1、,Digital Fundamentals Tenth Edition Floyd,Chapter 2, 2008 Pearson Education,The position of each digit in a weighted number system is assigned a weight based on the base or radix of the system. The radix of decimal numbers is ten, because only ten symbols (0 through 9) are used to represent any numb

2、er.,Summary,The column weights of decimal numbers are powers of ten that increase from right to left beginning with 100 =1:,Decimal Numbers,105 104 103 102 101 100.,For fractional decimal numbers, the column weights are negative powers of ten that decrease from left to right:,102 101 100. 10-1 10-2

3、10-3 10-4 ,Summary,Decimal Numbers,Express the number 480.52 as the sum of values of each digit.,Example,Solution,(9 x 103) + (2 x 102) + (4 x101) + (0 x 100) or 9 x 1,000 + 2 x 100 + 4 x10 + 0 x 1,Decimal numbers can be expressed as the sum of the products of each digit times the column value for t

4、hat digit. Thus, the number 9240 can be expressed as,480.52 = (4 x 102) + (8 x 101) + (0 x 100) + (5 x 10-1) +(2 x 10-2),Summary,Binary Numbers,For digital systems, the binary number system is used. Binary has a radix of two and uses the digits 0 and 1 to represent quantities.,The column weights of

5、binary numbers are powers of two that increase from right to left beginning with 20 =1:,25 24 23 22 21 20.,For fractional binary numbers, the column weights are negative powers of two that decrease from left to right:,22 21 20. 2-1 2-2 2-3 2-4 ,Summary,Binary Numbers,A binary counting sequence for n

6、umbers from zero to fifteen is shown.,0 0 0 0 0 1 0 0 0 1 2 0 0 1 0 3 0 0 1 1 4 0 1 0 0 5 0 1 0 1 6 0 1 1 0 7 0 1 1 1 8 1 0 0 0 9 1 0 0 1 10 1 0 1 0 11 1 0 1 1 12 1 1 0 0 13 1 1 0 1 14 1 1 1 0 15 1 1 1 1,Decimal Number,Binary Number,Notice the pattern of zeros and ones in each column.,Digital counte

7、rs frequently have this same pattern of digits:,Summary,Binary Conversions,The decimal equivalent of a binary number can be determined by adding the column values of all of the bits that are 1 and discarding all of the bits that are 0.,Convert the binary number 100101.01 to decimal.,Example,Solution

8、,Start by writing the column weights; then add the weights that correspond to each 1 in the number.,25 24 23 22 21 20. 2-1 2-2,32 16 8 4 2 1 . ,1 0 0 1 0 1. 0 1,32 +4 +1 + =,37,Summary,Binary Conversions,You can convert a decimal whole number to binary by reversing the procedure. Write the decimal w

9、eight of each column and place 1s in the columns that sum to the decimal number.,Convert the decimal number 49 to binary.,Example,Solution,The column weights double in each position to the right. Write down column weights until the last number is larger than the one you want to convert.,26 25 24 23

10、22 21 20.,64 32 16 8 4 2 1.,0 1 1 0 0 0 1.,Summary,You can convert a decimal fraction to binary by repeatedly multiplying the fractional results of successive multiplications by 2. The carries form the binary number.,Convert the decimal fraction 0.188 to binary by repeatedly multiplying the fraction

11、al results by 2.,Example,Solution,0.188 x 2 = 0.376 carry = 0,0.376 x 2 = 0.752 carry = 0,0.752 x 2 = 1.504 carry = 1,0.504 x 2 = 1.008 carry = 1,0.008 x 2 = 0.016 carry = 0,Answer = .00110 (for five significant digits),MSB,Binary Conversions,1,0,0,1,1,0,Summary,You can convert decimal to any other

12、base by repeatedly dividing by the base. For binary, repeatedly divide by 2:,Convert the decimal number 49 to binary by repeatedly dividing by 2.,Example,Solution,You can do this by “reverse division” and the answer will read from left to right. Put quotients to the left and remainders on top.,24,12

13、,6,3,1,0,Continue until the last quotient is 0,Answer:,Binary Conversions,Summary,Binary Addition,The rules for binary addition are,0 + 0 = 0 Sum = 0, carry = 0,0 + 1 = 0 Sum = 1, carry = 0,1 + 0 = 0 Sum = 1, carry = 0,1 + 1 = 10 Sum = 0, carry = 1,When an input carry = 1 due to a previous result, t

14、he rules are,1 + 0 + 0 = 01 Sum = 1, carry = 0,1 + 0 + 1 = 10 Sum = 0, carry = 1,1 + 1 + 0 = 10 Sum = 0, carry = 1,1 + 1 + 1 = 10 Sum = 1, carry = 1,Summary,Binary Addition,Add the binary numbers 00111 and 10101 and show the equivalent decimal addition.,Example,Solution,00111 7,10101 21,0,1,0,1,1,1,

15、1,0,1,28,=,Summary,Binary Subtraction,The rules for binary subtraction are,0 - 0 = 0,1 - 1 = 0,1 - 0 = 1,10 - 1 = 1 with a borrow of 1,Subtract the binary number 00111 from 10101 and show the equivalent decimal subtraction.,Example,Solution,00111 7,10101 21,0,/1,1,1,1,0,14,/1,/1,=,Summary,1s Complem

16、ent,The 1s complement of a binary number is just the inverse of the digits. To form the 1s complement, change all 0s to 1s and all 1s to 0s.,For example, the 1s complement of 11001010 is,00110101,In digital circuits, the 1s complement is formed by using inverters:,1 1 0 0 1 0 1 0,0 0 1 1 0 1 0 1,Sum

17、mary,2s Complement,The 2s complement of a binary number is found by adding 1 to the LSB of the 1s complement.,Recall that the 1s complement of 11001010 is,00110101 (1s complement),To form the 2s complement, add 1:,+1,00110110 (2s complement),1 1 0 0 1 0 1 0,0 0 1 1 0 1 0 1,1,0 0 1 1 0 1 1 0,Summary,

18、Signed Binary Numbers,There are several ways to represent signed binary numbers. In all cases, the MSB in a signed number is the sign bit, that tells you if the number is positive or negative.,Computers use a modified 2s complement for signed numbers. Positive numbers are stored in true form (with a

19、 0 for the sign bit) and negative numbers are stored in complement form (with a 1 for the sign bit).,For example, the positive number 58 is written using 8-bits as 00111010 (true form).,Sign bit,Magnitude bits,Summary,Signed Binary Numbers,Assuming that the sign bit = -128, show that 11000110 = -58

20、as a 2s complement signed number:,Example,Solution,1 1 0 0 0 1 1 0,Column weights: -128 64 32 16 8 4 2 1.,-128 +64 +4 +2 = -58,Negative numbers are written as the 2s complement of the corresponding positive number.,-58 = 11000110 (complement form),An easy way to read a signed number that uses this n

21、otation is to assign the sign bit a column weight of -128 (for an 8-bit number). Then add the column weights for the 1s.,The negative number -58 is written as:,Summary,Floating Point Numbers,Express the speed of light, c, in single precision floating point notation. (c = 0.2998 x 109),Example,Soluti

22、on,Floating point notation is capable of representing very large or small numbers by using a form of scientific notation. A 32-bit single precision number is illustrated.,S E (8 bits) F (23 bits),Sign bit,Magnitude with MSB dropped,Biased exponent (+127),In scientific notation, c = 1.001 1101 1110 1

23、001 0101 1100 0000 x 228.,In binary, c = 0001 0001 1101 1110 1001 0101 1100 00002.,S = 0 because the number is positive. E = 28 + 127 = 15510 = 1001 10112. F is the next 23 bits after the first 1 is dropped.,In floating point notation, c =,Summary,Arithmetic Operations with Signed Numbers,Using the

24、signed number notation with negative numbers in 2s complement form simplifies addition and subtraction of signed numbers.,Rules for addition: Add the two signed numbers. Discard any final carries. The result is in signed form. Examples:,00011110 = +30 00001111 = +15,00101101,= +45,00001110 = +14 111

25、01111 = -17,11111101,= -3,11111111 = -1 11111000 = -8,11110111,= -9,1,Summary,Arithmetic Operations with Signed Numbers,01000000 = +128 01000001 = +129,10000001,= -126,10000001 = -127 10000001 = -127,100000010,= +2,Note that if the number of bits required for the answer is exceeded, overflow will oc

26、cur. This occurs only if both numbers have the same sign. The overflow will be indicated by an incorrect sign bit.,Two examples are:,Wrong! The answer is incorrect and the sign bit has changed.,Summary,Arithmetic Operations with Signed Numbers,Rules for subtraction: 2s complement the subtrahend and

27、add the numbers. Discard any final carries. The result is in signed form.,00001111,= +15,1,2s complement subtrahend and add:,00011110 = +30 11110001 = -15,Repeat the examples done previously, but subtract:,00011111,= +31,00001110 = +14 00010001 = +17,00000111,= +7,1,11111111 = -1 00001000 = +8,Summa

28、ry,Hexadecimal Numbers,Hexadecimal uses sixteen characters to represent numbers: the numbers 0 through 9 and the alphabetic characters A through F.,0 1 2 3 4 5 6 7 8 9 10 11 12 13 1415,0 1 2 3 4 5 6 7 8 9 A B C D E F,0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111,Dec

29、imal,Hexadecimal,Binary,Large binary number can easily be converted to hexadecimal by grouping bits 4 at a time and writing the equivalent hexadecimal character.,Express 1001 0110 0000 11102 in hexadecimal:,Example,Solution,Group the binary number by 4-bits starting from the right. Thus, 960E,Summar

30、y,Hexadecimal Numbers,Hexadecimal is a weighted number system. The column weights are powers of 16, which increase from right to left.,.,1 A 2 F16,670310,Column weights,163 162 161 160,4096 256 16 1,.,Express 1A2F16 in decimal.,Example,Solution,Start by writing the column weights: 4096 256 16 1,1(40

31、96) + 10(256) +2(16) +15(1) =,0 1 2 3 4 5 6 7 8 9 10 11 12 13 1415,0 1 2 3 4 5 6 7 8 9 A B C D E F,0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111,Decimal,Hexadecimal,Binary,Summary,Octal Numbers,Octal uses eight characters the numbers 0 through 7 to represent numbers

32、. There is no 8 or 9 character in octal.,0 1 2 3 4 5 6 7 8 9 10 11 12 13 1415,0 1 2 3 4 5 6 7 10 1112 13 14 15 16 17,0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111,Decimal,Octal,Binary,Binary number can easily be converted to octal by grouping bits 3 at a time and wr

33、iting the equivalent octal character for each group.,Express 1 001 011 000 001 1102 in octal:,Example,Solution,Group the binary number by 3-bits starting from the right. Thus, 1130168,Summary,Octal Numbers,Octal is also a weighted number system. The column weights are powers of 8, which increase fro

34、m right to left.,.,3 7 0 28,198610,Column weights,83 82 81 80,512 64 8 1,.,Express 37028 in decimal.,Example,Solution,Start by writing the column weights: 512 64 8 1,3(512) + 7(64) +0(8) +2(1) =,0 1 2 3 4 5 6 7 8 9 10 11 12 13 1415,0 1 2 3 4 5 6 7 10 1112 13 14 15 16 17,0000 0001 0010 0011 0100 0101

35、 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111,Decimal,Octal,Binary,Summary,BCD,Binary coded decimal (BCD) is a weighted code that is commonly used in digital systems when it is necessary to show decimal numbers such as in clock displays.,0 1 2 3 4 5 6 7 8 9 10 11 12 13 1415,0000 0001 0010 0011

36、0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111,Decimal,Binary,BCD,0001 0001 0001 0001 0001 0001,0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 0000 0001 0010 0011 0100 0101,The table illustrates the difference between straight binary and BCD. BCD represents each decimal digit with a

37、4-bit code. Notice that the codes 1010 through 1111 are not used in BCD.,Summary,BCD,You can think of BCD in terms of column weights in groups of four bits. For an 8-bit BCD number, the column weights are: 80 40 20 10 8 4 2 1.,Question:,What are the column weights for the BCD number 1000 0011 0101 1

38、001?,Answer:,8000 4000 2000 1000 800 400 200 100 80 40 20 10 8 4 2 1,Note that you could add the column weights where there is a 1 to obtain the decimal number. For this case:,8000 + 200 +100 + 40 + 10 + 8 +1 = 835910,Summary,BCD,A lab experiment in which BCD is converted to decimal is shown.,Summar

39、y,Gray code,Gray code is an unweighted code that has a single bit change between one code word and the next in a sequence. Gray code is used to avoid problems in systems where an error can occur if more than one bit changes at a time.,0 1 2 3 4 5 6 7 8 9 10 11 12 13 1415,0000 0001 0010 0011 0100 010

40、1 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111,Decimal,Binary,Gray code,0000 0001 0011 0010 0110 0111 0101 0100 1100 1101 1111 1110 1010 1011 1001 1000,Summary,Gray code,A shaft encoder is a typical application. Three IR emitter/detectors are used to encode the position of the shaft. The encode

41、r on the left uses binary and can have three bits change together, creating a potential error. The encoder on the right uses gray code and only 1-bit changes, eliminating potential errors.,Binary sequence,Gray code sequence,Summary,ASCII,ASCII is a code for alphanumeric characters and control charac

42、ters. In its original form, ASCII encoded 128 characters and symbols using 7-bits. The first 32 characters are control characters, that are based on obsolete teletype requirements, so these characters are generally assigned to other functions in modern usage.,In 1981, IBM introduced extended ASCII,

43、which is an 8-bit code and increased the character set to 256. Other extended sets (such as Unicode) have been introduced to handle characters in languages other than English.,Summary,Parity Method,The parity method is a method of error detection for simple transmission errors involving one bit (or

44、an odd number of bits). A parity bit is an “extra” bit attached to a group of bits to force the number of 1s to be either even (even parity) or odd (odd parity).,The ASCII character for “a” is 1100001 and for “A” is 1000001. What is the correct bit to append to make both of these have odd parity?,Ex

45、ample,Solution,The ASCII “a” has an odd number of bits that are equal to 1; therefore the parity bit is 0. The ASCII “A” has an even number of bits that are equal to 1; therefore the parity bit is 1.,Summary,Cyclic Redundancy Check,The cyclic redundancy check (CRC) is an error detection method that

46、can detect multiple errors in larger blocks of data. At the sending end, a checksum is appended to a block of data. At the receiving end, the check sum is generated and compared to the sent checksum. If the check sums are the same, no error is detected.,Selected Key Terms,Byte Floating-point number

47、Hexadecimal Octal BCD,A group of eight bits,A number representation based on scientific notation in which the number consists of an exponent and a mantissa.,A number system with a base of 16.,A number system with a base of 8.,Binary coded decimal; a digital code in which each of the decimal digits,

48、0 through 9, is represented by a group of four bits.,Selected Key Terms,Alphanumeric ASCII Parity Cyclic redundancy check (CRC),Consisting of numerals, letters, and other characters,American Standard Code for Information Interchange; the most widely used alphanumeric code.,In relation to binary codes, the condition of evenness or oddness in the number