Number Systems

1
Number Systems

• Number of symbols base of the system
• Most intuitive -- base 10 (decimal system)
• counting on fingertips
• symbols -- 0 1 2 3 4 5 6 7 8 9
• Computers use base 2 (binary system)
• only two symbols - 0 / 1
• Many physical reasons for choosing
• Other useful bases
• 8 - octal
• 16 - hexadecimal

2
Representing a number in base B

• General expression
• A number U3 U2 U1 U0 in base B is effectively
• U3x B3 U2x B2 U1 x B1 U0 x B0
• Examples
• 234 in base 10 is
• 2 x 102 3 x 101 4 x 100
• 1011 in base 2 is
• 1 x 22 0 x 22 1 x 21 1 x 20
  1 x 101 1 x 100 in base 10

3
Why Dont Computers Use Base 10?

• Implementing Electronically
• Hard to store
• Hard to transmit
• Messy to implement digital logic functions
• Binary Electronic Implementation
• Easy to store with bistable elements
• Reliably transmitted on noisy and inaccurate
  wires
• Straightforward implementation of arithmetic
  functions

4
Octal and hexadecimal

• Octal uses 0-7 for representing numbers
• Hexadecimal uses 0-9, a, b, c, d, e and f as
  digits
• a decimal 10, bdecimal 11 and so on
• They are useful because
• translation from binary is easy
• Consider 101100110101
• rewrite as 101 100 110 101 and you get 5465 in
  octal
• rewrite as 1011 0011 0101 and you get b35 in hex
• Much more readable in octal and hex than in
  binary
• Also much easier to calculate equivalent decimal
  value
• It will really pay to learn to interpret hex
  numbers in this course

5
Encoding Byte Values

• Byte 8 bits
• Binary 000000002 to 111111112
• Decimal 010 to 25510
• Hexadecimal 0016 to FF16
• Base 16 number representation
• Use characters 0 to 9 and A to F
• Write FA1D37B16 in C as 0xFA1D37B
• Or 0xfa1d37b

6
Machine Words

• Machine Has Word Size
• Nominal size of integer-valued data
• Including addresses
• Most current machines are 32 bits (4 bytes) or 64
  bits
• 32 bits Limits addresses to 4GB
• Becoming too small for memory-intensive
  applications
• High-end systems are 64 bits (8 bytes)
• Potentially address ? 1.8 X 1019 bytes
• Machines support multiple data formats
• Fractions or multiples of word size
• Always integral number of bytes

7
Data Representations

• Sizes of C Objects (in Bytes)
• C Data Type Compaq Alpha Typical 32-bit Intel
  IA32
• int 4 4 4
• long int 8 4 4
• char 1 1 1
• short 2 2 2
• float 4 4 4
• double 8 8 8
• long double 8 8 10/12
• char 8 4 4
• Or any other pointer

8
Byte Ordering

• How should bytes within multi-byte word be
  ordered in memory?
• Conventions
• Suns, Macs are Big Endian machines
• Least significant byte has highest address
• Alphas, PCs are Little Endian machines
• Least significant byte has lowest address

9
Byte Ordering Example

• Big Endian
• Least significant byte has highest address
• Little Endian
• Least significant byte has lowest address
• Example
• Variable x has 4-byte representation 0x01234567
• Address given by x is 0x100

Big Endian
01
23
45
67
Little Endian
67
45
23
01
10
Reading Byte-Reversed Listings

• Disassembly
• Text representation of binary machine code
• Generated by program that reads the machine code
• Example Fragment

Address Instruction Code Assembly Rendition
8048365 5b pop ebx
8048366 81 c3 ab 12 00 00 add
0x12ab,ebx 804836c 83 bb 28 00 00 00 00 cmpl
0x0,0x28(ebx)

• Deciphering Numbers
• Value 0x12ab
• Pad to 4 bytes 0x000012ab
• Split into bytes 00 00 12 ab
• Reverse ab 12 00 00

11
Representing Integers
Decimal 15213 Binary 0011 1011 0110 1101 Hex
3 B 6 D

• int A 15213
• int B -15213
• long int C 15213

Twos complement representation (Covered later)
12
Representing Pointers
Alpha P

• int B -15213
• int P B

Alpha Address Hex 1 F F F F F
C A 0 Binary 0001 1111 1111 1111 1111
1111 1100 1010 0000
Sun P
Sun Address Hex E F F F F B
2 C Binary 1110 1111 1111 1111 1111
1011 0010 1100
Linux P
Linux Address Hex B F F F F 8
D 4 Binary 1011 1111 1111 1111 1111
1000 1101 0100
Different compilers machines assign different
locations to objects
13
Representing Floats

• Float F 15213.0

IEEE Single Precision Floating Point
Representation Hex 4 6 6 D B
4 0 0 Binary 0100 0110 0110 1101 1011
0100 0000 0000 15213 1110 1101 1011
01
IEEE Single Precision Floating Point
Representation Hex 4 6 6 D B
4 0 0 Binary 0100 0110 0110 1101 1011
0100 0000 0000 15213 1110 1101 1011
01
Not same as integer representation, but
consistent across machines
Can see some relation to integer representation,
but not obvious
14
Representing Strings
char S6 "15213"

• Strings in C
• Represented by array of characters
• Each character encoded in ASCII format
• Standard 7-bit encoding of character set
• Other encodings exist, but uncommon
• Character 0 has code 0x30
• Digit i has code 0x30i
• String should be null-terminated
• Final character 0
• Compatibility
• Byte ordering not an issue
• Data are single byte quantities
• Text files generally platform independent
• Except for different conventions of line
  termination character(s)!

Linux/Alpha S
Sun S
15
Boolean Algebra

• Developed by George Boole in 19th Century
• Algebraic representation of logic
• Encode True as 1 and False as 0

16
Application of Boolean Algebra

• Applied to Digital Systems by Claude Shannon
• 1937 MIT Masters Thesis
• Reason about networks of relay switches
• Encode closed switch as 1, open switch as 0

Connection when AB AB
AB
17
Algebra

• Integer Arithmetic
• Addition is sum operation
• Multiplication is product operation
• is additive inverse
• 0 is identity for sum
• 1 is identity for product
• Boolean Algebra
• ?0,1, , , , 0, 1? forms a Boolean algebra
• Or is sum operation
• And is product operation
• is complement operation (not additive
  inverse)
• 0 is identity for sum
• 1 is identity for product

18

Boolean Algebra ? Integer Ring

• Commutativity
• A B B A A B B A
• A B B A A B B A
• Associativity
• (A B) C A (B C) (A B) C
  A (B C)
• (A B) C A (B C) (A B) C A
  (B C)
• Product distributes over sum
• A (B C) (A B) (A C) A (B C)
  A B B C
• Sum and product identities
• A 0 A A 0 A
• A 1 A A 1 A
• Zero is product annihilator
• A 0 0 A 0 0
• Cancellation of negation
• ( A) A ( A) A

19

Boolean Algebra ? Integer Ring

• Boolean Sum distributes over product
• A (B C) (A B) (A C) A (B C)
  ? (A B) (B C)
• Boolean Idempotency
• A A A A A ? A
• A is true or A is true A is true
• A A A A A ? A
• Boolean Absorption
• A (A B) A A (A B) ? A
• A is true or A is true and B is true A is
  true
• A (A B) A A (A B) ? A
• Boolean Laws of Complements
• A A 1 A A ? 1
• A is true or A is false
• Ring Every element has additive inverse
• A A ? 0 A A 0

20
Properties of and

• Boolean Ring
• ?0,1, , , ?, 0, 1?
• Identical to integers mod 2
• ? is identity operation ? (A) A
• A A 0
• Property Boolean Ring
• Commutative sum A B B A
• Commutative product A B B A
• Associative sum (A B) C A (B C)
• Associative product (A B) C A (B C)
• Prod. over sum A (B C) (A B) (B C)
• 0 is sum identity A 0 A
• 1 is prod. identity A 1 A
• 0 is product annihilator A 0 0
• Additive inverse A A 0

21
Relations Between Operations

• DeMorgans Laws
• Express in terms of , and vice-versa
• A B (A B)
• A and B are true if and only if neither A nor B
  is false
• A B (A B)
• A or B are true if and only if A and B are not
  both false
• Exclusive-Or using Inclusive Or
• A B (A B) (A B)
• Exactly one of A and B is true
• A B (A B) (A B)
• Either A is true, or B is true, but not both

22
General Boolean Algebras

• Operate on Bit Vectors
• Operations applied bitwise
• All of the Properties of Boolean Algebra Apply

01101001 01010101 01000001
01101001 01010101 01111101
01101001 01010101 00111100
01010101 10101010
01000001
01111101
00111100
10101010
23
Bit-Level Operations in C

• Operations , , , Available in C
• Apply to any integral data type
• long, int, short, char
• View arguments as bit vectors
• Arguments applied bit-wise
• Examples (Char data type)
• 0x41 --gt 0xBE
• 010000012 --gt 101111102
• 0x00 --gt 0xFF
• 000000002 --gt 111111112
• 0x69 0x55 --gt 0x41
• 011010012 010101012 --gt 010000012
• 0x69 0x55 --gt 0x7D
• 011010012 010101012 --gt 011111012

24
Contrast Logic Operations in C

• Contrast to Logical Operators
• , , !
• View 0 as False
• Anything nonzero as True
• Always return 0 or 1
• Early termination
• Examples (char data type)
• !0x41 --gt 0x00
• !0x00 --gt 0x01
• !!0x41 --gt 0x01
• 0x69 0x55 --gt 0x01
• 0x69 0x55 --gt 0x01
• p p (avoids null pointer access)

25
Shift Operations

• Left Shift x ltlt y
• Shift bit-vector x left y positions
• Throw away extra bits on left
• Fill with 0s on right
• Right Shift x gtgt y
• Shift bit-vector x right y positions
• Throw away extra bits on right
• Logical shift
• Fill with 0s on left
• Arithmetic shift
• Replicate most significant bit on right
• Useful with twos complement integer
  representation

01100010
Argument x
00010000
ltlt 3
00010000
00010000
00011000
Log. gtgt 2
00011000
00011000
00011000
Arith. gtgt 2
00011000
00011000
10100010
Argument x
00010000
ltlt 3
00010000
00010000
00101000
Log. gtgt 2
00101000
00101000
11101000
Arith. gtgt 2
11101000
11101000
26
Cool Stuff with Xor
void funny(int x, int y) x x y
/ 1 / y x y / 2 / x x
y / 3 /

• Bitwise Xor is form of addition
• With extra property that every value is its own
  additive inverse
• A A 0

y
x
B
A
Begin
1
2
3
End
27
Storing negative integers

• sign-magnitude
• of n bits in the word, the most significant bit
  is sign
• 1 indicates negative number, 0 a positive number
• remaining n-1 bits determine the magnitude
• 0 has two representations
• The range of valid values is -(2(n-1)-1) to
  (2(n-1)-1)
• Examples
• 8 bit words can represent -127 to 127
• 16 bit words can represent -32767 to 32767
• Advantages
• Very straightforward
• Simple to understand
• Disadvantage
• different machinery for addition and subtraction
• multiple representations of zero.

28

• Ones complement
• The most significant bit is effectively sign bit
• The range of numbers the same as signed magnitude
• Positive numbers stored as such
• Negative numbers are bit-wise inverted
• Example
• 4 bit storage range -7 to 7
• -7 1000, 7 0111, 3 0011, -4 1011
• 8 bit storage range -127 t0 128
• Zero has two representations
• 0000 0000
• 1111 1111
• Hardly ever used for actual storage
• Useful for understanding 2s compliment and some
  other operations

29
Twos Complement
Unsigned
Twos Complement
short int x 15213 short int y -15213
Sign Bit

• C short 2 bytes long
• Sign Bit
• For 2s complement, most significant bit
  indicates sign
• 0 for nonnegative
• 1 for negative

30

• Twos compliment is effectively ones compliment
  with a 1 added to it.
• It is a numbers additive inverse
• Number added to its own twos compliment results
  in zero
• Same circuitry can do arithmetic operations for
  positive and negative numbers
• The range is (-2n-1) to (2n-1-1)
• Example 13 (-6)
• 13 in 4 bit binary is 1101
• 6 is 0110
• Its ones complement is 1001
• Twos complement is 1010
• Add 1101 and 1010 in 4 bits, ignoring the carry
• The answer is 0111, with carry of 1, which in
  decimal is 7

31
Power-of-2 Multiply with Shift

• Operation
• u ltlt k gives u 2k
• Both signed and unsigned
• Examples
• u ltlt 3 u 8
• u ltlt 5 - u ltlt 3 u 24
• Most machines shift and add much faster than
  multiply

k
u
  
Operands w bits
2k

0
0
1
0
0
0


u 2k
True Product wk bits
0
0
0

UMultw(u , 2k)
0
0
0


Discard k bits w bits
TMultw(u , 2k)
32
Unsigned Power-of-2 Divide with Shift

• Quotient of Unsigned by Power of 2
• u gtgt k gives ? u / 2k ?
• Uses logical shift

k
u
Binary Point

Operands
2k
/
0
0
1
0
0
0


u / 2k
Division
.

0

Result
? u / 2k ?

0

33
Signed Power-of-2 Divide with Shift

• Quotient of Signed by Power of 2
• x gtgt k gives ? x / 2k ?
• Uses arithmetic shift
• Rounds wrong direction when u lt 0

34
Correct Power-of-2 Divide

• Quotient of Negative Number by Power of 2
• Want ? x / 2k ? (Round Toward 0)
• Compute as ? (x2k-1)/ 2k ?
• In C (x (1ltltk)-1) gtgt k
• Biases dividend toward 0
• Case 1 No rounding

k
Dividend
u
1

0
0
0

2k 1
0
0
0
1
1
1


Binary Point
1

1
1
1

Divisor
2k
/
0
0
1
0
0
0


? u / 2k ?
.
1

0
1
1

1
1
1
1

Biasing has no effect
35
Correct Power-of-2 Divide (Cont.)
Case 2 Rounding
k
Dividend
x
1


2k 1
0
0
0
1
1
1


1


Binary Point
Incremented by 1
Divisor
2k
/
0
0
1
0
0
0


? x / 2k ?
.
1

0
1
1

1

Biasing adds 1 to final result
Incremented by 1
36
Fractional Binary Numbers
2i
2i1
4

2
1
1/2

1/4
1/8
2j

• Representation
• Bits to right of binary point represent
  fractional powers of 2
• Represents rational number

37
Frac. Binary Number Examples

• Value Representation
• 5-3/4 101.112
• 2-7/8 10.1112
• 63/64 0.1111112
• Observations
• Divide by 2 by shifting right
• Multiply by 2 by shifting left
• Numbers of form 0.1111112 just below 1.0
• 1/2 1/4 1/8 1/2i ? 1.0

38
Representable Numbers

• Limitation
• Can only exactly represent numbers of the form
  x/2k
• Other numbers have repeating bit representations
• Value Representation
• 1/3 0.0101010101012
• 1/5 0.00110011001100112
• 1/10 0.000110011001100112