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Lecture 4: Number Systems (Chapter 3)

Lecture 4: Number Systems (Chapter 3). (1) Data Types Section 3-1 (2) Complements Section 3-2 (3) Fixed Point Representations Section 3-3 (4) Floating Point Representations Section 3-4 (5) Other Binary Codes Section 3-5 (6) Error Detection Codes Section 3-6. Data Types.

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Lecture 4: Number Systems (Chapter 3)

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  1. Lecture 4: Number Systems (Chapter 3) (1) Data Types Section 3-1 (2) Complements Section 3-2 (3) Fixed Point Representations Section 3-3 (4) Floating Point Representations Section 3-4 (5) Other Binary Codes Section 3-5 (6) Error Detection Codes Section 3-6

  2. Data Types Information that a Computer is dealing with: Data Numeric Data Numbers (Integer, real) Non-numeric Data Letters, Symbols Relationship between data elements Data Structures Linear Lists, Trees, Rings, etc Program (Instructions)

  3. Data Types: Numeric Data Representation Nonpositional number system Roman number system Positional number system Each digit position has a value called a weight associated with it Examples: Decimal, Octal, Hexadecimal, Binary Base (or radix) R number Uses R distinct symbols for each digit Example A R = a n-1 a n-2 ... a 1 a 0.a -1 …a -m V(A R) = SUM (a k * R k) for k = -m to n-1 R = 10 Decimal number system R = 2 Binary R = 8 Octal R = 16 Hexadecimal

  4. Data Types: Numeric Data Representation Why a Positional Number System for Digital Computers??? Major Consideration is the COST and TIME Cost of building hardware Arithmetic and Logic Unit, CPU,Communications Time to processing Arithmetic - Addition of Numbers - Table for Addition Non-positional Number System Table for addition is infinite --> Impossible to build, very expensive even if it can be built Positional Number System Table for Addition is finite --> Physically realizable, but cost wise the smaller the table size, the less expensive --> Binary is favorable to Decimal

  5. Positive (Unsigned) Binary Numbers Unsigned binary numbers are typically used to represent computer addresses or other values that are guaranteed not to be negative. An n-bit unsigned binary integer A = an-1 an-2... a1 a0 has a value of For example, 1011 = 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 = 8 + 2 + 1 = 11 An n-bit unsigned binary integer has a range from 0 to 2n - 1.

  6. Octal and Hexadecimal Numbers Octal, base-8, numbers were used in the early days of computing to represent binary numbers Octal numbers are made by grouping binary numbers together three bits at a time Hexadecimal, base-16, numbers are the representation of choice today Hex numbers are made by grouping binary numbers together four bits at a time For example: Octal: 7 2 5 1 7 5 2 2 . Binary: 1 1 1 01 0 1 01 0 0 11 1 1 10 1 0 10 0 1 0 Hex: E A 9 F 5 2

  7. Negative (Signed) Binary Numbers Positional representation using n bits X = X n X n-1 X n-2 …X 1 X 0 . X -1 X -2 ...X -m Sign-magnitude format Left most bit position (X n) is the sign bit -- only bit that is complemented 0 for positive number 1 for negative number Remaining n-1 bits represent the magnitude Min: - (2 n - 2 -m) = 1111 1111 . 1111 1111 Max: + (2 n - 2 -m) = 0111 1111 . 1111 1111 Zero: - 0 = 1000 0000 . 0000 0000 Zero: +0 = 0000 0000 . 0000 0000

  8. Complements of Numbers Two types of complements for base R number system: R’s complement (R-1)’s complement The (R-1)’s Complement Subtract each digit of a number from (R-1) Examples: 9’s complement of 835 10 is 164 10 1’s complement of 1010 2 is 0101 2 (bit by bit complement operation) The R’s Complement Add 1 to the low-order digit of its (R-1)’s complement Examples: 10’s complement of 835 10 is 164 10 + 1 = 165 10 2’s complement of 1010 2 is 0101 2 + 1 = 0110 2

  9. Negative (Signed) Binary Numbers Ones complement format Negative numbers are represented by a bit-by-bit complementation of the (positive) magnitude (the process of negation) Sign bit interpreted as in sign-magnitude format Examples (8-bit words): +42 = 0 00101010 -42 = 1 11010101 Min: - (2 n - 2 -m) = 1111 1111 . 1111 1111 Max: + (2 n - 2 -m) = 0111 1111 . 1111 1111 Zero: - 0 = 1111 1111 . 1111 1111 Zero: +0 = 0000 0000 . 0000 0000

  10. Negative (Signed) Binary Numbers Twos complement format Negative numbers, -X, are represented by the pseudo- positive number: 2n - |X| An n-bit unsigned binary integer A = an-1 an-2... a1 a0 has a value of For example: 1011 = -1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 = -8 + 2 + 1 = -5 With 2n digits: 2 n-1 -1 positive numbers 2 n -1 negative numbers Given the representation for +X, the representation for -X is found by taking the 1s complement of +X and adding 1

  11. Negative (Signed) Binary Numbers Twos complement format Most significant bit is the “sign bit”. Number representation is not symmetric. Only one representation for zero. Easy to negate, add, and subtract numbers. A little bit trickier for multiply and divide. Min: - (2 n) = 1000 0000 . 0000 0000 Max: + (2 n - 2 -m) = 0111 1111 . 1111 1111 Zero: = 0000 0000 . 0000 0000

  12. Signed 2’s Complement Addition Add the two numbers, including their sign bit, and discard any carry out of left-most(sign) bit Examples: 6 0 0110 -6= 1 1010 + 9 0 1001 + 9= 0 1001 15 0 1111 3= 0 0011 6 0 0110 -9 1 0111 + -9 1 0111 + -9 1 0111 -3 1 1101 -18 (1) 0 1110 9 0 1001 + 9 0 1001 18 1 0010

  13. Detecting 2’s Complement Overflow When adding two's complement numbers, overflow will only occur if the numbers being added have the same sign the sign of the result is different If we perform the addition an-1 an-2 ... a1 a0 +bn-1bn-2… b1 b0 ---------------------------------- =sn-1sn-2… s1 s0 Overflow can be detected as where cn-1and cn are the carry in and carry out of the most significant bit.

  14. Signed 2’s Complement Subtraction To subtract two's complement numbers we first negate the second number and then add the corresponding bits of both numbers. Examples: 3 = 0011 -3 = 1101 -3 = 1101 3 = 0011 - 2 = 0010 - -2 = 1110 - 2 = 0010 - -2 = 1110 become: 3 = 0011 -3 = 1101 -3 = 1101 3 = 0011 + -2 = 1110 + 2 = 0010 + -2 = 1110 + 2 = 0010 1 = 0001 -1 = 1111 -5 = 1011 5 = 0101

  15. Sign-Extension / Zero-Extension Sign-extension is used for signed immediates and signed values from memory To sign-extend an n bit number to n+m bits, copy the sign-bit m times. For example, with n = 4 and m = 4, 1011 = -4 0101 = 5 11111011 = -4 00000101 = 5 Zero-extension is used for logical operations and unsigned values from memory To zero-extend an n bit number to n+m bits, copy zero m times. For example, with n = 4 and m = 4, 1011 = 11 0101 = 5 00001011 = 11 00000101 = 5

  16. Floating Point Number Representation The location of the fractional point is not fixed to a certain location --> The range of the representable numbers is wide --> high precision F = EM m n e k e k-1 ... e 0 m n-1 m n-2 ... m 0 . m -1 ... m -m sign exponent mantissa Mantissa Signed fixed point number, either an integer or a fractional number Exponent Designates the position of the radix point

  17. Floating Point Number Representation Decimal Value: V = M * R E Where: M= Mantissa E= Exponent R= Radix (10) Example (decimal): 1234.5678 Exponent Mantissa Sign Value Sign Value 0 4 0 0.12345678 ==> 0.12345678 x 10 +4

  18. Floating Point Number Representation Example (binary): + 1001.11 (= 9.75) Make a fractional number, counting the number of shifts: + .100111 ==> 4 shifts Exponent Mantissa Sign Value Sign Value 0 100 0 1001111 Or for a 16-bit number with a sign, 5-bit exponent, 10-bit mantissa: 0 00100 1001111000

  19. Other Representations- Gray Codes Characterized by having their representations of the binary integers different in only one digit between consecutive integers Useful in analog-digital conversion. Decimal Gray Binary Decimal Gray Binary 0 0 0 0 0 0 0 0 0 8 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 9 1 1 0 1 1 0 0 1 2 0 0 1 1 0 0 1 0 10 1 1 1 1 1 0 1 0 3 0 0 1 0 0 0 1 1 11 1 1 1 0 1 0 1 1 4 0 1 1 0 0 1 0 0 12 1 0 1 0 1 1 0 0 5 0 1 1 1 0 1 0 1 13 1 0 1 1 1 1 0 1 6 0 1 0 1 0 1 1 0 14 1 0 0 1 1 1 1 0 7 0 1 0 0 0 1 1 1 15 1 0 0 0 1 1 1 1

  20. Other Representations- ASCII Characters 4MSBs 3LSBs 0 1 1 3 4 5 6 7 0 (hex) NUL DLE SP 0 @ P ‘ p 1 SOH DC1 ! 1 A Q a q 2 STX DC2 " 2 B R b r 3 ETX DC3 # 3 C S c s 4 EOT DC4 $ 4 D T d t 5 ENQ NAK % 5 E U e u 6 ACK SYN & 6 F V f v 7 BEL ETB ’ 7 G W g w 8 BS CAN ( 8 H X h x 9 HT EM ) 9 I Y i y A LF SUB * : J Z j z B VT ESC + ; K [ k { C FF FS , < L \ l | D CR GS - = M ] m } E SO RS . > N ^ n ~ F SI US / ? O _ o DEL

  21. Error Detecting Codes- Parity Parity System Simplest method for error detection One parity bit attached to the information Even Parity and Odd Parity Even Parity One bit is attached to the information so that the total number of 1 bits is an even number 1011001 0 1010010 1 ==> B even = B n-1 (+) B n-2 (+) … B 0 Odd Parity One bit is attached to the information so that the total number of 1 bits is an odd number 1011001 1 1010010 0 ==> B odd = B n-1 (+) B n-2 (+) … B 0 (+) 1

  22. Error Detecting Codes- Parity Even Parity Generator Circuit B 0 B 1 B 2 B 3 B 4 B even B 5 B 6 Even Parity Checker Circuit B 0 B 1 B 2 B 3 B 4 B 5 ERROR B 6 B even

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