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Chapter 2 Bits, Data Types, and Operations Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. How do we represent data in a computer? At the lowest level, a computer is an electronic machine. • works by controlling the flow of electrons Easy to recognize two conditions: 1. presence of a voltage – we’ll call this state “1” 2. absence of a voltage – we’ll call this state “0” Could base state on value of voltage, but control and detection circuits more complex. • compare turning on a light switch to measuring or regulating voltage We’ll see examples of these circuits in the next chapter. Wael Qassas/AABU 2-2 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Computer is a binary digital system. Digital system: • finite number of symbols Binary (base two) system: • has two states: 0 and 1 Basic unit of information is the binary digit, or bit. Values with more than two states require multiple bits. • A collection of two bits has four possible states: 00, 01, 10, 11 • A collection of three bits has eight possible states: 000, 001, 010, 011, 100, 101, 110, 111 • A collection of n bits has 2n possible states. Wael Qassas/AABU 2-3 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. What kinds of data do we need to represent? • Numbers – unsigned, signed, integers, floating point, complex, rational, irrational, … • Text – characters, strings, … • Images – pixels, colors, shapes, … • Sound • Logical – true, false • Instructions • … Data type: • representation and operations within the computer We’ll start with numbers… Wael Qassas/AABU 2-4 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Unsigned Integers Non-positional notation • could represent a number (“5”) with a string of ones (“11111”) • problems? Weighted positional notation • like decimal numbers: “329” • “3” is worth 300, because of its position, while “9” is only worth 9 329 102 101 100 3x100 + 2x10 + 9x1 = 329 most significant 22 101 21 least significant 20 1x4 + 0x2 + 1x1 = 5 Wael Qassas/AABU 2-5 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Unsigned Integers (cont.) An n-bit unsigned integer represents 2n values: from 0 to 2n-1. 22 21 20 0 0 0 0 0 0 1 1 0 1 0 2 0 1 1 3 1 0 0 4 1 0 1 5 1 1 0 6 1 1 1 7 Wael Qassas/AABU 2-6 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Converting Binary (2’s C) to Decimal 1. If leading bit is one, take two’s complement to get a positive number. 2. Add powers of 2 that have “1” in the corresponding bit positions. 3. If original number was negative, add a minus sign. X = 01101000two = 26+25+23 = 64+32+8 = 104ten n 2n 0 1 2 3 4 5 6 7 8 9 10 1 2 4 8 16 32 64 128 256 512 1024 Assuming 8-bit 2’s complement numbers. Wael Qassas/AABU 2-7 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. More Examples X = 00100111two = 25+22+21+20 = 32+4+2+1 = 39ten X = -X = = = X= 11100110two 00011010 24+23+21 = 16+8+2 26ten -26ten n 2n 0 1 2 3 4 5 6 7 8 9 10 1 2 4 8 16 32 64 128 256 512 1024 Assuming 8-bit 2’s complement numbers. Wael Qassas/AABU 2-8 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Converting Decimal to Binary (2’s C) First Method: Division 1. Divide by two – remainder is least significant bit. 2. Keep dividing by two until answer is zero, writing remainders from right to left. 3. Append a zero as the MS bit; if original number negative, take two’s complement. X = 104ten X = 01101000two 104/2 52/2 26/2 13/2 6/2 3/2 = = = = = = 52 r0 26 r0 13 r0 6 r1 3 r0 1 r1 1/2 = 0 r1 bit 0 bit 1 bit 2 bit 3 bit 4 bit 5 bit 6 Wael Qassas/AABU 2-9 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Converting Decimal to Binary (2’s C) n 2n Second Method: Subtract Powers of Two 1. Change to positive decimal number. 2. Subtract largest power of two less than or equal to number. 3. Put a one in the corresponding bit position. 4. Keep subtracting until result is zero. 5. Append a zero as MS bit; if original was negative, take two’s complement. X = 104ten 104 - 64 = 40 40 - 32 = 8 8-8 = 0 0 1 2 3 4 5 6 7 8 9 10 1 2 4 8 16 32 64 128 256 512 1024 bit 6 bit 5 bit 3 X = 01101000two Wael Qassas/AABU 2-10 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Unsigned Binary Arithmetic Base-2 addition – just like base-10! • add from right to left, propagating carry carry 10010 + 1001 11011 10010 + 1011 11101 1111 + 1 10000 10111 + 111 Subtraction, multiplication, division,… Wael Qassas/AABU 2-11 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Signed Integers With n bits, we have 2n distinct values. • assign about half to positive integers (1 through 2n-1) and about half to negative (- 2n-1 through -1) • that leaves two values: one for 0, and one extra Positive integers • just like unsigned – zero in most significant bit 00101 = 5 Negative integers • sign-magnitude – set top bit to show negative, other bits are the same as unsigned 10101 = -5 • one’s complement – flip every bit to represent negative 11010 = -5 • in either case, MS bit indicates sign: 0=positive, 1=negative Wael Qassas/AABU 2-12 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Negative numbers (Sign magnitude) This technique uses additional binary digit to represent the sign, 0 to represent positive, 1 to represent negative. Ex.: 5 = 0101 -5 = 1101 Also 5 = 00000101 -5= 10000101 Wael Qassas/AABU 2-13 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Negative numbers (1’s Complement) 1’s comp is another method used to represent negative numbers. In this method we invert every bit in the positive number in order to represent its negative. Ex.: 9 = 01001 -9= 10110 Again remember that we use additional digit to represent the sign We may represent 9 as 00001001 ; zeros on left have no value -9 in 1’s comp will be 11110110 Wael Qassas/AABU 2-14 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Negative numbers (2’s complement) This is the method which is used in most computers to represent integer numbers nowadays. Remember always to represent a positive number using any of the previous methods is the same, all what is needed is a 0 on the left to show that the number is positive To represent a negative number in 2’s Comp , first we find the 1’s Comp, then add 1 to the result Ex: How we represent -9 in 2’s comp 1- 9 in binary= 01001 2- invert = 10110 3 add 1 = 10111; -9 in 2’s Comp. Wael Qassas/AABU 2-15 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Two’s Complement Representation If number is positive or zero, • normal binary representation, zeroes in upper bit(s) If number is negative, • start with positive number • flip every bit (i.e., take the one’s complement) • then add one 00101 (5) 11010 (1’s comp) + 1 11011 (-5) 01001 (9) (1’s comp) + 1 (-9) Wael Qassas/AABU 2-16 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Two’s Complement Problems with sign-magnitude and 1’s complement • two representations of zero (+0 and –0) • arithmetic circuits are complex How to add two sign-magnitude numbers? – e.g., try 2 + (-3) How to add to one’s complement numbers? – e.g., try 4 + (-3) Two’s complement representation developed to make circuits easy for arithmetic. • for each positive number (X), assign value to its negative (-X), such that X + (-X) = 0 with “normal” addition, ignoring carry out 00101 (5) + 11011 (-5) 00000 (0) 01001 (9) + (-9) 00000 (0) Wael Qassas/AABU 2-17 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Two’s Complement Shortcut To take the two’s complement of a number: • copy bits from right to left until (and including) the first “1” • flip remaining bits to the left 011010000 100101111 + 1 100110000 011010000 (1’s comp) (flip) (copy) 100110000 Wael Qassas/AABU 2-18 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Two’s Complement MS bit is sign bit – it has weight –2n-1. Range of an n-bit number: -2n-1 through 2n-1 – 1. • The most negative number (-2n-1) has no positive counterpart. -23 22 21 20 -23 22 21 20 0 0 0 0 0 1 0 0 0 -8 0 0 0 1 1 1 0 0 1 -7 0 0 1 0 2 1 0 1 0 -6 0 0 1 1 3 1 0 1 1 -5 0 1 0 0 4 1 1 0 0 -4 0 1 0 1 5 1 1 0 1 -3 0 1 1 0 6 1 1 1 0 -2 0 1 1 1 7 1 1 1 1 -1 Wael Qassas/AABU 2-19 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Operations: Arithmetic and Logical Recall: a data type includes representation and operations. We now have a good representation for signed integers, so let’s look at some arithmetic operations: • Addition • Subtraction • Sign Extension We’ll also look at overflow conditions for addition. Multiplication, division, etc., can be built from these basic operations. Logical operations are also useful: • AND • OR • NOT Wael Qassas/AABU 2-20 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Addition As we’ve discussed, 2’s comp. addition is just binary addition. • assume all integers have the same number of bits • ignore carry out • for now, assume that sum fits in n-bit 2’s comp. representation 01101000 (104) 11110110 (-10) + 11110000 (-16) + (-9) 01011000 (88) (-19) Assuming 8-bit 2’s complement numbers. Wael Qassas/AABU 2-21 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Subtraction Negate subtrahend (2nd no.) and add. • assume all integers have the same number of bits • ignore carry out • for now, assume that difference fits in n-bit 2’s comp. representation 01101000 (104) 11110101 (-11) - 00010000 (16) - 11110111 (-9) 01101000 (104) 11110101 (-11) + 11110000 (-16) + (9) 01011000 (88) (-2) Assuming 8-bit 2’s complement numbers. Wael Qassas/AABU 2-22 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sign Extension To add two numbers, we must represent them with the same number of bits. If we just pad with zeroes on the left: 4-bit 8-bit 0100 (4) 00000100 (still 4) 1100 (-4) 00001100 (12, not -4) Instead, replicate the MS bit -- the sign bit: 4-bit 8-bit 0100 (4) 00000100 (still 4) 1100 (-4) 11111100 (still -4) Wael Qassas/AABU 2-23 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Overflow If operands are too big, then sum cannot be represented as an n-bit 2’s comp number. 01000 (8) + 01001 (9) 10001 (-15) 11000 (-8) + 10111 (-9) 01111 (+15) We have overflow if: • signs of both operands are the same, and • sign of sum is different. Another test -- easy for hardware: • carry into MS bit does not equal carry out Wael Qassas/AABU 2-24 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Int x=2000000000; Int y=1000000000; Int z=x+y; Cout<<z; What is the max number represented in Short type Wael Qassas/AABU 2-25 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Logical Operations Operations on logical TRUE or FALSE • two states -- takes one bit to represent: TRUE=1, FALSE=0 View n-bit number as a collection of n logical values • operation applied to each bit independently A 0 0 1 1 B 0 1 0 1 A AND B 0 0 0 1 A 0 0 1 1 B 0 1 0 1 A OR B 0 1 1 1 A 0 1 NOT A 1 0 Wael Qassas/AABU 2-26 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Examples of Logical Operations AND • useful for clearing bits AND with zero = 0 AND with one = no change 11000101 AND 00001111 00000101 OR • useful for setting bits OR with zero = no change OR with one = 1 NOT • unary operation -- one argument • flips every bit OR NOT 11000101 00001111 11001111 11000101 00111010 Wael Qassas/AABU 2-27 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. X=5 Y=6 Z=X&Y Cout<< Z Z=X|Y Cout<< Z Z= ~X Cout<< Z X=-6 Z= ~X Cout<< Z Wael Qassas/AABU 2-28 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Hexadecimal Notation It is often convenient to write binary (base-2) numbers as hexadecimal (base-16) numbers instead. • fewer digits -- four bits per hex digit • less error prone -- easy to corrupt long string of 1’s and 0’s Binary Hex Decimal Binary Hex Decimal 0000 0001 0010 0011 0100 0101 0110 0111 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 1000 1001 1010 1011 1100 1101 1110 1111 8 9 A B C D E F 8 9 10 11 12 13 14 15 Wael Qassas/AABU 2-29 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Converting from Binary to Hexadecimal Every four bits is a hex digit. • start grouping from right-hand side 011101010001111010011010111 3 A 8 F 4 D 7 This is not a new machine representation, just a convenient way to write the number. Wael Qassas/AABU 2-30 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Hexadecimal As we noticed to read a long binary number is confusing So another numbering system was invented ( base 16) We know base 10 ,base 2, and now base 16 Numbers in (base 16) are : 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F There is a direct relation between base 2 and base 16, 24=16 , so the conversion from binary to hexadecimal is quite easy. Each 4 binary digits are converted to one hexadecimal digit. Wael Qassas/AABU 2-31 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Real numbers(Fixed Point) All what we discussed before was about integers. What about real numbers (ex.: 6.125) To represent real numbers we take first the integer part and convert is as we learned before, and then take the fraction part and convert it using the following algorithm 1- multiply fraction by 2 2- take the integer of the result 3- Repeat 1 and 2 until the fraction is zero, or until u reach get enough digits. Ex. : 6.125 6= 110 0.125x2 = 0.25 0.25x2 = 0.5 0.5x2 = 1.0 6.125= 110.001 Wael Qassas/AABU 2-32 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Another example: Convert 9.2 to binary. 9=1001 0.2x2 = 0.4 0.4x2 = 0.8 0.8x2 = 1.6 ; take the fraction only 0.6x2 = 1.2 0.2x2 = 0.4 ; let us stop here, 5 digits after the point The binary equivalent will be: 1001.00110 Wael Qassas/AABU 2-33 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Convert from binary Again let us take the previous results 22 21 20 2-1 2-2 2-3 11 0.0 0 1 = 4+2+.125=6.125 Ex. 2: 23 22 21 20 2-1 2-2 2-3 2-4 2-5 1 0 0 1. 0 0 1 1 0 = 8+1+ .125+.0625 = 9.1875 Why not 9.2 !?? Wael Qassas/AABU 2-34 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Float x=0;y=0.1; While (x!=1) {cout<<“Hi”; X=x+y; } Wael Qassas/AABU 2-35 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Float x=1,y=3; X=x/y*y -1; Cout<<x; Wael Qassas/AABU 2-36 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Floating numbers Nowadays numbers with decimal point are represended in the computer using IEEE 754 float number format. This format has to subtypes : • Float: 32 bit number can represent up to 1038 . • Double: 64 bit number can represent up to 10310 with more number of digits after the point. We will not cover this topic here. It will be covered in Computer architecture course. Wael Qassas/AABU 2-37 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Very Large and Very Small: Floating-Point Large values: 6.023 x 1023 -- requires 79 bits Small values: 6.626 x 10-34 -- requires >110 bits Use equivalent of “scientific notation”: F x 2E Need to represent F (fraction), E (exponent), and sign. IEEE 754 Floating-Point Standard (32-bits): 1b 8b S Exponent 23b Fraction N 1S 1.fraction 2exponent 127 , 1 exponent 254 N 1S 0.fraction 2 126 , exponent 0 Wael Qassas/AABU 2-38 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1000= 100.0 x21 =10.00 x22 = 1.000 x23 =0.1000 x24 10000= 1x24 101000=101x23= 1.01x25 0.125= 0.001= 1x2-3 2127= x.xxxx 10 38 Wael Qassas/AABU 2-39 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Floating Point Example Single-precision IEEE floating point number: 10111111010000000000000000000000 sign exponent fraction • Sign is 1 – number is negative. • Exponent field is 01111110 = 126 (decimal). • Fraction is 0.100000000000… = 0.5 (decimal). Value = -1.5 x 2(126-127) = -1.5 x 2-1 = -0.75. Wael Qassas/AABU 2-40 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. ِAnother example -0.125 in IEEE 754 binary floating format 1) 0.125 0.001 in fixed point binary 2) 0.001 = 1.000000000 x 2 -3 after normalization 3) exponent = 127+ -3 = 124 4) 124 = 0111 1100 in binary 5) 1 0111 1100 0000 0000 0000 0000 0000 000 Wael Qassas/AABU 2-41 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Another example: Write 9.2 in IEEE floating point format. The binary equivalent will be: 1) 1001.0011001100110011… 2) 1.0010011001100110011…… x 23 3) Exponent = 127+3=130 4) Exponent in binary =1000 0010 5) 0 1000 0010 00100110011001100110011 Wael Qassas/AABU 2-42 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Floating-Point Operations Will regular 2’s complement arithmetic work for Floating Point numbers? (Hint: In decimal, how do we compute 3.07 x 1012 + 9.11 x 108?) Wael Qassas/AABU 2-43 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Text: ASCII Characters ASCII: Maps 128 characters to 7-bit code. • both printable and non-printable (ESC, DEL, …) characters 00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f nul soh stx etx eot enq ack bel bs ht nl vt np cr so si 10 11 12 13 14 15 16 17 18 19 1a 1b 1c 1d 1e 1f dle dc1 dc2 dc3 dc4 nak syn etb can em sub esc fs gs rs us 20 21 22 23 24 25 26 27 28 29 2a 2b 2c 2d 2e 2f sp ! " # $ % & ' ( ) * + , . / 30 31 32 33 34 35 36 37 38 39 3a 3b 3c 3d 3e 3f 0 1 2 3 4 5 6 7 8 9 : ; < = > ? 40 41 42 43 44 45 46 47 48 49 4a 4b 4c 4d 4e 4f @ A B C D E F G H I J K L M N O 50 51 52 53 54 55 56 57 58 59 5a 5b 5c 5d 5e 5f P Q R S T U V W X Y Z [ \ ] ^ _ 60 61 62 63 64 65 66 67 68 69 6a 6b 6c 6d 6e 6f ` a b c d e f g h i j k l m n o 70 71 72 73 74 75 76 77 78 79 7a 7b 7c 7d 7e 7f p q r s t u v w x y z { | } ~ del Wael Qassas/AABU 2-44 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A = $ 41 in hexa decimal = 65 in decimal a = $ 61 in hexa decimal = 97 in decimal Space = $20 = 32 Wael Qassas/AABU 2-45 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Interesting Properties of ASCII Code What is relationship between a decimal digit ('0', '1', …) and its ASCII code? What is the difference between an upper-case letter ('A', 'B', …) and its lower-case equivalent ('a', 'b', …)? Given two ASCII characters, how do we tell which comes first in alphabetical order? Are 128 characters enough? (http://www.unicode.org/) No new operations -- integer arithmetic and logic. Wael Qassas/AABU 2-46 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Other Data Types Text strings • sequence of characters, terminated with NULL (0) • typically, no hardware support Image • array of pixels monochrome: one bit (1/0 = black/white) color: red, green, blue (RGB) components (e.g., 8 bits each) other properties: transparency • hardware support: typically none, in general-purpose processors MMX -- multiple 8-bit operations on 32-bit word Sound • sequence of fixed-point numbers Wael Qassas/AABU 2-47 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Images •Normally Each pixel is represented as 24 bits (3 bytes) •The file size of an image is proportional to the image dimensions and the color quality 60 •Example 50 •Number of pixels is 60x50=3000 pixel •If each pixel is represented using 3 bytes RGB the file size will be 3000x3= 9KByte •if the picture is Monocrome, then the file size is 3000x1_bit= 3000 bit = 3000/8 Byte 400 Byte •If the number of colors used is 256 then each pixel is represented as 8 bit (1 Byte) ( remember 28 =256) the image size is 3000 Bytes Wael Qassas/AABU 2-48 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Digital Camera 0.3 MP= 480x640 1.3 MP = 1280x1024 2 MP =1200 x1600 3.15 MP= 1536 x 2048 5 MP = 8.1 MP= Wael Qassas/AABU 2-49 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Image compression •But some times we see that the size of the Image file is not as we calculated? •Data representing Images can be compressed •Two types of compression: • Lossless compression: GIF • Loosely compression: JPG Wael Qassas/AABU 2-50 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Cameras and Mega Pixel? 2MP = 1600x1200 =192000 3.15MP =2048x1536 =3145000 Wael Qassas/AABU 2-51 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. video Frame per second Fps 10 fps 15 fps 30 fps Wael Qassas/AABU 2-52 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. LC-2 Data Types Some data types are supported directly by the instruction set architecture. For LC-2, there is only one supported data type: • 16-bit 2’s complement signed integer • Operations: ADD, AND, NOT Other data types are supported by interpreting 16-bit values as logical, text, fixed-point, etc., in the software that we write. Wael Qassas/AABU 2-53