1、Chapter 2 Bits, Data Types, and Operations,2-2,How do we represent data in a computer?,At the lowest level, a computer is an electronic machine. works by controlling the flow of electronsEasy to recognize two conditions: presence of a voltage well call this state “1” absence of a voltage well call t
2、his 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,2-3,Computer is a binary digital system.,Basic unit of information is the binary digit, or bit. Values with more than two states re
3、quire 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.,Binary (base two) system: has two states: 0 and 1,Digital system: finite
4、number of symbols,2-4,What kinds of data do we need to represent?,Numbers signed, unsigned, 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 com
5、puter Well start with numbers,2-5,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 9,3x100 + 2x10 + 9x1 = 329,1
6、x4 + 0x2 + 1x1 = 5,most significant,least significant,2-6,Unsigned Integers (cont.),An n-bit unsigned integer represents 2n values: from 0 to 2n-1.,2-7,Unsigned Binary Arithmetic,Base-2 addition just like base-10! add from right to left, propagating carry,10010 10010 1111+ 1001 + 1011 + 111011 11101
7、 1000010111+ 111,carry,Subtraction, multiplication, division,2-8,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 lik
8、e unsigned zero in most significant (MS) bit 00101 = 5 Negative integers sign-magnitude set MS bit to show negative, other bits are the same as unsigned 10101 = -5 ones complement flip every bit to represent negative 11010 = -5 in either case, MS bit indicates sign: 0=positive, 1=negative,2-9,Twos C
9、omplement,Problems with sign-magnitude and 1s 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 ones complement numbers? e.g., try 4 + (-3) Twos complement representation developed to make circuit
10、s 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) 01001 (9)+ 11011 (-5) + (-9)00000 (0) 00000 (0),2-10,Twos Complement Representation,If number is positive or zero, normal binary represe
11、ntation, zeroes in upper bit(s) If number is negative, start with positive number flip every bit (i.e., take the ones complement) then add one,00101 (5) 01001 (9)11010 (1s comp) (1s comp)+ 1 + 1 11011 (-5) (-9),2-11,Twos Complement Shortcut,To take the twos complement of a number: copy bits from rig
12、ht to left until (and including) the first “1” flip remaining bits to the left,011010000 011010000100101111 (1s comp) + 1 100110000 100110000,(copy),(flip),2-12,Twos Complement Signed Integers,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 nu
13、mber (-2n-1) has no positive counterpart.,2-13,Converting Binary (2s C) to Decimal,If leading bit is one, take twos complement to get a positive number. Add powers of 2 that have “1” in the corresponding bit positions. If original number was negative, add a minus sign.,X = 01101000two= 26+25+23 = 64
14、+32+8= 104ten,Assuming 8-bit 2s complement numbers.,2-14,More Examples,Assuming 8-bit 2s complement numbers.,X = 00100111two= 25+22+21+20 = 32+4+2+1= 39ten,X = 11100110two -X = 00011010= 24+23+21 = 16+8+2= 26tenX = -26ten,2-15,Converting Decimal to Binary (2s C),First Method: Division Find magnitude
15、 of decimal number. (Always positive.) Divide by two remainder is least significant bit. Keep dividing by two until answer is zero, writing remainders from right to left. Append a zero as the MS bit; if original number was negative, take twos complement.,X = 104ten 104/2 = 52 r0 bit 052/2 = 26 r0 bi
16、t 126/2 = 13 r0 bit 213/2 = 6 r1 bit 36/2 = 3 r0 bit 43/2 = 1 r1 bit 5X = 01101000two 1/2 = 0 r1 bit 6,2-16,Converting Decimal to Binary (2s C),Second Method: Subtract Powers of Two Find magnitude of decimal number. Subtract largest power of two less than or equal to number. Put a one in the corresp
17、onding bit position. Keep subtracting until result is zero. Append a zero as MS bit; if original was negative, take twos complement.,X = 104ten 104 - 64 = 40 bit 640 - 32 = 8 bit 58 - 8 = 0 bit 3X = 01101000two,2-17,Operations: Arithmetic and Logical,Recall: a data type includes representation and o
18、perations. We now have a good representation for signed integers, so lets look at some arithmetic operations: Addition Subtraction Sign Extension Well also look at overflow conditions for addition. Multiplication, division, etc., can be built from these basic operations. Logical operations are also
19、useful: AND OR NOT,2-18,Addition,As weve discussed, 2s 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 2s comp. representation,01101000 (104) 11110110 (-10)+ 11110000 (-16) + (-9)01011000 (98) (-19),Assu
20、ming 8-bit 2s complement numbers.,2-19,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 2s comp. representation,01101000 (104) 11110110 (-10)- 00010000 (16) - (-9)01101000 (104) 11110110 (
21、-10)+ 11110000 (-16) + (9)01011000 (88) (-1),Assuming 8-bit 2s complement numbers.,2-20,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:Instead, replicate the MS bit - the sign bit:,4-bit 8-bit 0100 (4) 00000100 (still 4)
22、 1100 (-4) 00001100 (12, not -4),4-bit 8-bit 0100 (4) 00000100 (still 4) 1100 (-4) 11111100 (still -4),2-21,Overflow,If operands are too big, then sum cannot be represented as an n-bit 2s comp number.We have overflow if: signs of both operands are the same, and sign of sum is different. Another test
23、 - easy for hardware: carry into MS bit does not equal carry out,01000 (8) 11000 (-8)+ 01001 (9) + 10111 (-9)10001 (-15) 01111 (+15),2-22,Logical Operations,Operations on logical TRUE or FALSE two states - takes one bit to represent: TRUE=1, FALSE=0View n-bit number as a collection of n logical valu
24、es operation applied to each bit independently,2-23,Examples of Logical Operations,AND useful for clearing bits AND with zero = 0 AND with one = no changeOR useful for setting bits OR with zero = no change OR with one = 1NOT unary operation - one argument flips every bit,11000101 AND 00001111 000001
25、01,11000101 OR 00001111 11001111,NOT 11000101 00111010,2-24,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 1s and 0s,2-25,Converting from B
26、inary to Hexadecimal,Every four bits is a hex digit. start grouping from right-hand side,011101010001111010011010111,7,D,4,F,8,A,3,This is not a new machine representation, just a convenient way to write the number.,2-26,Fractions: Fixed-Point,How can we represent fractions? Use a “binary point” to
27、separate positive from negative powers of two - just like “decimal point.” 2s comp addition and subtraction still work. if binary points are aligned,No new operations - same as integer arithmetic.,2-27,Very Large and Very Small: Floating-Point,Large values: 6.023 x 1023 - requires 79 bits Small valu
28、es: 6.626 x 10-34 - requires 110 bitsUse equivalent of “scientific notation”: F x 2E Need to represent F (fraction), E (exponent), and sign. IEEE 754 Floating-Point Standard (32-bits):,S,Exponent,Fraction,1b,8b,23b,2-28,Floating Point Example,Single-precision IEEE floating point number:1011111101000
29、0000000000000000000Sign 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.,sign,exponent,fraction,2-29,Floating-Point Operations,Will regular 2s complement arithmetic work for Floating Point
30、numbers? (Hint: In decimal, how do we compute 3.07 x 1012 + 9.11 x 108?),2-30,Text: ASCII Characters,ASCII: Maps 128 characters to 7-bit code. both printable and non-printable (ESC, DEL, ) characters,2-31,Interesting Properties of ASCII Code,What is relationship between a decimal digit (0, 1, ) and
31、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.,
32、2-32,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 non
33、e, in general-purpose processors MMX - multiple 8-bit operations on 32-bit word Sound sequence of fixed-point numbers,2-33,LC-3 Data Types,Some data types are supported directly by the instruction set architecture.For LC-3, there is only one hardware-supported data type: 16-bit 2s complement signed integer Operations: ADD, AND, NOTOther data types are supported by interpreting 16-bit values as logical, text, fixed-point, etc., in the software that we write.,
