inst eecs berkeley edu cs61c CS61C Machine Structures Lecture 2 Number Representation 2010 01 22 There is one handout today at the front and back of the room Lecturer SOE Dan Garcia www cs berkeley edu ddgarcia Great book The Universal History of Numbers by Georges Ifrah CS61C L02 Number Representation 1 Garcia Spring 2010 UCB Review Continued rapid improvement in computing 2X every 2 0 years in memory size every 1 5 years in processor speed every 1 0 year in disk capacity Moore s Law enables processor 2X transistors chip every 2 yrs 5 classic components of all computers a b c d e Control Datapath Memory Input Output What ll be the most important part of a computer Processor in the future CS61C L02 Number Representation 2 Garcia Spring 2010 UCB Putting it all in perspective If the automobile had followed the same development cycle as the computer a Rolls Royce would today cost 100 get a million miles per gallon and explode once a year killing everyone inside Robert X Cringely CS61C L02 Number Representation 3 Garcia Spring 2010 UCB Data input Analog Digital Real world is analog To import analog information we must do two things Sample E g for a CD every 44 100ths of a second we ask a music signal how loud it is Quantize For every one of these samples we figure out where on a 16 bit 65 536 tic mark yardstick it lies www joshuadysart com journal archives digital sampling gif CS61C L02 Number Representation 4 Garcia Spring 2010 UCB Digital data not nec born Analog hof povray org CS61C L02 Number Representation 5 Garcia Spring 2010 UCB BIG IDEA Bits can represent anything Characters 26 letters 5 bits 25 32 upper lower case punctuation 7 bits in 8 ASCII standard code to cover all the world s languages 8 16 32 bits Unicode www unicode com Logical values 0 False 1 True colors Ex Red 00 Green 01 Blue 11 locations addresses commands MEMORIZE N bits at most 2N things CS61C L02 Number Representation 6 Garcia Spring 2010 UCB How many bits to represent a 1 b 9 3 14 so that s 011 001 100 c 64 Since Macs are 64 bit machines d Every bit the machine has e CS61C L02 Number Representation 7 Garcia Spring 2010 UCB What to do with representations of numbers Just what we do with numbers Add them Subtract them Multiply them Divide them Compare them Example 10 7 17 1 1 1 0 1 0 0 1 1 1 1 0 0 0 1 so simple to add in binary that we can build circuits to do it subtraction just as you would in decimal Comparison How do you tell if X Y CS61C L02 Number Representation 8 Garcia Spring 2010 UCB What if too big Binary bit patterns above are simply representatives of numbers Strictly speaking they are called numerals Numbers really have an number of digits with almost all being same 00 0 or 11 1 except for a few of the rightmost digits Just don t normally show leading digits If result of add or cannot be represented by these rightmost HW bits overflow is said to have occurred 00000 00001 00010 11110 11111 unsigned CS61C L02 Number Representation 9 Garcia Spring 2010 UCB How to Represent Negative C s unsigned int C99 s uintN t Numbers Binary So far unsigned numbers odometer 00000 00001 01111 10000 11111 Obvious solution define leftmost bit to be sign 1 0 Rest of bits can be numerical value of number Binary odometer Representation called sign and magnitude 00000 00001 01111 11111 10001 10000 CS61C L02 Number Representation 10 META Ain t no free lunch Garcia Spring 2010 UCB Shortcomings of sign and magnitude Arithmetic circuit complicated Special steps depending whether signs are the same or not Also two zeros 0x00000000 0ten 0x80000000 0ten What would two 0s mean for programming Also incrementing binary odometer sometimes increases values and sometimes decreases Therefore sign and magnitude abandoned CS61C L02 Number Representation 11 Garcia Spring 2010 UCB Administrivia Upcoming lectures Next three lectures Introduction to C Lab overcrowding Remember you can go to ANY discussion none or one that doesn t match with lab or even more than one if you want Overcrowded labs consider finishing at home and getting checkoffs in lab or bringing laptop to lab If you re checked off in 1st hour you get an extra point on the labs Enrollment It will work out don t worry Exams are all open book no need to memorize Soda locks doors 6 30pm on weekends Look at class website newsgroup often http inst eecs berkeley edu cs61c ucb class cs61c CS61C L02 Number Representation 12 Iclickerskinz com Garcia Spring 2010 UCB Great DeCal courses I supervise UCBUGG 3 units P NP UC Berkeley Undergraduate Graphics Group Tue 5 7pm or Wed 4 6pm in 200 Sutardja Dai Learn to create a short 3D animation No prereqs but they might have too many students so admission not guaranteed http ucbugg berkeley edu MS DOS X 2 units P NP Macintosh Software Developers for OS X Mon 5 7pm in 200 Sutardja Dai Learn to program the Macintosh or iPhone or iPod Touch No prereqs other than interest http msdosx berkeley edu CS61C L02 Number Representation 13 Garcia Spring 2010 UCB Another try complement the bits 710 001112 710 110002 Example Called One s Complement Note positive numbers have leading 0s negative numbers have leadings 1s Binary odometer 00000 00001 01111 10000 11110 11111 What is 00000 Answer 11111 How many positive numbers in N bits How many negative numbers CS61C L02 Number Representation 14 Garcia Spring 2010 UCB Shortcomings of One s complement Arithmetic still a somewhat complicated Still two zeros 0x00000000 0ten 0xFFFFFFFF 0ten Although used for a while on some computer products one s complement was eventually abandoned because another solution was better CS61C L02 Number Representation 15 Garcia Spring 2010 UCB Standard Negative Representation Problem is the negative mappings overlap with the positive ones the two 0s Want to shift the negative mappings left by one Solution For negative numbers complement then add 1 to the result As with sign and magnitude one s compl leading 0s positive leading 1s negative 000000 xxx is 0 111111 xxx is 0 except 1 1111 is 1 not 0 as in sign mag This representation is Two s Complement This makes the hardware simple C s int aka a signed integer Also C s short long long C99 s intN t CS61C L02 Number Representation 16 Garcia Spring 2010 UCB Two s Complement Formula Can represent positive and negative numbers in terms of the bit value times a power of 2 d31 x 231 d30 x 230 d2 x 22 d1 x 21 d0 x 20 Example 1101two in a nibble 1x 23 1x22 0x21 1x20 23 22 0 20 8 4 0 1 8 5 3ten CS61C L02 Number Representation 17 Example 3 to 3 to
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