PowerPoint PresentationReviewPutting it all in perspective…Decimal Numbers: Base 10Numbers: positional notationHexadecimal Numbers: Base 16Decimal vs. Hexadecimal vs. BinaryWhat to do with representations of numbers?BIG IDEA: Bits can represent anything!!How to Represent Negative Numbers?Shortcomings of sign and magnitude?Another try: complement the bitsShortcomings of One’s complement?Standard Negative Number Representation2’s Complement Number “line”: N = 5Two’s Complement FormulaTwo’s Complement shortcut: NegationWhat if too big?Peer Instruction QuestionNumber summary...Reference slidesKilo, Mega, Giga, Tera, Peta, Exa, Zetta, Yottakibi, mebi, gibi, tebi, pebi, exbi, zebi, yobiThe way to remember #sWhich base do we use?Two’s Complement for N=32Two’s comp. shortcut: Sign extensionPreview: Signed vs. Unsigned VariablesAdministriviaGreat DeCal courses I supervise (2 units)CS61C L02 Number Representation (1)Garcia, Spring 2008 © UCBLecturer SOE Dan Garciawww.cs.berkeley.edu/~ddgarciainst.eecs.berkeley.edu/~cs61c CS61C : Machine StructuresLecture #2 – Number Representation2007-01-25There are two handouts today at the front and back of the room!Great book The Universal Historyof Numbersby Georges IfrahCS61C L02 Number Representation (2)Garcia, Spring 2008 © UCBReview•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 ~1.5 yrs)•5 classic components of all computers Control Datapath Memory Input OutputProcessor}CS61C L02 Number Representation (3)Garcia, Spring 2008 © UCBPutting 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. CringelyCS61C L02 Number Representation (4)Garcia, Spring 2008 © UCBDecimal Numbers: Base 10Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9Example:3271 = (3x103) + (2x102) + (7x101) + (1x100)CS61C L02 Number Representation (5)Garcia, Spring 2008 © UCBNumbers: positional notation•Number Base B B symbols per digit:•Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9Base 2 (Binary): 0, 1•Number representation: •d31d30 ... d1d0 is a 32 digit number•value = d31 B31 + d30 B30 + ... + d1 B1 + d0 B0•Binary: 0,1 (In binary digits called “bits”)•0b11010 = 124 + 123 + 022 + 121 + 020 = 16 + 8 + 2= 26•Here 5 digit binary # turns into a 2 digit decimal #•Can we find a base that converts to binary easily?#s often written0b…CS61C L02 Number Representation (6)Garcia, Spring 2008 © UCBHexadecimal Numbers: Base 16•Hexadecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F•Normal digits + 6 more from the alphabet•In C, written as 0x… (e.g., 0xFAB5)•Conversion: BinaryHex•1 hex digit represents 16 decimal values•4 binary digits represent 16 decimal values1 hex digit replaces 4 binary digits•One hex digit is a “nibble”. Two is a “byte”•2 bits is a “half-nibble”. Shave and a haircut…•Example:•1010 1100 0011 (binary) = 0x_____ ?CS61C L02 Number Representation (7)Garcia, Spring 2008 © UCBDecimal vs. Hexadecimal vs. BinaryExamples:1010 1100 0011 (binary) = 0xAC310111 (binary) = 0001 0111 (binary) = 0x170x3F9 = 11 1111 1001 (binary)How do we convert between hex and Decimal?00 0 000001 1 000102 2 001003 3 001104 4 010005 5 010106 6 011007 7 011108 8 100009 9 100110 A 101011 B 101112 C 110013 D 110114 E 111015 F 1111MEMORIZE!Examples:1010 1100 0011 (binary) = 0xAC310111 (binary) = 0001 0111 (binary) = 0x170x3F9 = 11 1111 1001 (binary)How do we convert between hex and Decimal?CS61C L02 Number Representation (8)Garcia, Spring 2008 © UCBWhat 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•…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 ? 1 0 1 0+ 0 1 1 1-------------------------1 0 0 0 111CS61C L02 Number Representation (9)Garcia, Spring 2008 © UCBBIG 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:•locations / addresses? commands?•MEMORIZE: N bits at most 2N thingsRed (00) Green (01) Blue (11)CS61C L02 Number Representation (10)Garcia, Spring 2008 © UCBHow to Represent Negative Numbers?•So far, unsigned numbers•Obvious solution: define leftmost bit to be sign! •0 +, 1 – •Rest of bits can be numerical value of number•Representation called sign and magnitude•MIPS uses 32-bit integers. +1ten would be:0000 0000 0000 0000 0000 0000 0000 0001•And –1ten in sign and magnitude would be:1000 0000 0000 0000 0000 0000 0000 0001CS61C L02 Number Representation (11)Garcia, Spring 2008 © UCBShortcomings 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?•Therefore sign and magnitude abandonedCS61C L02 Number Representation (12)Garcia, Spring 2008 © UCBAnother try: complement the bits•Example: 710 = 001112 –710 = 110002•Called One’s Complement•Note: positive numbers have leading 0s, negative numbers have leadings 1s.00000 00001 01111...111111111010000 ...•What is -00000 ? Answer: 11111•How many positive numbers in N bits?•How many negative numbers?CS61C L02 Number Representation (13)Garcia, Spring 2008 © UCBShortcomings of One’s complement?•Arithmetic still a somewhat complicated.•Still two zeros• 0x00000000 = +0ten• 0xFFFFFFFF = -0ten •Although used for awhile on some computer products, one’s complement was eventually abandoned because another solution was better.CS61C L02 Number Representation (14)Garcia, Spring 2008 © UCBStandard Negative Number Representation•What is result for unsigned numbers if tried to subtract large number from a small one?•Would try to borrow from string of
View Full Document
Unlocking...