Review – Binary numbers
Microcomputer Architecture and Interfacing
Colorado School of Mines
Professor William Hoff
1
Unsigned Binary Numbers
• Representation
• List the set of 3
bit numbers
… __ __ _ _ _ _
32 16 8 4 2 1
• Determining the value of an N-bit number
B  bn 1 2n 1  bn 2 2n 2    b1 21  b0 20
• Determine the
value of 100101
• Largest and smallest N-bit numbers
• Addition is done just like decimal
Microcomputer Architecture and Interfacing
Colorado School of Mines
• Add:
101101
+ 001110
Professor William Hoff
2
Twos Complement Binary
• The most significant bit (MSB) is negative
__ __ _ _ _ _
-32 16 8 4 2 1
• Determining the value of an N-bit number
B  bn 1 2n 1  bn 2 2n 2    b1 21  b0 20
• List the set of 3
bit numbers
• Determine the
value of 100101
• Largest and smallest N-bit numbers
• Addition is done just like unsigned (ignore
carry out of MSB position)
Microcomputer Architecture and Interfacing
Colorado School of Mines
• Add: 101101
001110
Professor William Hoff
3
Binary Arithmetic
• Negating twos complement numbers
– Just flip all the bits and add 1
– Example:
• +3: 011
• -3:
• Detecting overflow when adding two N-bit
numbers
– Unsigned
• Overflow occurs when there is a “1” carried out of the MSB
– Twos complement
• Overflow occurs when you add two positive numbers and get a negative
number, or add two negative numbers and get a positive result
Microcomputer Architecture and Interfacing
Colorado School of Mines
Professor William Hoff
4
Hexadecimal Numbers
• Representation
…
___
__ __ __ _
65536 4096 256 16 1
• Determining the value of an (unsigned) N-digit
hex number
H  hn 116n 1  hn 216n 2    h1161  h0160
Table 1.1 Prefixes for number bases
Base
Prefix
Example
Binary
Octal
Decimal
Hexadecimal
(shorthand hex)
Microcomputer Architecture and Interfacing
%
@
$
%10001010
@123467
12345678
$392
Colorado School of Mines
Professor William Hoff
Value Digit
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
A
11
B
12
C
13
D
14
E
15
F
5
Examples
• Which is the bigger number?
Hex: AB
Unsigned binary: 10100001
• Add these 2’s complement binary numbers, using 8-bit
arithmetic:
0100 0000
1100 0000
Microcomputer Architecture and Interfacing
Colorado School of Mines
Professor William Hoff
6