Integer Representation Bits, binary numbers, and bytes Fixed-width - - PowerPoint PPT Presentation
Wellesley CS 240 Integer Representation Bits, binary numbers, and bytes Fixed-width representation of integers: unsigned and signed Modular arithmetic and overflow 1 positional number representation 2 4 0 = 2 x 10 2 + 4 x 10 1 + 0 x 10 0
Integer Representation
Bits, binary numbers, and bytes Fixed-width representation of integers: unsigned and signed Modular arithmetic and overflow
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Wellesley CS 240
– –
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100 10 1 102 101 100 2 1
= 2 x 102 + 4 x 101 + 0 x 100
position weight
When ambiguous, subscript with base:
10110 Dalmatians (movie) 101ten 1012-Second Rule (folk wisdom for food safety) 101two
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8 4 2 1 23 22 21 20 3 2 1
= 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 position weight
irony
1910 = ?2 10012 = ?10 24010 = ?2 110100112 = ?10 1012 + 10112 = ?2 10010112 x 210 = ?2
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Show powers, strategies.
One wire carries one bit. How many wires to represent a given number? What if I want to build a computer (and not change the hardware later)?
1 0 0 1 1 0 0 0 1 0 0 1
Smallest unit of data
used by a typical modern computer
Binary 000000002 -- 111111112 Decimal 00010 -- 25510 Hexadecimal 0016 -- FF16
Programmer’s hex notation (C, etc.): 0xB4 = B416 = B4hex
Octal (base 8) also useful.
Why do 240 students often confuse Halloween and Christmas?
0000 1 1 0001 2 2 0010 3 3 0011 4 4 0100 5 5 0101 6 6 0110 7 7 0111 8 8 1000 9 9 1001 A 10 1010 B 11 1011 C 12 1100 D 13 1101 E 14 1110 F 15 1111
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What do you call 4 bits? a.k.a. octet
Natural (fixed size) unit of data used by processor.
– Word size determines:
10 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
MSB: most significant bit LSB: least significant bit
(size in bytes)
Java Data Type C Data Type
32-bit word 64-bit word
boolean 1 1 byte char 1 1 char 2 2 short short int 2 2 int int 4 4 float float 4 4 long int 4 8 double double 8 8 long long long 8 8 long double 8 16
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Depends on word size
A C-style string is represented by a series of bytes (chars).
— One-byte ASCII codes for each character. — ASCII = American Standard Code for Information Interchange
32 space 48 64 @ 80 P 96 ` 112 p 33 ! 49 1 65 A 81 Q 97 a 113 q 34 ” 50 2 66 B 82 R 98 b 114 r 35 # 51 3 67 C 83 S 99 c 115 s 36 $ 52 4 68 D 84 T 100 d 116 t 37 % 53 5 69 E 85 U 101 e 117 u 38 & 54 6 70 F 86 V 102 f 118 v 39 ’ 55 7 71 G 87 W 103 g 119 w 40 ( 56 8 72 H 88 X 104 h 120 x 41 ) 57 9 73 I 89 Y 105 I 121 y 42 * 58 : 74 J 90 Z 106 j 122 z 43 + 59 ; 75 K 91 [ 107 k 123 { 44 , 60 < 76 L 92 \ 108 l 124 | 45
= 77 M 93 ] 109 m 125 } 46 . 62 > 78 N 94 ^ 110 n 126 ~ 47 / 63 ? 79 O 95 _ 111
del
Fixed-width integer encodings
Unsigned
⊂
non-negative integers only Signed ⊂ both negative and non-negative integers n bits offer only 2n distinct values. Terminology:
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0110010110101001
“Most-significant” bit(s)
“Least-significant” bit(s)
MSB LSB
(4-bit) unsigned integer representation
n-bit unsigned integers: minimum = maximum =
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8 4 2 1 23 22 21 20 3 2 1
= 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 position weight
modular arithmetic, overflow
15 0000 0001 0011 1111 1110 1100 1011 1010 1000 0111 0110 0100 0010 0101 1001 1101 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
unsigned overflow = "wrong" answer = wrap-around
= carry 1 out of MSB = math answer too big to fit
x+y in n-bit unsigned arithmetic is (x + y) mod 2N in math
4-bit unsigned integers
11 + 2 13 + 5
Unsigned addition overflows if and only if a carry bit is dropped.
1011 + 0010 1101 + 0101
sign-magnitude
Most-significant bit (MSB) is sign bit
0 means non-negative 1 means negative
Remaining bits are an unsigned magnitude 8-bit sign-magnitude: Anything weird here?
00000000 represents _____ 01111111 represents _____ 10000101 represents _____ 10000000 represents _____
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00000100 +10000011 Arithmetic?
Example: 4 - 3 != 4 + (-3)
Zero?
(4-bit) two's complement signed integer representation
4-bit two's complement integers:
minimum = maximum =
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22 21 20
= 1 x -23 + 0 x 22 + 1 x 21 + 1 x 20
two’s complement vs. unsigned
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2n-1 2n-2 … 22 21 20
… 22 21 20 two's complement places unsigned places
What's the difference?
n-bit minimum = n-bit maximum =
8-bit representations
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1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1
n-bit two's complement numbers: minimum = maximum =
4-bit unsigned vs. 4-bit two’s complement
22 0000 0001 0011 1111 1110 1100 1011 1010 1000 0111 0110 0100 0010 0101 1001 1101 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0000 0001 0011 1111 1110 1100 1011 1010 1000 0111 0110 0100 0010 0101 1001 1101 + 1 + 2 + 3 + 4 + 5 + 6 + 7 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1
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difference = ___ = 2___
1 0 1 1
1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 1 x -23 + 0 x 22 + 1 x 21 + 1 x 20
4-bit unsigned 4-bit two's complement
two’s complement addition
23 0000 0001 0011 1111 1110 1100 1011 1010 1000 0111 0110 0100 0010 0101 1001 1101 + 1 + 2 + 3 + 4 + 5 + 6 + 7 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1
2 + 3 0010 + 0011
Modular Arithmetic
+ 3 1110 + 0011
+ -3 1110 + 1101 2 + -3 0010 + 1101
two’s complement overflow
24 0000 0001 0011 1111 1110 1100 1011 1010 1000 0111 0110 0100 0010 0101 1001 1101 + 1 + 2 + 3 + 4 + 5 + 6 + 7 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1
6 + 3 0110 + 0011 Modular Arithmetic
+ 2 1111 + 0010
Addition overflows if and only if the arguments have the same sign but the result does not. if and only if the carry in and carry out of the sign bit differ.
Some CPUs/languages raise exceptions on overflow. C and Java cruise along silently... Feature? Oops?
Ariane 5 Rocket, 1996
Exploded due to cast of 64-bit floating-point number to 16-bit signed number. Overflow.
"... a Model 787 airplane … can lose all alternating current (AC) electrical power … caused by a software counter internal to the GCUs that will overflow after 248 days of continuous power. We are issuing this AD to prevent loss of all AC electrical power, which could result in loss of control of the airplane."
Boeing 787, 2015
A few reasons two’s complement is awesome
Addition, subtraction, hardware Sign Negative one Complement rules
Another derivation
How should we represent 8-bit negatives?
00000001 00000010 00000011 + + + 00000000 00000000 00000000
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