Lecture 4: Number Systems (Chapter 3) (1) Data TypesSection3-1 (2) ComplementsSection3-2 (3) Fixed Point RepresentationsSection3-3 (4) Floating Point RepresentationsSection3-4 (5) Other Binary CodesSection3-5 (6) Error Detection CodesSection3-6

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)

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 = 10Decimal number system R = 2Binary R = 8Octal R = 16Hexadecimal

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

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 = a n-1 a n-2... a 1 a 0 has a value of For example, 1011 = 1 x x x x 2 0 = = 11 An n-bit unsigned binary integer has a range from 0 to 2 n - 1. Positive (Unsigned) Binary Numbers

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: Binary: Hex:E A 9 F 5 2

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 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 )= Max:+ (2 n - 2 -m )= Zero:- 0= Zero:+0=

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 is ’s complement of is (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 is = ’s complement of is =

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= = Min:- (2 n - 2 -m )= Max:+ (2 n - 2 -m )= Zero:- 0= Zero:+0=

Negative (Signed) Binary Numbers Twos complement format Negative numbers, -X, are represented by the pseudo- positive number:2 n - |X| An n-bit unsigned binary integer A = a n-1 a n-2... a 1 a 0 has a value of For example:1011= -1 x x x x 2 0 = = -5 With 2 n 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

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 )= Max:+ (2 n - 2 -m )= Zero:=

Signed 2’s Complement Addition Add the two numbers, including their sign bit, and discard any carry out of left-most(sign) bit Examples: = = = (1)

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 a n-1 a n-2... a 1 a 0 + b n-1 b n-2 … b 1 b = s n-1 s n-2 … s 1 s 0 Overflow can be detected as where c n-1 and c n are the carry in and carry out of the most significant bit.

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 = = = = = = = = 1110 become: 3 = = = = = = = = = = = = 0101

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 = = = = 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 = = = = 5

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 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

Floating Point Number Representation Decimal Value: V = M * R E Where: M= Mantissa E= Exponent R= Radix (10) Example (decimal): ExponentMantissa SignValueSignValue ==> x 10 +4

Floating Point Number Representation Example (binary): (= 9.75) Make a fractional number, counting the number of shifts: ==> 4 shifts ExponentMantissa SignValueSignValue Or for a 16-bit number with a sign, 5-bit exponent, 10-bit mantissa:

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. DecimalGray BinaryDecimalGrayBinary

Other Representations- ASCII Characters 4MSBs 3LSBs (hex)NUL DLE SP P ‘ p 1SOH DC1 ! 1A Qaq 2STX DC2 " 2B Rbr 3ETX DC3 # 3C Scs 4EOT DC4 $ 4D Tdt 5ENQ NAK % 5E Ueu 6ACK SYN & 6FVfv 7BEL ETB ’ 7GWgw 8BS CAN (8HXhx 9HT EM ) 9IYiy ALF SUB * :JZjz BVT ESC + ; K[k{ CFF FS, <L\l| DCR GS - =M]m} ESO RS. >N^n~ FSI US / ? O_oDEL

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 ==> 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 ==> B odd = B n-1 (+) B n-2 (+) … B 0 (+) 1

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