1. Digital Design: From Gates to Intelligent Machines
Bruce F Katz
Da Vinci Engineering Press

2. Number Systems Numbers and Numerals A number is a quantity
A numeral is a representation of a number
Example (all representations of the quantity 5)
5, V (Roman), 101 (binary), (Babylonian)
Not equivalent in ease of computation, however

3. Number Systems Positional Number Systems
A quantity is a weighted sum of powers of a base b
Compactness of representation and ease of computation
Additional characteristics
digits to the left of radix point are integral
digits to the right of the radix point are fractional

4. Number Systems Examples of Positional Number Systems base 2 base 10
= 1* * * *100
base 8
= 4*82 + 1*81 + 7*80 + 2* *8-2
= /8 + 3/64
base 2
= 1*23 + 1*21 + 1*2-1 =

5. Number Systems Commonly used bases decimal (base 10) binary (base 2)
octal
(base 8)
hexadecimal
(base 16) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
00000
00001
00010
00011
00100
00101
00110
00111
01000
01001
01010
01011
01100
01101
01110
01111
10000 17 20 A B C D E F

6. Number Systems Which base is best for human use?
Base 12 has the most divisors among the small numbers but we usually use base 10. Why?

7. Number Systems Conversion between bases (special case) Principle
If one base is an integer power of another base, can group by
this integer to perform conversion.
Examples
= ?8
solution: group by 3 bits
{101}{111}2 = 578
F416 = ?2
solution: each hex digit represents 4 bits
F416 = {1111}{0100}2 =

8. Number Systems Conversion to and from base 10 To base 10
Use definition of a positional number
Example: to base 10
= 1*23 + 1*22 + 0*21 + 1*21 + 1* * *2-3 =
From base 10
Use reformulation of definition of a positional number
Successive divisions by the base will yield then digits as remainders
Example: 125 to base 3
125/3 = 41 remainder 2
41/3 = 13 remainder 2
13/3 = 4 remainder 1
4/3 = 1 remainder 1
1/3 = 0 remainder 1
therefore answer is

9. Number Systems Binary Number Systems Motivation
Correspondence between 0 and 1 and logical values (true and false)
Ease of constructing binary circuits
Powers of 2 n 2n
significance 4 8 10 16 20 30 40
256
1,024
65,536
1,048,576
1,073,741,824
1.099E12
one nibble (1/2 a byte)
one byte; one ASCII char
quantity abbreviated by 1K (kilo)
two bytes; one UNICODE char
quantity abbreviated by 1M (mega)
quantity abbreviated by 1G (giga)
quantity abbreviated by 1T (tera)

10. Number Systems Binary Addition and Subtraction Subtraction Addition
Same as decimal addition with binary carries
carry
addend1
addend2
sum
Subtraction
Same as decimal subtraction with binary borrows
borrow
minuend
subtrahend
difference

11. Number Systems Binary Addition and Subtraction Tables
Important: These tables are the foundation for computer arithmetic!
addition
addend 1
addend 2
carry in
sum
carry out 1
subtraction
minuend
subtrahend
borrow in
difference
out 1

12. Number Systems Binary Multiplication Example
Easier than decimal multiplication because always multiplying by 0 or 1
Example *

13. Number Systems Representing Negative Numbers
System 1: Signed Magnitude
Leftmost bit represents negative sign
Examples
= 87
= - 87
Advantage
Simplicity
Disadvantage
Mathematical operations clumsy, e.g. addition:
if (signs same)then {
     add magnitudes
     give result this sign }
else     /* signs different */
     compare magnitudes
     subtract smaller from larger
     give result sign of the larger

14. Number Systems Representing Negative Numbers
System 2: 2’s complement
Positive numbers identical, negative numbers 2n - positive version, where n is the number of bits in the representation
Examples
= 17
= = - 17
Trick for computing negative number
Flip all the bits and add 1
after flip
after adding 1
Note: Negative numbers will always begin with 1, positive with 0

15. Number Systems Representing Negative Numbers Addition method
System 2: 2’s complement
Addition method
Just add! (and ignore any bits > 2n-1)
(43)
(-17)
Overflow condition:
If add 2 positive and get a negative or vice versa
Example
(-113)
overflow!

16. Number Systems Codes Example in base 2
A way of representing a set of quantities or a set of symbols within a given base
Example in base 2
decimal
Binary
BCD
gray
even parity 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
00000
10001
10010
00011
10100
00101
00110
10111
11000
01001
01010
11011
01100
11101
11110
01111

17. Number Systems
Codes
BCD
Each decimal digit is encoded by four binary digits
Motivation
ease of conversion
Examples
Gray Coding
Each successive number differs by only 1 bit from previous
counting with CMOS
Karnaugh maps

18. Number Systems Codes ASCII
Parity
An extra bit is added to make the string always even or odd
Motivation
error checking
ASCII
b6b5b4
b3b2b1b0
000
001
010
011
100
101
110
111
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
NUL
SOH
STX
ETX
EOT
ENQ
ACK
BEL BS HT LF VT FF CR SO SI
DLE
DC1
DC2
DC3
DC4
NAK
SYN
ETB
CAN EM
SUB
ESC FS GS RS US SP ! “ # $ %
& ‘ ( ) * + , - . / 1 2 3 4 5 6 7 8 9 : ;
< =
> ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~
DEL

19. Number Systems Codes 4 hex digits encode 216 characters Example
Unicode
4 hex digits encode 216 characters
Example

20. Number Systems Summary of topics Numbers and numerals
Positional number systems
Conversion between bases
Binary number systems
Binary addition, subtraction, and multiplication
Representation of negative numbers
Codes