(精品)ch-02.ppt

《(精品)ch-02.ppt》由会员分享，可在线阅读，更多相关《(精品)ch-02.ppt（50页珍藏版）》请在淘文阁 - 分享文档赚钱的网站上搜索。

1、2 2Number Number SystemsSystems1q Understand the concept of number systems.q Distinguish between non-positional and positional number systems.q Describe the decimal,binary,hexadecimal and octal system.qConvert a number in binary,octal or hexadecimal to a number in the decimal system.q Convert a numb

2、er in the decimal system to a number in binary,octal and hexadecimal.q Convert a number in binary to octal and vice versa.q Convert a number in binary to hexadecimal and vice versa.q Find the number of digits needed in each system to represent a particular value.ObjectivesAfter studying this chapter

3、,the student should be able After studying this chapter,the student should be able to:to:22-1 INTRODUCTIONA A number number systemsystem defines defines how how a a number number can can be be represented represented using using distinct distinct symbols.symbols.A A number number can can be be repre

4、sented represented differently differently in in different different systems.systems.For For example,example,the the two two numbers numbers(2A)(2A)1616 and and(52)(52)8 8 both both refer refer to to the the same same quantity,quantity,(42)(42)1010,but but their their representations representations

5、 are are different.different.Several Several number number systems systems have have been been used used in in the the past past and and can can be be categorized categorized into into two two groups:groups:positionalpositional and and non-positionalnon-positional systems.systems.Our Our main main g

6、oal goal is is to to discuss discuss the the positional positional number number systems,systems,but but we we also also give give examples examples of of non-positional systems.non-positional systems.32-2 POSITIONAL NUMBER SYSTEMSIn In a a positional positional number number systemsystem,the the po

7、sition position a a symbol symbol occupies occupies in in the the number number determines determines the the value value it it represents.In this system,a number represented as:represents.In this system,a number represented as:has the value of:has the value of:in which S is the set of symbols,b is

8、the in which S is the set of symbols,b is the basebase(or (or radixradix).).4S=0,1,2,3,4,5,6,7,8,9The decimal system(base 10)The word decimal is derived from the Latin root decem(ten).In this system the base b=10 and we use ten symbolsThe symbols in this system are often referred to as decimal digit

9、s or just digits.5IntegersFigure 2.1 Place values for an integer in the decimal system6Example 2.1The The following following shows shows the the place place values values for for the the integer integer+224+224 in in the the decimal system.decimal system.Note Note that that the the digit digit 2 2

10、in in position position 1 1 has has the the value value 20,20,but but the the same same digit digit in in position position 2 2 has has the the value value 200.200.Also Also note note that that we we normally normally drop the plus sign,but it is implicit.drop the plus sign,but it is implicit.7Examp

11、le 2.2The The following following shows shows the the place place values values for for the the decimal decimal number number 7508.7508.We We have have used used 1,1,10,10,100,100,and and 1000 1000 instead instead of of powers powers of of 10.10.Note Note that that the the digit digit 2 2 in in posi

12、tion position 1 1 has has the the value value 20,20,but but the the same same digit digit in in position position 2 2 has has the the value value 200.200.Also Also note note that that we we normally normally drop the plus sign,but it is implicit.drop the plus sign,but it is implicit.()Values8RealsEx

13、ample 2.3The following shows the place values for the real number+24.13.The following shows the place values for the real number+24.13.9The word binary is derived from the Latin root bini(or two by two).In this system the base b=2 and we use only two symbols,The binary system(base 2)S=0,1The symbols

14、 in this system are often referred to as binary digits or bits(binary digit).10IntegersFigure 2.2 Place values for an integer in the binary system11Example 2.4The The following following shows shows that that the the number number(11001)(11001)2 2 in in binary binary is is the the same as 25 in deci

15、mal.The subscript 2 shows that the base is 2.same as 25 in decimal.The subscript 2 shows that the base is 2.The equivalent decimal number is N=16+8+0+0+1=25.The equivalent decimal number is N=16+8+0+0+1=25.12RealsExample 2.5The The following following shows shows that that the the number number(101.

16、11)(101.11)2 2 in in binary binary is is equal equal to the number 5.75 in decimal.to the number 5.75 in decimal.13The word hexadecimal is derived from the Greek root hex(six)and the Latin root decem(ten).In this system the base b=16 and we use sixteen symbols to represent a number.The set of symbol

17、s isThe hexadecimal system(base 16)S=0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F Note that the symbols A,B,C,D,E,F are equivalent to 10,11,12,13,14,and 15 respectively.The symbols in this system are often referred to as hexadecimal digits.14IntegersFigure 2.3 Place values for an integer in the hexadecimal syste

18、m15Example 2.6The The following following shows shows that that the the number number(2AE)16(2AE)16 in in hexadecimal hexadecimal is is equivalent to 686 in decimal.equivalent to 686 in decimal.The equivalent decimal number is N=512+160+14=686.The equivalent decimal number is N=512+160+14=686.16The

19、word octal is derived from the Latin root octo(eight).In this system the base b=8 and we use eight symbols to represent a number.The set of symbols isThe octal system(base 8)S=0,1,2,3,4,5,6,7 17IntegersFigure 2.3 Place values for an integer in the octal system18Example 2.7The The following following

20、 shows shows that that the the number number(1256)(1256)8 8 in in octal octal is is the the same same as 686 in decimal.as 686 in decimal.Note that the decimal number is N=512+128+40+6=686.Note that the decimal number is N=512+128+40+6=686.19Table 2.1 shows a summary of the four positional number sy

21、stems discussed in this chapter.Summary of the four positional systems20Table 2.2 shows how the number 0 to 15 is represented in different systems.21We need to know how to convert a number in one system to the equivalent number in another system.Since the decimal system is more familiar than the oth

22、er systems,we first show how to covert from any base to decimal.Then we show how to convert from decimal to any base.Finally,we show how we can easily convert from binary to hexadecimal or octal and vice versa.Conversion22Any base to decimal conversionFigure 2.5 Converting other bases to decimal23Ex

23、ample 2.8The The following following shows shows how how to to convert convert the the binary binary number number(110.11)(110.11)2 2 to decimal:(110.11)to decimal:(110.11)2 2=6.75.=6.75.24Example 2.9The The following following shows shows how how to to convert convert the the hexadecimal hexadecima

24、l number number(1A.23)(1A.23)1616 to decimal.to decimal.Note Note that that the the result result in in the the decimal decimal notation notation is is not not exact,exact,because because 3 3 161622 =0.01171875.0.01171875.We We have have rounded rounded this this value value to to three three digits

25、 digits(0.012).(0.012).25Example 2.10The following shows how to convert(23.17)The following shows how to convert(23.17)8 8 to decimal.to decimal.This This means means that that(23.17)8(23.17)8 19.234 19.234 in in decimal.decimal.Again,Again,we we have have rounded up 7 8rounded up 7 822=0.109375.=0.

26、109375.26Decimal to any baseFigure 2.6 Converting other bases to decimal(integral part)27Figure 2.7 Converting the integral part of a number in decimal to other bases28Example 2.11The The following following shows shows how how to to convert convert 35 35 in in decimal decimal to to binary.binary.We

27、 We start start with with the the number number in in decimal,decimal,we we move move to to the the left left while while continuously continuously finding finding the the quotients quotients and and the the remainder remainder of of division division by 2.The result is 35=(100011)by 2.The result is

28、 35=(100011)2 2.29Example 2.12The The following following shows shows how how to to convert convert 126 126 in in decimal decimal to to its its equivalent equivalent in in the the octal octal system.system.We We move move to to the the right right while while continuously continuously finding findin

29、g the the quotients quotients and and the the remainder remainder of of division division by 8.The result is 126=(176)by 8.The result is 126=(176)8 8.30Example 2.13The The following following shows shows how how we we convert convert 126 126 in in decimal decimal to to its its equivalent equivalent

30、in in the the hexadecimal hexadecimal system.system.We We move move to to the the right right while while continuously continuously finding finding the the quotients quotients and and the the remainder remainder of of division division by 16.The result is 126=(7E)by 16.The result is 126=(7E)161631Fi

31、gure 2.8 Converting the fractional part of a number in decimal to other bases32Figure 2.9 Converting the fractional part of a number in decimal to other bases33Example 2.14Convert the decimal number 0.625 to binary.Convert the decimal number 0.625 to binary.Since Since the the number number 0.625 0.

32、625=(0.101)(0.101)2 2 has has no no integral integral part,part,the the example shows how the fractional part is calculated.example shows how the fractional part is calculated.34Example 2.15The The following following shows shows how how to to convert convert 0.634 0.634 to to octal octal using usin

33、g a a maximum maximum of of four four digits.digits.The The result result is is 0.634 0.634=(0.5044)(0.5044)8 8.Note Note that we multiple by 8(base octal).that we multiple by 8(base octal).35Example 2.16The The following following shows shows how how to to convert convert 178.6 178.6 in in decimal

34、decimal to to hexadecimal hexadecimal using using only only one one digit digit to to the the right right of of the the decimal decimal point.point.The The result result is is 178.6 178.6=(B2.9)(B2.9)1616 Note Note that that we we divide divide or or multiple multiple by by 16(base hexadecimal).16(b

35、ase hexadecimal).36Example 2.17An An alternative alternative method method for for converting converting a a small small decimal decimal integer integer(usually(usually less less than than 256)256)to to binary binary is is to to break break the the number number as as the the sum sum of of numbers n

36、umbers that that are are equivalent equivalent to to the the binary binary place place values values shown:shown:37Example 2.18A A similar similar method method can can be be used used to to convert convert a a decimal decimal fraction fraction to to binary when the denominator is a power of two:bin

37、ary when the denominator is a power of two:The answer is then(0.011011)The answer is then(0.011011)2 238Binary-hexadecimal conversionFigure 2.10 Binary to hexadecimal and hexadecimal to binary conversion39Example 2.19Show Show the the hexadecimal hexadecimal equivalent equivalent of of the the binar

38、y binary number number(110011100010)(110011100010)2 2.SolutionSolutionWe first arrange the binary number in 4-bit patterns:We first arrange the binary number in 4-bit patterns:100 1110 0010Note Note that that the the leftmost leftmost pattern pattern can can have have one one to to four four bits.bi

39、ts.We We then then use use the the equivalent equivalent of of each each pattern pattern shown shown in in Table Table 2.2 2.2 on on page page 25 25 to change the number to hexadecimal:(4E2)16.to change the number to hexadecimal:(4E2)16.40Example 2.20What is the binary equivalent of(24C)What is the

40、binary equivalent of(24C)1616?SolutionSolutionEach hexadecimal digit is converted to 4-bit patterns:Each hexadecimal digit is converted to 4-bit patterns:2 0010,4 0100,and C 1100The result is(001001001100)The result is(001001001100)2 2.41Binary-octal conversionFigure 2.10 Binary to octal and octal t

41、o binary conversion42Example 2.21Show the octal equivalent of the binary number(101110010)Show the octal equivalent of the binary number(101110010)2 2.SolutionSolutionEach Each group group of of three three bits bits is is translated translated into into one one octal octal digit.digit.The The equiv

42、alent of each 3-bit group is shown in Table 2.2 on page 25.equivalent of each 3-bit group is shown in Table 2.2 on page 25.The result is(562)The result is(562)8 8.101 110 01043Example 2.22What is the binary equivalent of for(24)What is the binary equivalent of for(24)8 8?SolutionSolutionWrite each o

43、ctal digit as its equivalent bit pattern to getWrite each octal digit as its equivalent bit pattern to get2 010 and 4 100The result is(010100)The result is(010100)2 2.44Octal-hexadecimal conversionFigure 2.12 Octal to hexadecimal and hexadecimal to octal conversion45Example 2.23Find Find the the min

44、imum minimum number number of of binary binary digits digits required required to to store store decimal integers with a maximum of six digits.decimal integers with a maximum of six digits.SolutionSolutionk k=6,b=6,b1 1=10,and b=10,and b2 2=2.Then=2.Thenx x=k k (logb (logb1 1/logb/logb2 2)=6 (1/0.30

45、103)6 (1/0.30103)=20.=20.The The largest largest six-digit six-digit decimal decimal number number is is 999,999 999,999 and and the the largest largest 20-bit 20-bit binary binary number number is is 1,048,575.1,048,575.Note Note that that the the largest largest number number that that can can be

46、be represented represented by by a a 19-bit 19-bit number number is is 524287,524287,which which is is smaller than 999,999.We definitely need twenty bits.smaller than 999,999.We definitely need twenty bits.462-3 NONPOSITIONAL NUMBER SYSTEMSAlthough Although non-positional non-positional number numb

47、er systemssystems are are not not used used in in computers,computers,we we give give a a short short review review here here for for comparison comparison with with positional positional number number systems.systems.A A non-non-positional positional number number system system still still uses use

48、s a a limited limited number number of of symbols symbols in in which which each each symbol symbol has has a a value.value.However,However,the the position position a a symbol symbol occupies occupies in in the the number number normally normally bears bears no no relation relation to to its its va

49、luethe valuethe value value of of each each symbol symbol is is fixed.fixed.To To find find the the value value of of a a number,number,we we add add the the value value of all symbols present in the representation.of all symbols present in the representation.47In this system,a number is represented

50、 as:In this system,a number is represented as:and has the value of:and has the value of:There There are are some some exceptions exceptions to to the the addition addition rule rule we we just just mentioned,as shown in Example 2.24.mentioned,as shown in Example 2.24.48Example 2.24Roman Roman numera