Microsoft PowerPoint - STU_EC_Ch02.ppt

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1 樹德科技大學資訊工程系 Chapter 2: Number Systems Operations and Codes Shi-Huang Chen Sept Chapter Outline 2.1 Decimal Numbers 2.2 Binary Numbers 2.3 Decimal-to-Binary Conversion 2.4 Binary Arithmetic 2.5 1's and 2's Complements of Binary Numbers 2.6 Signed Numbers 2.7 Arithmetic Operations with Signed Numbers 2.8 Hexadecimal Numbers 2.9 Octal Numbers 2.10 Binary Coded Decimal (BCD) 2.11 Digital Codes and Parity 2 1

2 Decimal Numbers -1 The decimal number system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 The decimal numbering system has a base of 10 with each position weighted by a factor of = 1 x x x 10-1 Decimal Point 3 Decimal Numbers -2 3 X X 10 0 is equal to

3 Decimal Numbers -3 The number 9240 can be expressed as or (9 x 10 3 ) + (2 x 10 2 ) + (4 x 10 1 ) + (0 x 10 0 ) 9 x 1, x x x 1 Express the number as the sum of values of each digit = (4 x 10 2 ) + (8 x 10 1 ) + (0 x 10 0 ) + (5 x 10-1 ) +(2 x 10-2 ) 5 Binary Numbers -1 The binary numbering system has 2 digits: 0 and 1 The binary numbering system has a base of 2 with each position weighted by a factor of Decimal Point = 1 x x x x x 2 0 =

4 Binary Numbers -2 4 = log Binary Numbers

5 Binary Numbers -4 Illustration of a simple binary counting application. 9 Binary Numbers -5 The binary number 1110 is equal to the decimal number

6 Binary Numbers -6 The binary number is equal to the decimal number Binary Numbers -7 Convert the binary number to decimal. Start by writing the column weights; then add the weights that correspond to each 1 in the number ½ ¼ ¼ = 37¼ 12 6

Decimal-to-Binary Conversion -1 Sum-of-weight method Binary weights 256 128 64 32 16 8 4 2 1 357 = 256 + 64 + 32 + 4 + 1 101100101 Binary weights

7 Decimal-to-Binary Conversion -1 Sum-of-weight method Binary weights = Binary weights = Decimal-to-Binary Conversion -2 Repeated division-by-2 method 14 7

8 Decimal-to-Binary Conversion -3 Converting Decimal fractions to Binary Using Sum-of-weights Binary weights = Repeated division by 2 yields the whole number while repeated multiplication by 2 of the fraction yields the binary fraction 15 Decimal-to-Binary Conversion -4 The decimal number 18 is equal to the binary number

9 Decimal-to-Binary Conversion -5 The decimal number 188 is equal to the binary number Decimal-to-Binary Conversion -6 Convert the decimal fraction to binary by repeatedly multiplying the fractional results by x 2 = carry = 0 MSB x 2 = carry = x 2 = carry = x 2 = carry = x 2 = carry = 0 Answer = (for five significant digits) 18 9

10 Binary Addition = 0 Sum of 0 with a carry of = 1 Sum of 1 with a carry of = 1 Sum of 1 with a carry of = 10 Sum of 0 with a carry of Binary Addition -2 The sum of equals

11 Binary Subtraction = = = = with a borrow of Binary Subtraction -2 The difference of equals

Binary Multiplication 0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 100110 1 x 1 = 1 x 101 100110 000000 38

12 Binary Multiplication 0 x 0 = 0 0 x 1 = 0 1 x 0 = x 1 = 1 x x Binary Division Use the same procedure as decimal division 24 12

13 1's Complements of Binary Numbers -1 Example of inverters used to obtain the 1 s complement of a binary number 25 1's Complements of Binary Numbers -2 The 1's complement of is

14 2's Complements of Binary Numbers -1 Example of obtaining the 2 s complement of a negative binary number. 27 2's Complements of Binary Numbers -2 The 2's complement of is

15 Signed Numbers -1 Sign-Magnitude Form The left-most bit in a signed binary number is the sign bit A 0 is for positive, and a 1 is for negative Ex: (8-bit) +25 = Sign bit Magnitude bits - 25 = Signed Numbers -2 Sign-Magnitude Form Express the decimal number 37 as an 8-bit number in sign-magnitude

16 Signed Numbers -3 1's Complement Form Ex: (8-bit) +25 = = (1's complement of 25) In the 1's Complement form, a negative number is the 1's complement of the corresponding positive number 31 Signed Numbers -4 1's Complement Form Determine the decimal values of the signed binary number in 1's complement

17 Signed Numbers -5 2's Complement Form Ex: (8-bit) +25 = = (2's complement of 25) In the 2's Complement form, a negative number is the 2's complement of the corresponding positive number 33 Signed Numbers -6 Floating-Point Numbers (1/2) 32 bits S Exponent (E) Mantissa (Fraction, F) 1 bit 8 bits 23 bits Biased Exp (+127) x 10 4 = = x = 141 =

18 Signed Numbers -7 Floating-Point Numbers (2/2) Number = (-1) S (1+F)(2 E-127 ) S E F Number = (-1) 1 ( )( ) = (-1)( )(2 18 ) = = 407, Arithmetic Operations with Signed Numbers -1 Addition Both numbers positive Positive number > Negative number Positive number < Negative number Both numbers negative Overflow condition Negative number is presented in 2 s complement form

19 Arithmetic Operations with Signed Numbers -2 Subtraction Take the 2 s complement of the subtrahend and add. Ex: 8 3 = 8 + ( 3) 12 ( 9) = = 25 + ( 19) 120 ( 30) = Arithmetic Operations with Signed Numbers -3 Multiplication Is equivalent to adding a number to itself a number of times equal to the multiplier Ex: 8x3 =

20 Arithmetic Operations with Signed Numbers -4 Division Dividend Divisor = Quotient Quotient is the number of times that the divisor will go into the dividend. Ex: 21 7 = = = 7 3 times 7 7 = 0 39 Hexadecimal Numbers

21 Hexadecimal Numbers -2 Counting in Hex. 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B, 2C,.. Bin.-to-Hex. Conversion Hex.-to-Bin. Conversion Hex.-to-Dec. Conversion Dec.-to-Hex. Conversion 41 Hexadecimal Numbers -3 Hex. Addition = 7A = = A = = 7 16 DF 16 + AC 16 = 18B 16 F 16 + C 16 = = = = B 16 with a 1 carry D 16 + A = = = 8 10 = 8 16 with a 1 carry 42 21

Hexadecimal Numbers -4 Hex. Subtraction (1/3) Getting the 2 s complement of a hexadecimal number, Method 1.

22 Hexadecimal Numbers -4 Hex. Subtraction (1/3) Getting the 2 s complement of a hexadecimal number, Method Hexadecimal Numbers -5 Hex. Subtraction (2/3) Getting the 2 s complement of a hexadecimal number, Method

Hexadecimal Numbers -6 Hex. Subtraction (3/3) Getting the 2 s complement of a hexadecimal number, Method 3.

23 Hexadecimal Numbers -6 Hex. Subtraction (3/3) Getting the 2 s complement of a hexadecimal number, Method Octal Numbers -1 Counting in Oct. 0, 1, 2, 3, 4, 5, 6, 7 10, 11, 12, 13, 14, 15, 16, 17, 20, 21,.. Oct.-to-Dec. Conversion Dec.-to-Oct. Conversion 46 23

24 Octal Numbers -2 Oct.-to-Bin. Conversion Bin.-to-Oct. Conversion 47 Binary Coded Decimal (BCD) The 8421 code BCD Addition 48 24

25 Digital Codes and Parity -1 The Gray code 49 Digital Codes and Parity -2 Bin.-to-Gray Code Bin. Gray Gray-to-Bin. Code Gray Bin

26 Digital Codes and Parity -3 A simplified illustration of how the Gray code solves the error problem in shaft position encoders. 51 Digital Codes and Parity -4 ASCII code 52 26

27 Digital Codes and Parity -5 Parity Method for Error Detection 53 27

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