1. CSC 3210 Computer Organization and Programming
Chapter 3
DIGITAL LOGIC AND BINARY NUMBERS
D.M. Rasanjalee Himali

2. Outline Binary Hardware Devices Decimal and Number Systems
ASCII Representation of Characters
Bitwise Logical Operations

3. Binary Hardware Devices
Bistable Device:
Devices that are either on or off (have two states)
Computer is a bistable device
Bistable devices have two states, which might be called:

4. Binary Hardware Devices
A binary digit, or bit represent two states, 0 and 1,or on and off:
Two such bits could represent four states:
Three bits would have eight states:
How many states can n bits represent? Clearly, 2n states.

5. Binary Hardware Devices
BCD (binary coded decimal) :
Use 4 bits to represent decimal
Only uses 10 of the 16 possibilities

6. Decimal and Binary Number Systems
A subscript indicates the number’s base
For binary, base = 2
Ex: 1410 = 11102
When storing an n-bit binary number, n bistable devices are needed
11102 could be stored in a four-bit memory device

7. Decimal and Binary Number Systems
Bytes:
A group of 8 bits is a byte
A byte can represent 28 = 256 possible states
Registers are usually a multiple of bytes
SPARC registers have 32 bits (4 bytes)
232 = 4,294,967,296

8. Decimal and Binary Number Systems
Numbers in computers are stored in memory
Each memory cell will have an address.
Memory addresses are also binary numbers
Memory Addresses :
often 32 bits, these days
if each memory address maps to 1 byte:
232 bytes = 4 GB
K = kilo = thousand
but 1KB actually means 1024 bytes
1MB = 1024 x 1024 bytes
1GB = 1024 x 1024 x 1024 bytes

9. Octal and Hexadecimal Numbers
The translation of the groups of three bits :
It is difficult for a human to work with long strings of 0’s and 1’s
Octal and Hexadecimal are ways to group bits together
Octal:
base 8
Hexadecimal:
base 16
Use 0, 1, 2, 3, …9 for the first 10 symbols
Use a, b, c, d, e, and f for the last 6
The translation of the groups of four bits :

10. Octal and Hexadecimal Numbers
Converting from Decimal to Binary:
Ex: what is in binary?
Continue to divide by 2 until the quotient is zero. The remainders are the binary digits, LSB first:
The resulting number is

11. Octal and Hexadecimal Numbers
Converting from Decimal to Octal:
Divide repeatedly by 8, with the remainders the octal digits, least significant digit first.
Ex: converting to octal yields the number :
Converting from Octal to Binary:
Expand each into groups of 3 bits:
Ex: converting 5668 to binary yields :

12. Octal and Hexadecimal Numbers
Converting from Binary to Hexadecimal:
Ex: Converting to hexadecimal yields 17616
Group into 4 bits, from the right
If there are not enough bits, pad with 0’s on the left
Converting from Hexadecimal to Binary:
Expand each into to groups of 4 bits:
Ex: converting F0E516 to binary yields :
f16 = 11112, 016 => 00002, e16 => 11102, 516 => 01012

13. Octal and Hexadecimal Numbers
Decimal to Any Number Base:
Take the decimal number, and divide by the new number base
Keep track of the quotient and remainder
Repeat until quotient = 0
Read number from the bottom to the top

14. Octal and Hexadecimal Numbers
Any Number Base to Decimal:
From right to left, multiply the digit of the number-to-convert by its baseposition
Sum all results
Binary is base 2:
Example: convert to decimal
= 1x24 + 0x23 + 1x22 + 1x21 + 0x20
= 1x16 + 0x8 + 1x4 + 1x2 + 0x1 = =
So = 2210

15. ASCII American Standard Code for Information Interchange
Use byte values to represent characters
The assembler allows double-quotes

16. ASCII

17. Bitwise Logical Operations
Binary variables representing the two states 0 and 1 may be combined in boolean expressions
There are several binary operations:
NOT
AND OR
XOR
NAND
NOR

18. Bitwise Logical Operations
NOT:
The NOT operation simply complements a binary value
not (a) a’
a not(a)
0 1
1 0
AND:
It is true only when both its arguments are true; otherwise, it is false :

19. Bitwise Logical Operations
OR:
true when either or both its arguments are true
XOR:
exclusive or function
true only when one of its inputs is different, one true and one false

20. Bitwise Logical Operations
NAND:
the logical complement of and
NOR:
the logical complement of or

21. Bitwise Logical Operations
Possible Boolean Functions:
single boolean variable has two possible states
for each of these two possible states there are four possible functions:
With two variables there are four possible states, and for each of these four possible states there are 16 possible boolean functions:

22. Logic Instruction Examples
mov 0x47, %l0
and %l0, 0xca, %l1
andn %l0, 0xca, %l1
or %l0, 0xca, %l1
orn %l0, 0xca, %l1
mov 0x55, %l0
not %l0 42
mov 0x21, %l0
and %l0, 0x3c, %l1
or %l0, 0x3c, %l1
mov 0x55, %l0
xnor %l0, 0x3c, %l1
xor %l0, 0x3c, %l1 20 5 3d cf
ffffff96
ffffff77 69
ffffffaa

23. A Few More Logic Examples
In all the examples below, these registers have the following initial values:
%l0 = 0x
%l1 = 0x9abcdef0
What are the values for %l1 after the instruction?
xor %l0, %l1, %l1
not %l0, %l1
and %l0, %l1, %l1
or %l0, %l1, %l1
edcba987
9abcdef8

24. SPARC Instruction Formats
The bitwise logic instructions provided in the SPARC architecture are:
Instructions that perform the operation and set the condition codes are also provided:
The assembler recognizes:
as:

25. SPARC Logical Instruction Example
C code:
Assembly code:
if (a > 0)
b++;
cmp %a_r, 0
ble next
nop
add %b_r, 1, %b_r
next:
%a_r and %b_r will be replaced by the actual registers, such as %r2 and %r3

26. Synthetic Instructions
The alternative forms of instructions, or instructions with one operand always %g0, are called synthetic instructions.
Some synthetic instructions:
mov:
actually is an or instruction
is recognized by the assembler as :
cir:
Is recognized by the assembler as :
cmp:
tst:
Since %g0 ignores any updates, only the condition codes are affected
Note that The %r0, or %g0 always discards anything written to it and always has a value of zero

27. Synthetic Instructions
tst:
provided for the following situation :
Instead, we may write:
which will expand into:
Assembly code using cmp instruction:
C code:

28. FLAGS
Individual bits are frequently used to represent boolean flags and a word may contain 32 such flags.
Operations on flags typically involve setting, clearing, and toggling.
Common flag operations and mnemonics :
A synthetic instruction, btst, is provided to test if any or no flags are set:
which the assembler expands to:
Ex: test if either flags 0x10 or 0x8 are set in register %a_r, we would write: