1010 2?= ?= CS 64 Lecture 2 Data Representation. Decimal Numbers: Base 10. Reading: FLD Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

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1 CS 64 Lecture 2 Data Representation Reading: FLD Decimal Numbers: Base 10 Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 ) + (1x10 0 ) ?= ?= 1

2 Numbers: positional notation Number Base B => B symbols per digit: Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Base 2 (Binary): 0, 1 Number representation: d 31 d d 2 d 1 d 0 is a 32 digit number value = d 31 x B 31 + d 30 x B d 2 x B 2 + d 1 x B 1 + d 0 x B 0 Binary: 0, = 1x x x x x x2 + 0x1 = = 90 An Example The number 2001 A base that converts to binary easily? 2

3 Anything Wrong with This Picture? Hexadecimal Numbers: Base 16 Hexadecimal: 0,1,2,3,4,5,6,7,8,9, A, B, C, D, E, F 0xEDF Conversion: Binary Hex 1 hex digit represents 16 decimal values 4 binary digits represent 16 decimal values => 1 hex digit replaces 4 binary digits Examples: (binary) =? (hex) (binary) = (binary) =? 3F9(hex) =? (binary) 3

4 Octal Numbers: Base 8 0,1,2,3,4,5,6,7 Conversion: Binary Octal 1 octal digit represents 8 decimal values 3 binary digits represent 8 decimal values => 1 octal digit replaces 3 binary digits Examples: (binary) =? (oct) How to convert Octal Hex Conversion from One to Another(1) 4

5 Conversion from One to Another(2) Examples of octal-to-binary and hexadecimal-tobinary conversion. Exercise I (binary) =? (hex) (binary) = (binary) =? (hex) =? (octal) 3F9(hex) =? (binary) =? (octal) A B C D E F

6 Some tricks Multiplication by a constant of power of 2 Shifting the number left by some bits The number of bits should be? 2 x 2 = x 4 = 8 010? 3 x 8 =?? 011? Much faster to implement a shift instruction(1 clock cycle) than multiple instruction (32 clock cycles) What about divisions? 4 / 2 =? Even or Odd numbers? An Example = = / 2 = 5 = / 2 = 2 = / 2 = 1 =

7 Decimal Binary Decimal Binary By successive halving, starting at the top and working downward, record the quotient and the remainder. 1492/2= /2= /2= /2= /2= 0..1 Record the reminders from right (LSD) to left (MSD) 7

8 Exercise Convert 96 from decimal to binary Convert 37 to binary, shift it left by one and convert back to decimal. What is the result? In 'C', an unsigned integer is usually 16 bits. What is the largest number that can be represented by an unsigned integer? Binary Decimal By Double-and-Add, scanning the binary number from left to right, start with the leftmost 1 8

9 A Short Review Data representation Hex, decimal, octal, binary Conversion between them Hex, octal, binary Decimal hex, octal, binary Multiply by a number that is a power of 2 Divide by a number that is a power of 2 Even and odd numbers Binary Arithmetic ADDITION SUBTRACTION MULTIPLICATION DIVISION 9

10 What to do with representations of numbers? Just what we do with numbers! Add them Subtract them Multiply them Divide them Compare them add in binary that we can build circuits to do it subtraction also just as you would in decimal Rules of Binary Addition = = = = 0, and carry 1 to the next more significant bit Example =

11 Rules of Binary Subtraction 0-0 = = 1, and borrow 1 from the next more significant bit 1-0 = = = Rules of Binary Multiplication 0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 1 x 1 = 1, and no carry or borrow bits Example =

12 Rules of Binary Division Binary division is the repeated process of subtraction, just as in decimal division. Example = Comparison How do you tell if X > Y? See if X - Y > 0 12

13 Representing Negative Numbers Clarification N BITS BINARY NUMBER 13

14 Attempt #1: sign and magnitude So far, unsigned numbers Obvious solution: sign and magnitude define leftmost bit (most significant bit) to be sign! 0 => +, 1 => - Rest of bits can be numerical value of number MIPS uses N= 32-bit integers. +1 ten would be: And - 1 ten in sign and magnitude would be: Shortcomings of sign and magnitude? Arithmetic circuit more complicated Special steps depending whether signs are the same or not Also, Two zeros (dirty zero) 0x = +0 ten 0x = -0 ten What would it mean for programming? Sign and magnitude abandoned! 14

15 Attempt #2: complement the bits Example: 7 10 = = Called one s Complement Note: positive numbers have leading 0s, negative numbers have leadings 1s What is ? How many positive numbers in N bits? How many negative ones? Shortcomings of ones complement? Arithmetic not too hard Still two zeros 0x = +0ten 0xFFFFFFFF = -0ten What would it mean for programming? A conventional binary addition & add any resulting carry back into the resulting sum One s complement eventually abandoned because another solution was better 15

16 Attempt #3: two s complement Avoid multiple representations of 0 and the need of examining the sign before operations The leftmost bit of a signed binary numeral indicates the sign 0 (+) To get the magnitude: all the bits are the numerical value of number 1 (-) To get the magnitude: all the bits are inverted then 1 is added to the result Example (8 bits binary number) 1 10 = = =

17 Two s Complement Formula Can represent positive and negative numbers in terms of the bit value times a power of 2: d 31 x d 30 x d 2 x d 1 x d 0 x 2 0 Example two = 1x x x x2 2 +0x2 1 +0x2 0 = = -2,147,483,648 ten + 2,147,483,644 ten = -4 ten Note: need to specify width: we use 32 bits Negation in Two s Complement Number Systems Invert every 0 to 1 and every 1 to 0, then add 1 to the result Sum of number and its one s complement must be two two = -1 ten Let x mean the inverted representation of x Then x + x = -1 x + x + 1 = 0 x + 1 = -x Example: -4 to +4 to -4 x : two x : two +1: two () : two +1: two 17

18 Two s Complement in MIPS two = 0 ten two = 1 ten two =2 ten two =2,147,483,645 ten two =2,147,483,646 ten two = 2,147,483,647 ten two = 2,147,483,648 ten two = 2,147,483,647 ten two = 2,147,483,646 ten two = 3 ten two = 2 ten two = 1 ten One zero, 1st bit >=0 or <0, called sign bit but one negative with no positive 2,147,483,648 ten Two s Complement Number line 2^N-1 non-negatives 2^N-1 negatives one zero how many positives? comparison? overflow?

19 Two s complement: Weird number Start with any number in two's complement representation, flip all the bits and add 1 we get the two's complement representation of the negative of that number, except for one -2 N-1 Example (assuming N=8) ( 128) (inverted bits) (add one) Two s complement: Sign Extension Convert 2 s complement number using N bits to more than N bits Simply replicate the most significant bit (sign bit) of smaller to fill new bits 2 s comp. positive number has infinite 0s 2 s comp. negative number has infinite 1s Decimal 4 bit 2 s complement 8 bit 2 s complement

20 Two s complement: Zero Extension Simply replicate 0 to fill new bits Decimal 4 bit 2 s complement 8 bit 2 s complement Two s complement: Arithmetic (carry) (15) (-5) =========== (10) (borrow) (15) ( 5) =========== (20) What about multiplication and division use 2 s complement to convert negative numbers to positive ones add sign to the results 20

21 What if too big? Binary bit patterns above are simply representatives of numbers Numbers really have an infinite number of digits with almost all being zero except for a few of the rightmost digits Just don t normally show leading zeros If result of add (or -,*/) cannot be represented by these rightmost HW bits, overflow is said to have occurred Detection of overflows 0111 (carry) 0111 (7) (3) ============= 1010 ( 6) invalid! Overflow of N bits binary representations NO --The left two bits of carry are 11 or 00 Overflow The left two bits of carry are 10 or 01 21

22 Put it together: Negative Numbers Format Positive or Negative?? Compute magnitude Zero?? (in 32 bit) Example -1(decimal) Sign & Magnitude The left most bit 0 = positive 1 = negative The rest bits as unsigned binary = (A) 0x x x One s Compliment The left most bit 0 = positive 1 = negative Positive =(A) If negative, compliment each bit 0x xFFFFFFFF 0xFFFFFFFE Two s Compliment The left most bit 0 = positive 1 = negative Positive =(A) If negative, compliment each bit and add 1 0x xFFFFFFFF Put it together: Negative Numbers Format Positive or Negative?? Compute magnitude Zero?? (in 32 bit) Example -1(decimal) Sign & Magnitude The left most bit 0 = positive 1 = negative The rest bits as unsigned binary = (A) 0x x x One s Compliment The left most bit 0 = positive 1 = negative Positive =(A) If negative, compliment each bit 0x xFFFFFFFF 0xFFFFFFFE Two s Compliment The left most bit 0 = positive 1 = negative Positive =(A) If negative, compliment each bit and add 1 0x xFFFFFFFF 22

23 The range of different formats Format (N bits) Positive numbers Negative numbers Zero?? (in 32 bit) Sign & Magnitude 1 to (2 N-1-1) -1 to -(2 N-1-1) 0x x One s Compliment 1 to (2 N-1-1) -1 to -(2 N-1-1) 0x xFFFFFFFF Two s Compliment 1 to (2 N-1-1) -1 to -(2 N-1 ) 0x The range of different formats Format (N bits) Positive numbers Negative numbers Zero?? (in 32 bit) Sign & Magnitude 1 to (2 N-1-1) -1 to -(2 N-1-1) 0x x One s Compliment 1 to (2 N-1-1) -1 to -(2 N-1-1) 0x xFFFFFFFF Two s Compliment 1 to (2 N-1-1) -1 to -(2 N-1 ) 0x

24 Homework 1 Assigned Due April 11 th 5pm, use the CS 64 HW box located in the HFH CS mailroom (2 nd floor) Part I: Fundamental of Logic Design (PLD), Chapter 1: Page 24-25, 1.5, 1.6, 1.7, 1.8, 1.36, 1.37 Part II: Consider the 16-bit hexadecimal number 0x98FD What is the value of the bit pattern (in decimal) if it is coded in unsigned integer? What is the value of the bit pattern (in decimal) if it is coded in signed integer (in two s compliment format) Consider the 16-bit hexadecimal number 0x067F What is the value of the bit pattern (in decimal) if it is coded in unsigned integer? What is the value of the bit pattern (in decimal) if it is coded in signed integer (in two s compliment format) 24

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