number system in digital logic design

1. 1
Digital Logic Design

2. 2
Digital Logic Design
° Digital
- Concerned with the interconnection among digital
components and modules
» Best Digital System example is General Purpose
Computer
° Logic Design
- Deals with the basic concepts and tools used to design
digital hardware consisting of logic circuits
» Circuits to perform arithmetic operations (+, -, x, ÷)

3. 3
DigiDigi
taltal
SigSig
nalsnals
° Decimal values are difficult to represent in electrical
systems. It is easier to use two voltage values than
ten.
° Digital Signals have two basic states:
1 (logic “high”, or H, or “on”)
0 (logic “low”, or L, or “off”)
° Digital values are in a binary format. Binary means
2 states.
° A good example of binary is a light (only on or off)
on off
Power switches have labels “1” for on and “0” for off.

4. 4
Digital Logic Design
° Bits and Pieces of DLD History
° George Boole
- Mathematical Analysis of Logic (1847)
- An Investigation of Laws of Thoughts; Mathematical
Theories of Logic and Probabilities (1854)
° Claude Shannon
- Rediscovered the Boole
- “ A Symbolic Analysis of Relay and Switching Circuits “
- Boolean Logic and Boolean Algebra were Applied to Digital
Circuitry
---------- Beginning of the Digital Age and/or Computer Age
World War II
Computers as Calculating Machines
Arlington (State Machines) “ Control “

5. 5
Motivation
° Microprocessors/Microelectronics have
revolutionized our world
• Cell phones, internet, rapid advances in medicine, etc.
° The semiconductor industry has grown tremendously

6. 6
Objectives
° Number System, Their Uses, Conversions
° Basic Building Blocks of Digital System
° Minimization
° Combinational And Sequential Logic
° Digital System/Circuit Analysis and Design
° State Minimizations
° Integrated Circuits
° Simulations

7. 7
Text Book
° Primary Text:
“Digital Design” By M. Morris Mano and Michael D.
Ciletti
° Complementary Material
“Logic and Computer Design Fundamentals” By M.
Morris Mano & Charles R Kime.

8. 8
Digital Logic Design
Lecture 1
Number Systems

9. 9
Number Systems
° Decimal is the number system that we use
° Binary is a number system that computers use
° Octal is a number system that represents groups of
binary numbers (binary shorthand). It is used in
digital displays, and in modern times in
conjunction with file permissions under Unix
systems.
° Hexadecimal (Hex) is a number system that
represents groups of binary numbers (binary
shorthand). Hex is primarily used in computing as
the most common form of expressing a human-
readable string representation of a byte (group of 8
bits).

10. 10
Overvie
w
° The design of computers
• It all starts with numbers
• Building circuits
• Building computing machines
° Digital systems
° Understanding decimal numbers
° Binary and octal numbers
• The basis of computers!
° Conversion between different number systems

11. 11
Analog vs. Digital
Consider a faucet
Digital
Water can be flowing or NOT flowing
from the faucet
Two States
• On
• Off
Analog
How much water is flowing from the
faucet?
Advantages of Digital
Replication
• Analog
Try replicating the exact flow
from a faucet
• Digital

12. 12
Advantages of Digital
o Error Correction/Detection
• Small errors don’t propagate
o Miniaturization of Circuits
o Programmability
• Digital computers are programmable
°Two discrete values are used in digital systems.
°How are discrete elements represented?
• Signals are the physical quantities used to represent discrete
elements of information in a digital system.
°Electric signals used:
• Voltage
• Current

13. 13
Advantages of Digital/Representation of Binary Values
Volts
- 1 .0
0 .0
1 .0
2 .0
3 .0
4 .0
5 .0
6 .0
H ig h
L o w
°Why are there voltage ranges instead
of exact voltages?
• Variations in circuit behavior & noise
oTwo possible values
• 1, 0
• On, Off
• True, False
• High, Low
• Heads, Tails
• Black, White

14. 14

15. 15
Digital Computer Systems
° Digital systems consider discrete amounts of data.
° Examples
• 26 letters in the alphabet
• 10 decimal digits
° Larger quantities can be built from discrete values:
• Words made of letters
• Numbers made of decimal digits (e.g. 239875.32)
° Computers operate on binary values (0 and 1)
° Easy to represent binary values electrically
• Voltages and currents.
• Can be implemented using circuits
• Create the building blocks of modern computers

16. 16
Understanding Decimal Numbers
° Decimal numbers are made of decimal digits:
(0,1,2,3,4,5,6,7,8,9)
° But how many items does a decimal number
represent?
• 8653 = 8x103
+ 6x102 +
5x101 +
3x100
° What about fractions?
• 97654.35 = 9x104
+ 7x103 +
6x102 +
5x101 +
4x100
+ 3x10-1 +
5x10-2
• In formal notation -> (97654.35)10
° Why do we use 10 digits, anyway?

17. 17
Understanding Octal
Numbers
° Octal numbers are made of octal digits:
(0,1,2,3,4,5,6,7)
° How many items does an octal number represent?
• (4536)8 = 4x83
+ 5x82 +
3x81 +
6x80
= (1362)10
° What about fractions?
• (465.27)8 = 4x82 +
6x81 +
5x80
+ 2x8-1 +
7x8-2
° Octal numbers don’t use digits 8 or 9
° Who would use octal number, anyway?

18. 18
Understanding Binary Numbers
° Binary numbers are made of binary digits (bits):
• 0 and 1
° How many items does an binary number represent?
• (1011)2 = 1x23
+ 0x22 +
1x21 +
1x20
= (11)10
° What about fractions?
• (110.10)2 = 1x22 +
1x21 +
0x20
+ 1x2-1 +
0x2-2
° Groups of eight bits are called a byte
• (11001001) 2
° Groups of four bits are called a nibble.
• (1101) 2

19. 19
Why Use Binary Numbers?
° Easy to represent 0 and 1 using
electrical values.
° Possible to tolerate noise.
° Easy to transmit data
° Easy to build binary circuits.
AND Gate
1
0
0

20. 20
BinaryBinary
Base 2 = Base 10
000 = 0
001 = 1
010 = 2
011 = 3
100 = 4
101 = 5
110 = 6
111 = 7
In Binary, there are only 0’s and 1’s. These numbers are called “Base-2” ( Example:
0102)
BinarytoDecimal
We count in “Base-10”
(0 to 9)
° Binary number has base 2
° Each digit is one of two
numbers: 0 and 1
° Each digit is called a bit
° Eight binary bits make a byte
° All 256 possible values of a
byte can be represented using
2 digits in hexadecimal
notation.

21. 21
Binary as a VoltageBinary as a Voltage
° Voltages are used to represent logic values:
° A voltage present (called Vcc or Vdd) = 1
° Zero Volts or ground (called gnd or Vss) = 0
A simple switch can provide a logic high or a logic low.

22. 22
A Simple SwitchA Simple Switch
° Here is a simple switch used to provide a logic value:
Vcc
Gnd, or 0
Vcc
Vcc, or 1
There are other ways to connect a switch.

23. 23
Binary digits
Bit: single binary digit
Byte: 8 binary digits
100101112
Bit
Byte
Radix

24. 24
Conversion Between Number Bases
Decimal(base 10)
Octal(base 8)
Binary(base 2)
Hexadecimal
(base16)
° Learn to convert between bases.
° Already demonstrated how to convert
from binary to decimal.
° Hexadecimal described in next
lecture.

25. 25
Number Systems
System Base Symbols
Used by
humans?
Used in
computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa-
decimal
16 0, 1, … 9,
A, B, … F
No No

26. 26
Con
ver
sio
n
Am
ong
Bas
es
The possibilities:
Hexadecimal
Decimal Octal
Binary

27. 27
Convert an Integer from Decimal to Another
Base
1. Divide decimal number by the base (e.g. 2)
2. The remainder is the lowest-order digit
3. Repeat first two steps until no divisor remains.
For each digit position:
Example for (13)10:
Integer
Quotient
13/2 = 6 + ½ a0 = 1
6/2 = 3 + 0 a1 = 0
3/2 = 1 + ½ a2 = 1
1/2 = 0 + ½ a3 = 1
Remainder Coefficient
Answer (13)10 = (a3 a2 a1 a0)2 = (1101)2

28. 28
Convert an Fraction from Decimal to Another
Base
1. Multiply decimal number by the base (e.g. 2)
2. The integer is the highest-order digit
3. Repeat first two steps until fraction becomes
zero.
For each digit position:
Example for (0.625)10:
Integer
0.625 x 2 = 1 + 0.25 a-1 = 1
0.250 x 2 = 0 + 0.50 a-2 = 0
0.500 x 2 = 1 + 0 a-3 = 1
Fraction Coefficient
Answer (0.625)10 = (0.a-1 a-2 a-3 )2 = (0.101)2

29. 29
The Growth of Binary
Numbersn 2n
0 20
=1
1 21
=2
2 22
=4
3 23
=8
4 24
=16
5 25
=32
6 26
=64
7 27
=128
n 2n
8 28
=256
9 29
=512
10 210
=1024
11 211
=2048
12 212
=4096
20 220
=1M
30 230
=1G
40 240
=1T
Mega
Giga
Tera

30. 30
Binary
Addition
° Binary addition is very simple.
° This is best shown in an example of adding two
binary numbers…
1 1 1 1 0 1
+ 1 0 1 1 1
---------------------
0
1
0
1
1
1111
1 1 00
carries

31. 31
Binary Subtraction
° We can also perform subtraction (with borrows in place of
carries).
° Let’s subtract (10111)2 from (1001101)2…
1 10
0 10 10 0 0 10
1 0 0 1 1 0 1
- 1 0 1 1 1
------------------------
1 1 0 1 1 0
borrows

32. 32
Binary
Multiplication
° Binary multiplication is much the same as decimal
multiplication, except that the multiplication
operations are much simpler…
1 0 1 1 1
X 1 0 1 0
-----------------------
0 0 0 0 0
1 0 1 1 1
0 0 0 0 0
1 0 1 1 1
-----------------------
1 1 1 0 0 1 1 0

33. 33
Convert an Integer from Decimal to
Octal
1. Divide decimal number by the base (8)
2. The remainder is the lowest-order digit
3. Repeat first two steps until no divisor remains.
For each digit position:
Example for (175)10:
Integer
Quotient
175/8 = 21 + 7/8 a0 = 7
21/8 = 2 + 5/8 a1 = 5
2/8 = 0 + 2/8 a2 = 2
Remainder Coefficient
Answer (175)10 = (a2 a1 a0)2 = (257)8

34. 34
Convert an Fraction from Decimal to
Octal
1. Multiply decimal number by the base (e.g. 8)
2. The integer is the highest-order digit
3. Repeat first two steps until fraction becomes
zero.
For each digit position:
Example for (0.3125)10:
Integer
0.3125 x 8 = 2 + 5 a-1 = 2
0.5000 x 8 = 4 + 0 a-2 = 4
Fraction Coefficient
Answer (0.3125)10 = (0.24)8

35. 35
Summary
° Binary numbers are made of binary digits (bits)
° Binary and octal number systems
° Conversion between number systems
° Addition, subtraction, and multiplication in binary