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1、会计学1数字数字(shz)电子技术英文电子技术英文第一页，共67页。Introductory ParagraphIntroductory Paragraphn nYou are familiar with the decimal number system You are familiar with the decimal number system because you use decimal numbers every day.Although because you use decimal numbers every day.Although decimal numbers are c
2、ommonplace,their weighted decimal numbers are commonplace,their weighted structure is often not understood.In this section,the structure is often not understood.In this section,the structure of decimal numbers is reviewed.Their structure of decimal numbers is reviewed.Their review will help you more
3、 easily understand the review will help you more easily understand the structure of the binary number system,which is structure of the binary number system,which is important in computers and digital electronics.important in computers and digital electronics.第2页/共67页第二页，共67页。Decimal ReviewDecimal Re
4、viewn nNumbers consist of a bunch of digits,each with a Numbers consist of a bunch of digits,each with a weightweight.These weights are all powers of the base,which is 10.We can rewrite this:To find the decimal value of a number,multiply each digit by its weight and sum the products.第3页/共67页第三页，共67页
5、。Nothing Special about 10!Nothing Special about 10!n nDecimal system(and the idea of“0”)was invented in India Decimal system(and the idea of“0”)was invented in India around 100-500AD.around 100-500AD.n nWhy did they use 10?Anything special about it?Why did they use 10?Anything special about it?n nNo
6、t really.Not really.n nProbably the fact that we have 10 fingers influenced this.Probably the fact that we have 10 fingers influenced this.n nWill a base other than 10 work?Will a base other than 10 work?n nSure.Sure.What about base 2?第4页/共67页第四页，共67页。2-2 BINARY NUMBERS二进制数二进制数二进制数二进制数第5页/共67页第五页，共6
7、7页。Introductory ParagraphIntroductory Paragraphn nThe binary number system is simply another way to The binary number system is simply another way to represent quantities.The binary system is less represent quantities.The binary system is less complicated than the decimal system because it has compl
8、icated than the decimal system because it has only two digits.It may seem more difficult at first only two digits.It may seem more difficult at first because it is unfamiliar to you.The decimal system because it is unfamiliar to you.The decimal system with its ten digits is a base-ten system;with it
9、s ten digits is a base-ten system;the binary the binary system with its two digits is a base-two systemsystem with its two digits is a base-two system.The two.The two digits(bits)are 1 and 0.The position of a 1 or 0 in a digits(bits)are 1 and 0.The position of a 1 or 0 in a binary number indicates i
10、ts weight,or value within binary number indicates its weight,or value within the number,just as the position of a decimal digit the number,just as the position of a decimal digit determines the value of that digit.determines the value of that digit.The weights in a The weights in a binary number are
11、 based on powers of twobinary number are based on powers of two.第6页/共67页第六页，共67页。Counting in BinaryCounting in Binaryn nA binary count of 0 through 15 is shown below.As you will A binary count of 0 through 15 is shown below.As you will see,4 bits are required to count from 0 to 15.see,4 bits are req
12、uired to count from 0 to 15.第7页/共67页第七页，共67页。Counting Tennis Balls Going into a box from a Conveyor Counting Tennis Balls Going into a box from a Conveyor BeltBeltn nThe counter counts the pulses from a sensor that detects The counter counts the pulses from a sensor that detects the passing of a bal
13、l and produces a sequence of logic levels the passing of a ball and produces a sequence of logic levels(digital waveforms)on each of its four parallel outputs.Each(digital waveforms)on each of its four parallel outputs.Each set of logic levels represent a 4-bit binary number.The set of logic levels
14、represent a 4-bit binary number.The decoder decodes each set of four bits and converts it to the decoder decodes each set of four bits and converts it to the corresponding decimal number in the 7-segment display.corresponding decimal number in the 7-segment display.第8页/共67页第八页，共67页。The Weighting Str
15、ucture of Binary NumbersThe Weighting Structure of Binary Numbersn nA binary number is a weighted numberA binary number is a weighted number（加权数）（加权数）.The right-.The right-most bit is the LSBmost bit is the LSB（最低有效（最低有效(y(y uxio)uxio)位）位）in a binary whole in a binary whole number and has a weight o
16、f 20=1.The weights increase from number and has a weight of 20=1.The weights increase from right right toto left by a power of two for each bit.The left-most left by a power of two for each bit.The left-most bit is the MSBbit is the MSB（最高有效（最高有效(y(y uxio)uxio)位）位）.n nFractional numbersFractional nu
17、mbers（小数）（小数）can also be represented in binary can also be represented in binary by placing bits by placing bits toto the right of the binary point.The left-most the right of the binary point.The left-most bit is the MSB in a binary fractional number and has a weight bit is the MSB in a binary fract
18、ional number and has a weight of 2-1=0.5.The fractional weights decreases from left of 2-1=0.5.The fractional weights decreases from left toto right by a negative power of two for each bit.right by a negative power of two for each bit.n nThe weight structure of a binary number isThe weight structure
19、 of a binary number is第9页/共67页第九页，共67页。Binary-to Decimal ConversionBinary-to Decimal Conversionn nThe decimal value of any binary number can be The decimal value of any binary number can be found by adding the weights of all bits that are 1 found by adding the weights of all bits that are 1 and disc
20、arding the weights of all bits that are 0.and discarding the weights of all bits that are 0.第10页/共67页第十页，共67页。2-3 DECIMAL-TO-BINARY 2-3 DECIMAL-TO-BINARY CONVERSIONCONVERSION第11页/共67页第十一页，共67页。Repeated Division-by-2 MethodRepeated Division-by-2 Methodn nA systematic method of converting whole number
21、s from A systematic method of converting whole numbers from decimal to binary is the repeated devision-by-2 process.decimal to binary is the repeated devision-by-2 process.第12页/共67页第十二页，共67页。Repeated Multiplication-by-2 MethodRepeated Multiplication-by-2 Methodn nA systematic method of converting fr
22、actional numbers A systematic method of converting fractional numbers from decimal to binary is the repeated multiplication-by-from decimal to binary is the repeated multiplication-by-2 process.2 process.第13页/共67页第十三页，共67页。2-4 BINARY ARITHMETIC2-4 BINARY ARITHMETIC第14页/共67页第十四页，共67页。Binary AdditionB
23、inary Additionn nThe four basic rules for adding binary digits(bits)are The four basic rules for adding binary digits(bits)are as follows:as follows:0+0=0 Sum of 0 with a carry of 00+1=1 Sum of 1 with a carry of 01+0=1 Sum of 1 with a carry of 01+1=0 Sum of 0 with a carry of 1Example Add 1111 and 11
24、00.第15页/共67页第十五页，共67页。Binary SubtractionBinary Subtractionn nThe four basic rules for subtraction binary digits The four basic rules for subtraction binary digits(bits)are as follows:(bits)are as follows:0-0=0 Difference of 0 with a borrow of 00-1=1 Difference of 1 with a borrow of 11-0=1 Difference
25、 of 1 with a borrow of 01-1=0 Difference of 0 with a borrow of 0Example Subtract 100 from 111.第16页/共67页第十六页，共67页。Binary MultiplicationBinary Multiplicationn nThe four basic rules for multiplication binary digits(bits)The four basic rules for multiplication binary digits(bits)are as follows:are as fo
26、llows:Example Multiply 1101 by 1010.第17页/共67页第十七页，共67页。Binary DivisionBinary Divisionn nDivision in binary follows the same procedure as Division in binary follows the same procedure as division in decimal.division in decimal.Example Divide 1100 by 100.第18页/共67页第十八页，共67页。2-5 1S AND 2S 2-5 1S AND 2S
27、COMPLEMENTS OF BINARY COMPLEMENTS OF BINARY NUMBERSNUMBERS二进制的反码二进制的反码二进制的反码二进制的反码(f(f n mn m)和补码和补码和补码和补码第19页/共67页第十九页，共67页。Finding the 1s Complement of a Binary NumberFinding the 1s Complement of a Binary Numbern nThe 1s complement of a binary number is found by The 1s complement of a binary numbe
28、r is found by changing all 1s to 0s and all 0s to 1s.changing all 1s to 0s and all 0s to 1s.Example Find the 1s complement of binary number 10110010.第20页/共67页第二十页，共67页。Finding the 2s Complement of a Binary Finding the 2s Complement of a Binary NumberNumbern nThe 2s complement of a binary number is f
29、ound by adding 1 The 2s complement of a binary number is found by adding 1 to the 1s complement.to the 1s complement.Example Find the 2s complement of binary number 10110010.第21页/共67页第二十一页，共67页。2-6 SIGNED NUMBERS2-6 SIGNED NUMBERS带符号数带符号数第22页/共67页第二十二页，共67页。The Sign BitThe Sign Bitn nThe left-most b
30、it in a signed binary number is the sign The left-most bit in a signed binary number is the sign bit,which tells you whether the number is positive or bit,which tells you whether the number is positive or negative.negative.A 0 is for positive,and a 1 is for negativeA 0 is for positive,and a 1 is for
31、 negative.第23页/共67页第二十三页，共67页。Sign-Magnitude SystemSign-Magnitude Systemn nWhen a signed binary number is represented in sign-When a signed binary number is represented in sign-magnitude,the left-most bit is the sign bit and the magnitude,the left-most bit is the sign bit and the remaining bits are
32、the magnitude bits.The magnitude bits remaining bits are the magnitude bits.The magnitude bits are in true(uncomplemented)binary for both positive and are in true(uncomplemented)binary for both positive and negative numbers.negative numbers.The decimal values are determined by summing the weights in
33、 all the magnitude bit positions where there are 1s.The sign is determined by examination of the sign bit.第24页/共67页第二十四页，共67页。Sign-Magnitude SystemSign-Magnitude Systemn nExample 2-15 Determine the decimal value of this Example 2-15 Determine the decimal value of this signed binary number expressed
34、in sign-magnitude:signed binary number expressed in sign-magnitude:10010101.10010101.第25页/共67页第二十五页，共67页。1s Complement System1s Complement Systemn nPositive numbers in the 1s complement system are Positive numbers in the 1s complement system are represented the same way as the positive sign-represen
35、ted the same way as the positive sign-magnitude numbers.magnitude numbers.n nNegative numbers,however,are the 1s complements of Negative numbers,however,are the 1s complements of the corresponding positive numbers.the corresponding positive numbers.nThe decimal values of positive numbers are determi
36、ned by summing the weights in all bit positions where there are 1s.The decimal values of negative numbers are determined by summing the weights in all bit positions where there are 1s,and adding 1 to the result.The weight of the sign bit is given a negative value.第26页/共67页第二十六页，共67页。1s Complement Sy
37、stem1s Complement Systemn nWhy?Why?第27页/共67页第二十七页，共67页。1s Complement System1s Complement Systemn nExample 2-16 Determine the decimal value of the Example 2-16 Determine the decimal value of the signed binary numbers expressed in 1s complement:signed binary numbers expressed in 1s complement:(a)00010
38、111 (b)11101000.(a)00010111 (b)11101000.第28页/共67页第二十八页，共67页。2s Complement System2s Complement Systemn nPositive numbers in the 2s complement system are Positive numbers in the 2s complement system are represented the same way as in sign-magnitude and 1s represented the same way as in sign-magnitude
39、and 1s complement plement systems.n nNegative numbers are the 2s complements of the Negative numbers are the 2s complements of the corresponding positive numbers.corresponding positive numbers.The decimal values are determined by summing the weights in all bit positions where there are 1s.The weight
40、 of the sign bit is given a negative value.第29页/共67页第二十九页，共67页。2s Complement System2s Complement Systemn nExample 2-17 Determine the decimal value of the Example 2-17 Determine the decimal value of the signed binary numbers expressed in 2s complement:signed binary numbers expressed in 2s complement:
41、(a)01010110 (b)10101010.(a)01010110 (b)10101010.第30页/共67页第三十页，共67页。2s Complement Advantage2s Complement Advantagen nTo convert to decimalTo convert to decimaln nThe 2s complement system simply requires a summation of weights The 2s complement system simply requires a summation of weights regardless
42、of whether the number is positive or negative.regardless of whether the number is positive or negative.n nThe sign-magnitude system requires two steps sum the The sign-magnitude system requires two steps sum the weights of the magnitude bits and examine the sign bit to weights of the magnitude bits
43、and examine the sign bit to determine if the number is positive or negative.determine if the number is positive or negative.n nThe 1s complement system requires adding 1 to the The 1s complement system requires adding 1 to the summation of weights for negative numbers but not for summation of weight
44、s for negative numbers but not for positive numbers.positive numbers.n nAlso,the 1s complement system is not used because two Also,the 1s complement system is not used because two representations of zero(00000000 or 11111111)are possible.representations of zero(00000000 or 11111111)are possible.第31页
45、/共67页第三十一页，共67页。The DevelopmentThe Developmentn nThe 2s complement of an n-bit number x is defined byThe 2s complement of an n-bit number x is defined byFor example,if n=8,then第32页/共67页第三十二页，共67页。The DevelopmentThe Developmentn nandand1s complement第33页/共67页第三十三页，共67页。Range of Signed Integer NumbersR
46、ange of Signed Integer Numbersn nThe number of different combinations of n bits isThe number of different combinations of n bits isFor 2s complement signed numbers,the range of value for n-bit numbers is第34页/共67页第三十四页，共67页。By the WayBy the Wayn nThe 8-bit grouping has been given the special name The
47、 8-bit grouping has been given the special name bytebyte.Bit GroupingBit GroupingNameName4-bit4-bitNibbleNibble8-bit8-bitByteByte16-bit16-bitWordWord32-bit32-bitDouble wordDouble word第35页/共67页第三十五页，共67页。Floating-Point NumbersFloating-Point Numbers（浮点数）（浮点数）（浮点数）（浮点数）n nNot required!Not required!第36页
48、/共67页第三十六页，共67页。2-7 ARITHMETIC OPERATIONS WITH SIGNED NUMBERS带符号数的算术带符号数的算术(sunsh)运算运算第37页/共67页第三十七页，共67页。AdditionAddition（加法（加法（加法（加法(jif(jif)）n nThe two numbers in an addition are the addendThe two numbers in an addition are the addend（加数（加数(ji(ji sh)sh)）and the augend and the augend（被加数（被加数(ji sh
49、)(ji sh)）.The results are.The results are the sum the sum（和）（和）and the carryand the carry（进位）（进位）.The addition process is stated as follows:add the two numbers and discard any final carry bit.Example(a)00000111+00000100=?(b)00001111+11111010=?(c)00010000+11101000=?(d)11111011+11110111=?第38页/共67页第三十八
50、页，共67页。Overflow ConditionOverflow Condition（溢出（溢出（溢出（溢出(y ch)(y ch)条件）条件）条件）条件）n nWhen two numbers are added and the number of bits When two numbers are added and the number of bits required to represent the sum exceeds the number of bits in required to represent the sum exceeds the number of bits i