1. CS 270: Computer Organization Bits, Bytes, and Integers
Instructor:
Professor Stephen P. Carl 1

2. Binary Representations
Base 2 Number Representation
Represent as
Represent as [0011]…2
Represent X 104 as X 213
Electronic Implementation
Easy to store with bistable elements
Reliably transmitted on noisy and inaccurate wires
0.0V
0.5V
2.8V
3.3V 1

3. Encoding Byte Values Byte = 8 bits Binary: 00000000 to 11111111
Decimal: 0 to 255
Note: first digit must not be 0 in C
Hexadecimal to FF16
Base 16 number representation
Use characters ‘0’ to ‘9’ and ‘A’ to ‘F’
Write FA1D37B16 in C as 0xFA1D37B
(Or 0xfa1d37b)
Note: Use subscripts when it isn’t clear what representation is being used. For example: 100 could be 1002 (4), (100), or (256)
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001 A 10
1010 B 11
1011 C 12
1100 D 13
1101 E 14
1110 F 15
1111
Hex
Decimal
Binary

4. Byte-Oriented Memory Organization
• • •
00•••0
FF•••F
Programs Refer to Virtual Addresses
Conceptually, a very large array of bytes
Actually implemented in a hierarchy of different memory types
Operating system provides private address spaces for each “process”, or program being executed
Program can clobber its own data, but not that of others
Early PC operating systems could have done this, but didn’t
Compiler + Run-Time System Control Allocation
Allocation determines where objects declared or created by the program should be stored
All allocation occurs within a single virtual address space

5. Machine Words All Machines define a “Word Size”
Nominal size of integer-valued data
This includes machine addresses
Most current machines use 32 bit words (how many bytes?)
Limits physical address space to 4GB
Becoming too small for memory-intensive applications
High-end systems use 64 bit words
Potential address space  1.8 X 1019 bytes
x86-64 machines support 48-bit addresses: 256 Terabytes
Machines support multiple data formats
Fractions or multiples of word size
Always integral number of bytes

6. Word-Oriented Memory Organization
32-bit
Words
64-bit
Words
Bytes
Addr.
0000
Addr = ??
0001
Addresses Specify Byte Locations
Address of first byte in word
Addresses of successive words differ by 4 (32-bit) or 8 (64-bit)
0002
0000
0004
0008
0012
Addr = ??
0003
0004
0000
0008
Addr = ??
0005
0006
0007
0008
Addr = ??
0009
0010
Addr = ??
0011
0012
Addr = ??
0013
0014
0015

7. Data Representations Sizes of C Objects (in Bytes)
Data Type 32-bit Intel IA x86-64
char
short
int 4 4 4
long 4 4 8
long long 8 8 8
float 4 4 4
double 8 8 8
long double 8 10/12 10/16
char * 4 4 8
Or any other pointer

8. Byte Ordering
How should bytes within multi-byte words be ordered in memory?
Two Conventions
Big Endian: Sun, PPC Mac, Internet
Least significant byte has highest address
Little Endian: Intel x86, MIPS
Least significant byte has lowest address
Former Sen. Dale Bumpers (D-Arkansas) proposed a third

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 0x
Address given by &x is 0x100
Big Endian
0x100
0x101
0x102
0x103 01 23 45 67 01 23 45 67
Little Endian
0x100
0x101
0x102
0x103 67 45 23 01 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
: 5b pop %ebx
: 81 c3 ab add $0x12ab,%ebx
804836c: 83 bb cmpl $0x0,0x28(%ebx)
Deciphering Numbers
Value: 0x12ab
Pad to 32 bits: 0x000012ab
Split into bytes: ab
Reverse: ab

11. Examining Data Representations
Code to Print Byte Representation of Data
Casting pointer to unsigned char * creates byte array
typedef unsigned char *pointer;
void show_bytes(pointer start, int len) {
int i;
for (i = 0; i < len; i++)
printf("0x%p\t0x%.2x\n",
start+i, start[i]);
printf("\n"); }
Printf directives:
%p: Print pointer
%x: Print Hexadecimal

12. show_bytes Execution Example
int a = 15213;
printf("int a = 15213;\n");
show_bytes((pointer) &a, sizeof(int));
Result (Linux):
int a = 15213;
0x11ffffcb8 0x6d
0x11ffffcb9 0x3b
0x11ffffcba 0x00
0x11ffffcbb 0x00

13. Representing Integers
Decimal: 15213
Binary:
Hex: B D
int A = 15213;
int B = ;
long int C = 15213; 6D 3B 00
IA32, x86-64 A 3B 6D 00
Sun A 6D 3B 00
IA32 C 6D 3B 00
x86-64 C 3B 6D 00
Sun C 00 93 C4 FF
IA32, x86-64 B C4 93 FF
Sun B
Two’s complement representation
(Covered later)

14. Representing Pointers
int B = ;
int *P = &B; FB 2C EF FF
Sun P FF BF D4 F8
IA32 P FF 7F 00 0C 89 EC
x86-64 P
Different compilers & machines assign different locations to objects

15. Representing Strings Strings in C Compatibility char S[6] = "15213";
Represented by array of characters
Each character encoded in ASCII format
Standard 7-bit encoding of character set
Character “0” has code 0x30
Digit i has code 0x30+i
Strings must be null-terminated
Final character = 0
Compatibility
Byte ordering not an issue
Linux/Alpha S
Sun S 32 31 35 33 00 32 31 35 33 00

16. Boolean Algebra Developed by George Boole in 19th Century
An algebraic representation of logic, which encodes “True” as 1 and “False” as 0
Basic operations: AND, OR, NOT, XOR
Given boolean variables P, Q, the operations mean the following:
P AND Q is 1 if both P and Q are 1, and 0 otherwise
P OR Q is 1 if either P or Q are 1, and 0 otherwise
NOT P is 1 if P is 0, and 0 otherwise
P XOR Q is 1 if P and Q are different, and 0 otherwise

17. Truth Tables in Boolean Algebra
And is represented by ‘&’ in C:
p & q = 1 when both p=1 and q=1
Or is represented by ‘|’ in C:
p | q = 1 when either p=1 or q=1

18. Boolean Algebra Not is represented by the ‘~’ in C:
~p = 1 when p=0
Exclusive-Or (Xor) is the ‘^’ in C:
p^q = 1 when either p=1 or q=1, but not both

19. Application of Boolean Algebra
Boolean algebra was first applied to Digital Systems by
Claude Shannon in his MIT Master’s Thesis, which showed
that boolean operations could be modeled in electronic
circuits (1937).
The stage was set for building computers out of digital electronics.

20. Application of Boolean Algebra
Allows us to reason about networks of relay switches
Encode closed switch as 1, open switch as 0
A&~B
Connection when
A&~B | ~A&B A ~A ~B B
~A&B
= A^B

21. General Boolean Algebras
Operate on Bit Vectors
Operations applied bitwise
All of the Properties of Boolean Algebra Apply
& | ^ ~

22. Representing & Manipulating Sets
Representation
Width w bit vector represents subsets of {0, …, w–1}
aj = 1 if j  A
{ 0, 3, 5, 6 }
{ 0, 2, 4, 6 }
Operations
& Intersection { 0, 6 }
| Union { 0, 2, 3, 4, 5, 6 }
^ Symmetric difference { 2, 3, 4, 5 }
~ Complement { 1, 3, 5, 7 }

23. Bit-Level Operations in C
Operations &, |, ~, ^ all available in C
Can apply to any “integral” data type
long, int, short, char, unsigned
View each argument as a bit vector
Operations applied bit-wise
Examples (Char data type)
~0x41 --> 0xBE ~ >
~0x00 --> 0xFF ~ >
0x69 & 0x55 --> 0x & >
0x69 | 0x55 --> 0x7D | >

24. Contrast: Logic Operations in C
Contrast these to the Logical Operators &&, ||, !
View 0 as “False”
Anything nonzero as “True”
Always return 0 or 1
Early termination
Examples (using char data type)
!0x41 --> 0x00
!0x00 --> 0x01
!!0x41 --> 0x01
0x69 && 0x55 --> 0x01
0x69 || 0x55 --> 0x01
p && *p (avoids null pointer access)

25. Shift Operations Left Shift: x << y Right Shift: x >> y
Shift bit-vector x left y positions
Throw away extra bits on left
Fill with 0’s on right
Right Shift: x >> y
Shift bit-vector x right y positions
Throw away extra bits on right
Logical shift
Fill with 0’s on left
Arithmetic shift
Replicate most significant bit on right
Undefined Behavior
Shift amount < 0 or  word size
Argument x
<< 3
Log. >> 2
Arith. >> 2
Argument x
<< 3
Log. >> 2
Arith. >> 2

26. Encoding Integers Two’s Complement Unsigned Sign Bit
short int x = ;
short int y = ;
Sign
Bit
C short 2 bytes long
Sign Bit
For 2’s complement, most significant bit indicates sign
0 for nonnegative
1 for negative

27. Encoding Example (Cont.)
y = :