal numbering system

hexadecimal number system

binary numbering system

binary numbers

example of binary