Introduction to Computing

<<< Previous Number systems, binary numbers, NOT, AND, OR and XOR logic operations Next >>> Introduction to Computing ­ CS101
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LESSON 8
BINARY NUMBERS & LOGIC OPERATIONS
The focus of the last Lesson was on the microprocessor
During that Lesson we learnt about the function of the central component of a computer, the
microprocessor
And its various sub-systems
Bus interface unit
Data & instruction cache memory
Instruction decoder
ALU
Floating-point unit
Control unit
Learning Goals for Today
To become familiar with number system used by the microprocessors - binary numbers
To become able to perform decimal-to-binary conversions
To understand the NOT, AND, OR and XOR logic operations ­ the fundamental operations that are
available in all microprocessors
BINARY
(BASE 2)
numbers
DECIMAL
(BASE 10)
numbers
Decimal (base 10) number system consists of 10 symbols or digits
0 12 3 4
5 67 8 9
Binary (base 2) number system consists of just two
01
Other popular number systems
Octal
base = 8
8 symbols (0,1,2,3,4,5,6,7)
base = 16
16 symbols (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)
Decimal (base 10) numbers are expressed in the
positional notation
The right-most is the least significant digit
4202 = 2x100 + 0x101 + 2x102 +
4x103
The left-most is the most significant digit
32 Introduction to Computing ­ CS101
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Decimal (base 10) numbers are expressed
in the positional notation
10
4202 = 2x100 + 0x101 + 2x102 + 4x103
10's multiplier
Decimal (base 10) numbers are expressed
in the positional notation
1
4202 = 2x100 + 0x101 + 2x102 + 4x103
1's multiplier
Decimal (base 10) numbers are expressed
in the positional notation
100
4202 = 2x100 + 0x101 + 2x102 + 4x103
100's multiplier
33 Introduction to Computing ­ CS101
VU
Binary (base 2) numbers are also expressed in
the positional notation
The right-most is the least significant
di i
10011 = 1x20 + 1x21 + 0x22 + 0x23 + 1x24
Binary (base 2) numbers are also expressed
in the positional notation
The left-most is the most significant digit
1
10011 = 1x20 + 1x21 + 0x22 + 0x23 + 1x24
1's multiplier
Binary (base 2) numbers are also expressed in
the positional notation
8
10011 = 1x20 + 1x21 + 0x22 + 0x23 + 1x24
8's multiplier
Binary (base 2) numbers are also expressed in
the positional notation
16
10011 = 1x20 + 1x21 + 0x22 + 0x23 + 1x24
16's multiplier
34 Introduction to Computing ­ CS101
VU
Counting in
Counting in
Decimal
Binary
10100
11110
20
30
0
1010
0
10
1
1011
10101
11111
11
21
31
1
10110
100000
32
10
1100
2
12
22
1101
10111
100001
23
33
11
3
13
100010
100
1110
11000
14
24
34
4
11001
100011
25
35
101
1111
5
15
10000
11010
100100
16
26
36
110
6
11011
.
.
111
10001
7
17
27
10010
11100
.
28
.
1000
8
18
10011
11101
.
29
1001
9
19
.
8.1 Why binary
Because this system is natural for digital computers
The fundamental building block of a digital computer ­ the switch ­ possesses two natural states, ON &
OFF.
It is natural to represent those states in a number system that has only two symbols, 1 and 0, i.e. the
binary number system
In some ways, the decimal number system is natural to us humans. Why?
bit
binary digit
Byte = 8 bits
Decimal
Binary conversion
Convert 75 to Binary
2
75
remainde
2
37
1
.
2
18
1
2
9
0
2
4
1
2
2
0
2
1
0
0
1
1001011
Check
1x20 + 1x21 + 0x22 + 1x23 + 0x24 + 0x25 + 1x26
1001011 =
=  1 + 2 + 0 + 8 + 0 + 0 + 64
=  75
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Convert 100 to Binary
remainder
100
2
2
50
0
0
2
25
1
2
12
0
6
2
0
2
3
1
2
1
1
0
1100100
That finishes our first topic - introduction to binary numbers and their conversion to and from decimal
numbers
Our next topic is ...
8.2 Boolean Logic Operations
Let x, y, z be Boolean variables. Boolean variables can only have binary values i.e., they can have
values which are either 0 or 1.
For example, if we represent the state of a light switch with a Boolean variable x, we will assign a value
of 0 to x when the switch is OFF, and 1 when it is ON
A few other names for the states of these Boolean variables
0
1
Off
On
Low
High
False
True
We define the following logic operations or functions among the Boolean variables
Name
Example
Symbolically
NOT
x´
y = NOT(x)
AND
x·y
z = x AND y
OR
x+y
z = x OR y
xy
XOR
z = x XOR y
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We'll define these operations with the help of truth tables
what is the truth table of a logic function
A truth table defines the output of a logic function for all possible inputs
Truth Table for the NOT Operation
(y true whenever x is false)
X
y = x´
0
1
Truth Table for the NOT Operation
X
y = x´
0
1
1
0
Truth Table for the AND Operation
(z true when both x & y true)
X
y
z=x·y
0
0
0
1
1
0
1
1
Truth Table for the AND Operation
X
y
z=x·y
0
0
0
0
1
0
1
0
0
1
1
1
Truth Table for the OR Operation
(z true when x or y or both true)
x
y
z=x+y
0
0
0
1
1
0
1
1
37 Introduction to Computing ­ CS101
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Truth Table for the OR Operation
x
y
z=x+y
0
0
0
0
1
1
1
0
1
1
1
1
Truth Table for the XOR Operation
(z true when x or y true, but not both)
z=xy
X
y
0
0
0
1
1
0
1
1
8.3 Truth Table for the XOR Operation
z=xy
X
y
0
0
0
0
1
1
1
0
1
1
1
0
Those 4 were the fundamental logic operations. Here are examples of a few more complex
situations
z = (x + y)´
z = y · (x + y)
z = (y · x) w
8.4 STRATEGY: Divide & Conquer
z = (x + y)´
z = (x + y)´
x
y
x+y
0
0
0
1
0
1
1
0
1
0
1
0
1
1
1
0
38 Introduction to Computing ­ CS101
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z = y · (x + y)
z = y · (x
x
y
x+y
+ y)
0
0
0
0
0
1
1
1
1
0
1
0
1
1
1
1
z = (y · x) w
z = (y · x)
x
y
W
y·x
w
0
0
0
0
0
0
0
1
0
1
0
1
0
0
0
0
1
1
0
1
1
0
0
0
0
1
0
1
0
1
1
1
0
1
1
1
1
1
1
0
Number of rows in a truth table?
2n
n = number of input variables
What have we learnt today?
About the binary number system, and how it differs from the decimal system
Positional notation for representing binary and decimal numbers
A process (or algorithm) which can be used to convert decimal numbers to binary numbers
Basic logic operations for Boolean variables, i.e. NOT, OR, AND, XOR, NOR, NAND, XNOR
Construction of truth tables (How many rows?)
Focus of the Next Lecture
Next Lesson will be the 3rd on Web dev
The focus of the one after that, the 10th lecture, however, will be on software. During that Lesson we
will try:
To understand the role of software in computing
To become able to differentiate between system and application software
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