Chapter 2 Bits, Data Types, and Operations

Chapter 2Bits, Data Types,and Operations

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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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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 tomeasuring or regulating voltage

We’ll see examples of these circuits in the next chapter.

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Computer is a binary digital system.

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.

Binary (base two) system:• has two states: 0 and 1

Digital system:• finite number of symbols

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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…

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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

329102 101 100

10122 21 20

3x100 + 2x10 + 9x1 = 329 1x4 + 0x2 + 1x1 = 5

mostsignificant

leastsignificant

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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

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Converting Binary (2’s C) to Decimal1. If leading bit is one, take two’s

complement to get a positive number.

2. Add powers of 2 that have “1” in thecorresponding bit positions.

3. If original number was negative,add a minus sign.

n 2n

0 1

1 2

2 4

3 8

4 16

5 32

6 64

7 128

8 256

9 512

10 1024

X = 01101000two

= 26+25+23 = 64+32+8= 104ten

Assuming 8-bit 2’s complement numbers.

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More Examples

n 2n

0 1

1 2

2 4

3 8

4 16

5 32

6 64

7 128

8 256

9 512

10 1024


X = 00100111two

= 25+22+21+20 = 32+4+2+1= 39ten

X = 11100110two

-X = 00011010= 24+23+21 = 16+8+2= 26ten

X = -26ten

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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 104/2 = 52 r0 bit 0

52/2 = 26 r0 bit 126/2 = 13 r0 bit 213/2 = 6 r1 bit 3

6/2 = 3 r0 bit 43/2 = 1 r1 bit 5

X = 01101000two 1/2 = 0 r1 bit 6

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Converting Decimal to Binary (2’s C)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 bit 6

40 - 32 = 8 bit 58 - 8 = 0 bit 3

X = 01101000two

n 2n

0 1

1 2

2 4

3 8

4 16

5 32

6 64

7 128

8 256

9 512

10 1024

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Unsigned Binary ArithmeticBase-2 addition – just like base-10!

• add from right to left, propagating carry

10010 10010 1111+ 1001 + 1011 + 111011 11101 10000

10111+ 111

carry

Subtraction, multiplication, division,…

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Signed IntegersWith 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 unsigned10101 = -5

• one’s complement – flip every bit to represent negative11010 = -5

• in either case, MS bit indicates sign: 0=positive, 1=negative

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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

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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 signWe may represent 9 as 00001001 ; zeros on left have no value-9 in 1’s comp will be 11110110

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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 positiveTo represent a negative number in 2’s Comp , first we find the 1’s Comp, then add 1 to the resultEx:How we represent -9 in 2’s comp1- 9 in binary= 010012- invert = 101103 add 1 = 10111; -9 in 2’s Comp.

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Two’s Complement RepresentationIf 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) 01001 (9)

11010 (1’s comp) (1’s comp)

+ 1 + 111011 (-5) (-9)

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Two’s ComplementProblems 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 makecircuits 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)

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Two’s Complement ShortcutTo 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 011010000100101111 (1’s comp)

+ 1100110000 100110000

(copy)(flip)

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Two’s ComplementMS 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

0 0 0 0 0

0 0 0 1 1

0 0 1 0 2

0 0 1 1 3

0 1 0 0 4

0 1 0 1 5

0 1 1 0 6

0 1 1 1 7

-23 22 21 20

1 0 0 0 -8

1 0 0 1 -7

1 0 1 0 -6

1 0 1 1 -5

1 1 0 0 -4

1 1 0 1 -3

1 1 1 0 -2

1 1 1 1 -1

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Operations: Arithmetic and LogicalRecall: 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

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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)


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SubtractionNegate 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.

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Sign ExtensionTo add two numbers, we must represent themwith 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-bit0100 (4) 00000100 (still 4)

1100 (-4) 00001100 (12, not -4)

4-bit 8-bit0100 (4) 00000100 (still 4)

1100 (-4) 11111100 (still -4)

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OverflowIf operands are too big, then sum cannot be represented as an n-bit 2’s comp number.

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

01000 (8) 11000 (-8)

+ 01001 (9) +10111 (-9)

10001 (-15) 01111 (+15)

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Int x=2000000000;

Int y=1000000000;

Int z=x+y;

Cout<<z;

What is the max number represented in Short type

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Logical OperationsOperations 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 B A AND B0 0 00 1 01 0 01 1 1

A B A OR B0 0 00 1 11 0 11 1 1

A NOT A0 11 0

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Examples of Logical OperationsAND

• useful for clearing bitsAND with zero = 0AND with one = no change

OR• useful for setting bits

OR with zero = no changeOR with one = 1

NOT• unary operation -- one argument• flips every bit

11000101AND 00001111

00000101

11000101OR 00001111

11001111

NOT 1100010100111010

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X=5Y=6Z=X&YCout<< Z

Z=X|YCout<< Z

Z= ~XCout<< Z

X=-6Z= ~XCout<< Z

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Hexadecimal NotationIt is often convenient to write binary (base-2) numbersas 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 Decimal0000 0 0

0001 1 1

0010 2 2

0011 3 3

0100 4 4

0101 5 5

0110 6 6

0111 7 7

Binary Hex Decimal1000 8 8

1001 9 9

1010 A 10

1011 B 11

1100 C 12

1101 D 13

1110 E 14

1111 F 15

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Converting from Binary to HexadecimalEvery four bits is a hex digit.

• start grouping from right-hand side

011101010001111010011010111

7D4F8A3

This is not a new machine representation,just a convenient way to write the number.

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HexadecimalAs we noticed to read a long binary number is confusingSo another numbering system was invented ( base 16)We know base 10 ,base 2, and now base 16Numbers in (base 16) are : 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,FThere 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.

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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. : 6.1256= 1100.125x2 = 0.250.25x2 = 0.50.5x2 = 1.0 6.125= 110.001

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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

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Convert from binaryAgain let us take the previous results

22 21 20 2-1 2-2 2-3

1 1 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 !??

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Float x=0;y=0.1;

While (x!=1)

{cout<<“Hi”;

X=x+y;

}

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Float x=1,y=3;

X=x/y*y -1;

Cout<<x;

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Floating numbersNowadays 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.

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Very Large and Very Small: Floating-PointLarge 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):

S Exponent Fraction

1b 8b 23b

0exponent,2fraction.01

254exponent1,2fraction.11126

127exponent

S

S

N

N

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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

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Floating Point ExampleSingle-precision IEEE floating point number:

10111111010000000000000000000000

• 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.

sign exponent fraction

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�ِ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

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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

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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?)

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Text: ASCII Characters ASCII: Maps 128 characters to 7-bit code.

• both printable and non-printable (ESC, DEL, …) characters

00 nul10 dle20 sp 30 0 40 @ 50 P 60 ` 70 p01 soh11 dc121 ! 31 1 41 A 51 Q 61 a 71 q02 stx12 dc222 " 32 2 42 B 52 R 62 b 72 r03 etx13 dc323 # 33 3 43 C 53 S 63 c 73 s04 eot14 dc424 $ 34 4 44 D 54 T 64 d 74 t05 enq15 nak25 % 35 5 45 E 55 U 65 e 75 u06 ack16 syn26 & 36 6 46 F 56 V 66 f 76 v07 bel17 etb27 ' 37 7 47 G 57 W 67 g 77 w08 bs 18 can28 ( 38 8 48 H 58 X 68 h 78 x09 ht 19 em 29 ) 39 9 49 I 59 Y 69 i 79 y0a nl 1a sub2a * 3a : 4a J 5a Z 6a j 7a z0b vt 1b esc2b + 3b ; 4b K 5b [ 6b k 7b {0c np 1c fs 2c , 3c < 4c L 5c \ 6c l 7c |0d cr 1d gs 2d - 3d = 4d M 5d ] 6d m 7d }0e so 1e rs 2e . 3e > 4e N 5e ^ 6e n 7e ~0f si 1f us 2f / 3f ? 4f O 5f _ 6f o 7f del

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A = $ 41 in hexa decimal = 65 in decimal

a = $ 61 in hexa decimal = 97 in decimal

Space = $20 = 32

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Interesting Properties of ASCII CodeWhat 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.

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Other Data TypesText 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 processorsMMX -- multiple 8-bit operations on 32-bit word

Sound• sequence of fixed-point numbers

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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•Example

•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

60

50

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Digital Camera0.3 MP= 480x640

1.3 MP = 1280x1024

2 MP =1200 x1600

3.15 MP= 1536 x 2048

5 MP =

8.1 MP=

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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

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Cameras and Mega Pixel?2MP = 1600x1200

=192000

3.15MP =2048x1536

=3145000

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videoFrame per second

Fps

10 fps

15 fps

30 fps

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LC-2 Data TypesSome data types are supported directly by theinstruction 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 interpreting16-bit values as logical, text, fixed-point, etc.,in the software that we write.

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Chapter 2 Bits, Data Types, and Operations

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