Digital Fundamentals

FloydFloydDigital Fundamentals, 9/eDigital Fundamentals, 9/e

Copyright ©2006 by Pearson Education, Inc.Copyright ©2006 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458Upper Saddle River, New Jersey 07458

All rights reserved.All rights reserved.Slide 1

Digital FundamentalsDigital Fundamentals

CHAPTER 2 CHAPTER 2 Number Systems, Operations, and CodesNumber Systems, Operations, and Codes




Number SystemsNumber Systems




Decimal Numbers Decimal Numbers

• The decimal number system has ten The decimal number system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

• The decimal numbering system has a The decimal numbering system has a base of 10 with each position weighted by base of 10 with each position weighted by a factor of 10:a factor of 10:




Binary Numbers 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:




Decimal-to-Binary ConversionDecimal-to-Binary Conversion




Decimal-to-Binary Conversion Decimal-to-Binary Conversion

• Sum-of-weights methodSum-of-weights method• Repeated division-by-2 methodRepeated division-by-2 method• Conversion of decimal fractions to Conversion of decimal fractions to

binarybinary




Binary ArithmeticBinary Arithmetic




Binary Arithmetic Binary Arithmetic

• Binary additionBinary addition• Binary subtractionBinary subtraction• Binary multiplicationBinary multiplication• Binary divisionBinary division




Complements of Binary NumbersComplements of Binary Numbers




Complements of Binary Numbers Complements of Binary Numbers

• 1’s complements1’s complements• 2’s complements2’s complements





• 1’s complement1’s complement





• 2’s complement2’s complement




Signed NumbersSigned Numbers




Signed Numbers 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





• Signed-magnitude form Signed-magnitude form – The sign bit is the left-most bit in a signed The sign bit is the left-most bit in a signed

binary numberbinary number– A 0 sign bit indicates a positive magnitudeA 0 sign bit indicates a positive magnitude– A 1 sign bit indicates a negative magnitudeA 1 sign bit indicates a negative magnitude





• 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





• 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)





• 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)




Arithmetic Operations with Signed NumbersArithmetic Operations with Signed Numbers

• AdditionAddition• SubtractionSubtraction• MultiplicationMultiplication• DivisionDivision





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..





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.





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.





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.





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.




Arithmetic Operations with Signed Numbers Arithmetic Operations with Signed Numbers

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.





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.





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.





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.





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.





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.




Hexadecimal NumbersHexadecimal Numbers




Hexadecimal Numbers Hexadecimal Numbers

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




Hexadecimal Numbers Hexadecimal Numbers

• Binary-to-hexadecimal conversionBinary-to-hexadecimal conversion• Hexadecimal-to-decimal conversionHexadecimal-to-decimal conversion• Decimal-to-hexadecimal conversionDecimal-to-hexadecimal conversion





• Binary-to-hexadecimal conversionBinary-to-hexadecimal conversion1.1. Break the binary number into 4-bit Break the binary number into 4-bit

groupsgroups2.2. Replace each group with the Replace each group with the

hexadecimal equivalenthexadecimal equivalent





• Hexadecimal-to-decimal conversionHexadecimal-to-decimal conversion1.1. Convert the hexadecimal to groups of 4-bit Convert the hexadecimal to groups of 4-bit

binarybinary2.2. Convert the binary to decimalConvert the binary to decimal





• Decimal-to-hexadecimal conversionDecimal-to-hexadecimal conversion– Repeated division by 16Repeated division by 16




Binary Coded Decimal (BCD)Binary Coded Decimal (BCD)




Binary Coded Decimal (BCD) Binary Coded Decimal (BCD)

Decimal and BCD digitsDecimal and BCD digits




Digital CodesDigital Codes




Digital Codes Digital Codes

• Gray codeGray code• ASCII codeASCII code





• Gray codeGray code




Digital Codes Digital Codes • ASCII code (control characters)ASCII code (control characters)




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













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




Error Detection and Correction CodesError Detection and Correction Codes




Error Detection and Correction Codes Error Detection and Correction Codes

• Parity error codesParity error codes• Hamming error codesHamming error codes





• Parity error codesParity error codes





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

data bitsdata bits

00000000000000000011100001110011011 0011011 00111100011110010101001010100101101010110101100110110011011010001101001001011100101110011001001100101001010100101010101101010111000011100001110011011001101111000111100011111111111111

00112233445566778899AABBCCDDEEFF

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Digital Fundamentals

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