Bits, Bytes and Boolean
CS 1 Introduction to Computers and Computer
Technology
Rick Graziani
Spring 2015
BIT – BInary digiT
• Bit (Binary Digit) = Basic unit of information, representing one of two
•
•
•
discrete states. The smallest unit of information within the computer.
The only thing a computer understands.
Abbreviation: b
Bit has one of two values:
– 0 (off) or 1 (on)
– 0 (False) or 1 (True)
OFF
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ON
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Bits
The boxes illustrate a position where magnetism may be set and
sensed; pluses (red) indicate magnetism of positive polarity (1 bit),
interpreted as “present” and minuses (blue) (0 bit).
0
1
1
0
1
0
0
0 1
0
1
1
0
1
0 1
• Two patterns are known as the
•
state of the bit.
For example, magnetic
encoding of information on
tapes, floppy disks, and hard
disks are done with positive or
negative polarity.
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Bits
• Bits are really only symbols.
• Used to display the one of two different, discrete states.
• Bits are used as:
– Storing data
• Numbers
• Text characters
• Images
• Sound
• Etc.
– Processing data
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Boolean Operations
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•
•
•
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Integrated Circuits (microchips) are used to store and manipulate (process)
bits.
This is done using Boolean operations (in honor of mathematician George
Boole, 1815-1864).
Boolean Operation: An operation that manipulates one or more true/false
values
Specific operations
– AND
– OR
– XOR (exclusive or)
– NOT
Using Truth Tables we can uses different sets of logic operations to store, add,
subtract, and more complicated operations with bit.
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Boolean Algebra and logical expressions
(Addendum)
• Boolean algebra (due to George Boole) - The mathematics of digital
•
•
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logic
– Useful in dealing with binary system of numbers.
– Used in the analysis and synthesis of logical expressions.
Logical expressions – Expressions constructed using logical-variables
and operators.
– Result is: True or False
Boolean algebra – In mathematics a variable uses one of the two
possible values: 1 or 0
May also be represented as:
– Truth or Falsehood of a statement
– On or Off states of a switch
– High (5V) or low (0V) of a voltage level
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Used in electronics
(Addendum)
• Electrical circuits are designed to represent logical
•
•
•
expressions
– Known as logic circuits.
Used to make important logical decisions in household
appliances, computers, communication devices, traffic
signals and microprocessors.
Three basic logic operations as listed below:
– OR operation
– AND operation
– NOT operation
A logic gate is an electronic circuit/device which makes
the logical decisions based on these operations.
Rick Graziani graziani@cabrillo.edu
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Logic gates
(Addendum)
•
•
•
Logic gates have:
– one or more inputs
– only one output
The output is active only for certain input combinations.
Logic gates are the building blocks of any digital circuit.
Rick Graziani graziani@cabrillo.edu
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Boolean Operations - AND
TRUE
TRUE
AND
= TRUE
• Truth tables (simple ones)
• AND operation
– Both input values must be TRUE for output to be TRUE
– Kermit is a frog AND Miss Piggy is an actress
– Inputs to AND operation represent truth of falseness of the
compound statement.
Rick Graziani graziani@cabrillo.edu
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Boolean Operations
• Gate:
•
– A device that computes a Boolean operation
– A device that produces the output of a Boolean operation when
given the operation’s input values.
Gates can be:
– Gears
– Relays
– Optic devices
– Electronic circuits (microchips)
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Boolean Operations – AND Gate
0
0
0
0
0
1
1
1
Rick Graziani graziani@cabrillo.edu
Inputs
Output
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
Truth Table
1
0 = FALSE
1 = TRUE
AND operation
• Both input values must
be TRUE for output to
be TRUE
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Boolean Operations - OR
TRUE
OR
True
= TRUE
• Truth tables (simple ones)
• OR operation
– Only one input values must be TRUE for output to be TRUE
– In Rick likes to surf OR Rick likes to go dancing.
– Taking both courses will also TRUE.
Rick Graziani graziani@cabrillo.edu
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Boolean Operations – OR Gate
0
0
0
0
1
1
1
1
0
1
1
Rick Graziani graziani@cabrillo.edu
1
Truth Table
Inputs
Output
0
0
0
0
1
1
1
0
1
1
1
1
0 = FALSE
1 = TRUE
OR operation
• At least one input value
must be TRUE for output
to be TRUE
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Boolean Operations - XOR
TRUE
XOR
False
= TRUE
• Truth tables (simple ones)
• XOR operation
– One and ONLY one input value can be TRUE for output to be
TRUE
– At noon Rick is going to surf the Hook XOR surf Liquor Stores (this
is a surf spot)
– Both cannot be true, as I cannot surf both spots at the same time.
Rick Graziani graziani@cabrillo.edu
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Boolean Operations – XOR Gate
0
0
0
0
1
1
1
1
0
1
1
Rick Graziani graziani@cabrillo.edu
0
Truth Table
Inputs
Output
0
0
0
0
1
1
1
0
1
1
1
0
0 = FALSE
1 = TRUE
XOR operation
• Only one input value is
TRUE for output to be
TRUE
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Boolean Operations – NOT Gate
0
1
1
0
Truth Table
Inputs
Output
0
1
1
0
0 = FALSE
1 = TRUE
NOT operation
• Only one input
• Opposite of input
NOT FALSE = TRUE
NOT TRUE = FALSE
Rick Graziani graziani@cabrillo.edu
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http://www.cs.kent.edu/~volkert/F1010051/notes/logsim.html
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Another way to write it…
0 = FALSE; 1 = TRUE
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Binary Math
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Base 10 (Decimal) Number System
Digits (10): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Number of:
104
103
10,000’s 1,000’s
102
100’s
1
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101
10’s
1
9
0
100
1’s
1
2
3
9
0
9
0
20
Base 10 (Decimal) Number System
Digits (10): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Number of:
104
103
10,000’s 1,000’s
4
1
1
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0
0
102
100’s
1
3
0
0
101
10’s
0
8
0
1
100
1’s
8
2
9
0
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Rick’s Number System Rules
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•
•
•
All digits start with 0
A Base-n number system has n number of digits:
– Decimal: Base-10 has 10 digits
– Binary: Base-2 has 2 digits
– Hexadecimal: Base-16 has 16 digits
The first column is always the number of 1’s
Each of the following columns is n times the previous
column (n = Base-n)
– Base 10: 10,000
1,000
100
10
1
– Base 2:
16
8
4
2
1
– Base 16: 65,536
4,096
256
16
1
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Counting in Decimal (0,1,2,3,4,5,6,7,8,9)
1,000’s 100’s 10’s 1’s
0
1
2
3
...
9
1
0
1
1
...
1
8
1
9
2
0
2
1
2
2
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1,000’s 100’s 10’s 1’s
. . .
2
9
3
0
3
1
...
9
9
1
0
0
1
0
1
...
9
9
9
1
0
0
0
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Counting in Binary (0, 1)
8’s
4’s
Dec
1
1
0
1
1
1
0
1
2
3
4
5
6
7
0
0
0
8
1
1
1
2’s 1’s
0
1
1
0
1
1
0
0
0
1
Rick Graziani graziani@cabrillo.edu
8’s
4’s
2’s 1’s
1
0
0
1
9
1
0
1
0
10
1
0
1
1
11
1
1
0
0
12
1
1
0
1
13
1
1
1
0
14
1
1
1
1
15
Dec
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Binary Math (more later)
0
+0
0
0
+1
1
111
+ 1
1000
1
+1
10
……
Rick Graziani graziani@cabrillo.edu
10
+1
11
11
+1
100
00000000
+
0
00000000
100
+ 1
101
->
101
+ 1
110
11111110
+
1
11111111
25
Base 2 (Binary) Number System
Digits (2): 0, 1
Number of:
27
26
25
24
128’s 64’s 32’s 16’s
Dec.
2
10
17
70
130
255
Rick Graziani graziani@cabrillo.edu
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22
21
20
8’s 4’s 2’s 1’s
1
0
1
1
0
0
26
Base 2 (Binary) Number System
Digits (2): 0, 1
Number of:
27
26
25
24
128’s 64’s 32’s 16’s
Dec.
2
10
17
1
70
1
0
0
130
1
0
0
0
255
1
1
1
1
Rick Graziani graziani@cabrillo.edu
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22
21
20
8’s 4’s 2’s 1’s
1
0
0
0
1
0
0
1
0
1
1
1
0
1
1
1
0
0
1
0
0
1
27
Converting between Decimal and Binary
Digits (2): 0, 1
Number of:
27
26
25
24
128’s 64’s 32’s 16’s
Dec.
1
0
0
1
0
0
0
0
0
1
0
0
0
172
192
Rick Graziani graziani@cabrillo.edu
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22
21
20
8’s 4’s 2’s 1’s
0
1
0
0
1
0
0
0
1
0
0
0
0
0
0
0
28
Converting between Decimal and Binary
Digits (2): 0, 1
Number of:
27
26
25
24
128’s 64’s 32’s 16’s
Dec.
70
1
0
0
40
1
0
0
0
0
0
0
128
1
0
0
0
172
1
0
1
0
192
1
1
0
0
Rick Graziani graziani@cabrillo.edu
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22
21
20
8’s 4’s 2’s 1’s
0
1
0
0
1
0
1
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
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Computers do Binary
0
•
•
•
1
Bits have two values: OFF and ON
The Binary number system (Base-2) can represent OFF
and ON very well since it has two values, 0 and 1
– 0 = OFF
– 1 = ON
Understanding Binary to Decimal conversion is critical in
computer science, computer networking, digital media, etc.
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Rick’s Program
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Rick’s Program
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Rick’s Program
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Decimal Math - Addition
10,000’s 1,000’s
1
+
1
10’s
1’s
1
1
6
5
1
0
1
6
5
9
5
-----------------------------
3
Rick Graziani graziani@cabrillo.edu
100’s
3
1
0
5
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Binary Math - Addition
1’s
Dec
1
1
0
1
0
1
1
0
1
1
-----------------------------
58
+ 27
----85
64’s 32’s 16’s
1
1
8’s
1
4’s
2’s
1
1
+
1
0
1
0
1
0
1
Double check using Decimal.
Rick Graziani graziani@cabrillo.edu
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Half Adder Gate – Adding two bits
XOR
Inputs: A, B
S = Sum
C = Carry
AND
A
+
Rick Graziani graziani@cabrillo.edu
B
=
2’s
1’s
36
Half Adder Gate – Adding two bits
0
0
Inputs: A, B
S = Sum
C = Carry
XOR
0
0
AND
A
0
+
Rick Graziani graziani@cabrillo.edu
B
0
=
=
C
S
2’s
1’s
0
0
0
+ 0
---0
37
Half Adder Gate – Adding two bits
0
1
Inputs: A, B
S = Sum
C = Carry
XOR
1
0
AND
A
0
+
Rick Graziani graziani@cabrillo.edu
B
1
=
=
C
S
2’s
1’s
0
1
0
+ 1
---1
38
Half Adder Gate – Adding two bits
1
0
Inputs: A, B
S = Sum
C = Carry
XOR
1
0
AND
A
1
+
Rick Graziani graziani@cabrillo.edu
B
0
=
=
C
S
2’s
1’s
0
1
1
+ 0
---1
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Half Adder Gate – Adding two bits
1
1
Inputs: A, B
S = Sum
C = Carry
XOR
0
1
AND
A
1
+
Rick Graziani graziani@cabrillo.edu
B
1
=
=
C
S
2’s
1’s
1
0
1
+ 1
---1 0
40
Marble Adding Machine
•
http://www.youtube.com/watch?v=GcDshWmhF4A&NR=1
&feature=fvwp
Rick Graziani graziani@cabrillo.edu
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Flip-flops
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•
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Flip-flop: A circuit built from gates that can store one bit, uses feedback.
A means of storing bits such as RAM
Modern computers use technologies with:
– greater miniaturization
– faster response times
– additional circuitry
DRAM (Dynamic RAM)
SDRAM (Synchronous DRAM)
PCs currently use DDR (double data rate) for RAM, DDR1, DDR2 and DDR3
– Type of SDRAM
– Each type has types of DIMM (dual in-line memory module) slots
(different number of pins)
Rick Graziani graziani@cabrillo.edu
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Example of Flip Flops storing bits (FYI)
•
•
•
S = Set
R = Reset
DRAM (Dynamic RAM)
– Each bit of data is stored in a separate capacitor within an integrated
circuit.
– Since real capacitors leak charge, the information eventually fades
unless the capacitor charge is refreshed periodically.
Rick Graziani graziani@cabrillo.edu
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Types of RAM
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•
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•
Understanding RAM Types: DRAM, SDRAM, DIMM, SIMM & More
– http://proprofs.com/mwiki/index.php?title=Understanding_RAM_Types:_DR
AM_SDRAM_DIMM_SIMM_And_More
RAM - From Wikipedia, the free encyclopedia
– http://en.wikipedia.org/wiki/RAM
DDR2 SDRAM - From Wikipedia, the free encyclopedia
– http://en.wikipedia.org/wiki/DDR2_SDRAM
Dynamic random access memory - From Wikipedia, the free encyclopedia
– http://en.wikipedia.org/wiki/Dynamic_random_access_memory
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Bits, Bytes and Boolean
CS 1 Introduction to Computers and Computer
Technology
Rick Graziani