Download Topic: Manipulating Data

Lesson Plan 4
Information Technology
Topic:
Manipulating Data
Objectives:
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Describe how data are stored and manipulated within the computer
Binary, octal and hexadecimal number systems;
Binary addition
Sign and magnitude
BCD (binary coded decimal)
Two’s complement
Representation of characters, ASCII
A PowerPoint slide (Number Systems1.ppt) which would be given to students on their flash
drives at the end of the week.
Content:
What is a Number System?
A number system is the notational scheme used to represent quantities and is defined by the
number of symbols needed to represent an infinite quantity of values. For Example: Decimal or
Base 10 is a number system that uses 10 symbols – 0,1,2,3,4,5,6,7,8,9.
Number Values
 In the initial stages, the symbol has only the value assigned
 In the first grouping, the symbol’s value is compounded by one group of the base
 Each additional grouping increases in value exponentially by the base
Number Systems | 1
Standard Number Values
104
103
102
10,000
1,000 100
3
9
1
3
0
101
100
10
1
Number Values
8
5
(3x102)+(8x101)+(5x100) or 385
7
(7x100) or 7
8
2
8
8
0
0
0
1
5
0
1
0
9
9
1
2
6
8
5
4
Other Number Systems
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There are several number systems used throughout history.
The Babylonians used the sexagesimal system, based on the number 60,
The Romans used (for some purposes) the duodecimal system, based on the number 12.
The Mayas used the vigesimal system, based on the number 20.
Number Systems used today
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Binary
Octal
Decimal
Hexadecimal
–
-
Base 2
Base 8
Base 10
Base 16
The principles governing each of the number systems are the same. The only difference is the
number of symbols used in the system.
Number Systems | 2
Binary
Octal
Decimal
Hexadecimal
(Base 2)
(Base 8)
(Base 10)
(Base 16)
0
0
0
0
1
1
1
1
10
2
2
2
11
3
3
3
100
4
4
4
101
5
5
5
110
6
6
6
111
7
7
7
1000
10
8
8
1001
11
9
9
1010
12
10
A
1011
13
11
B
1100
14
12
C
1101
15
13
D
1110
16
14
E
1111
17
15
F
10000
20
16
10
11111111
377
255
FF
1.11E+10
3721
2001
7D1
For example, additions:
Binary
Octal
Hexadecimal
Number Systems | 3
The class would be shown how to convert from decimal to binary and vice versa. Examples were
shown and then the following exercises would be given:
Class Activity
Converting from decimal to binary:
a) 63
b) 25 c) 54 d) 81 e) 12
Converting from binary to decimal:
a) 10110111
b) 10101010 c) 10001111
Class Activity
1. Compute the following Binary additions.
a) 1101 + 0111
b) 1011011 + 1000110
d) 10111111 + 00101110 e) 10111 + 10011
c) 1101101 + 1010101
f) 110111 + 101101
Home Work
2. Compute the following Octal additions.
a) 1725 + 0576
b) 1016637 + 1564711
d) 13133777 + 00666666 e) 215401 + 565247
c) 7654123 + 2657321
f) 645355 + 236500
Class Activity
3. Compute the following Hexadecimal additions.
a) 1F24 + 028A
b) 0028F1F + BBAD02
d) 69879999 + A2897995
c) 124B0D + EE20C1
Negative Numbers
There are several systems used to represent negative and positive integers:
 Sign and Magnitude

One’s Complement
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Two’s Complement
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Binary Coded Decimal
Number Systems | 4
Sign and Magnitude
It uses one bit (usually the leftmost) to indicate the
sign. "0" indicates a positive integer, and "1" indicates
a negative integer. The rest of the bits are used for the
magnitude of the number. For example, -2510 is
represented as:
1001 1001
The sign "1" means negative
The magnitude is 25 (in 7-bit binary)
Examples:
Convert the following sign
and magnitude 8-bit:
1) -6310
2) -4210
3) -7010
4) 3010
5) 4310
Steps:
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Convert the number given to binary, ignoring the sign of the number
If a specific number of bits is required, add zeros on the left of the binary value
Change the leftmost bit of the answer to binary 1, if a minus sign is required or 0 for a
plus sign.
Question
With 8-bit sign-magnitude representation, what positive integers can be
represented and what negative integers can be represented?
Answer:
-12710 ... -0, 0 ... 12710
The leftmost bit is used for the sign, which leaves seven bits for the magnitude.
The magnitude uses 7-bit unsigned binary, which can represent 010 (as 000
0000) up to 12710 (as 111 1111). The eighth bit makes these positive or
negative, resulting in -12710 ... -0, 0 ... 12710.
One’s Compliment
In ones' complement, the inverse of a number is formed by complementing each bit.
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Convert the number to binary, ignoring the sign of the number
Add any required zeros on the left of the binary value so that it would meet the bit
requirement.
Change all 0s to 1s and all 1s to 0s
Number Systems | 5
Two’s Compliment
This is a binary number that is created by reversing all the bits and then adding 1 if the number is
negative. In computing, the two's complement often is used to represent negative numbers. For
example, to find the 8-bit two's complement of "-1", which is "00000001", reverse each of the
bits to get "11111110" and then add "1" to get "11111111".
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Convert the value from decimal to
binary
Present the answer using the
instructed number of bits by adding
zeros to the left
If the number is positive, then this is
the answer, if not, then follow the
other steps
Change all 0s to 1s and all 1s to 0s to
invert the bits of the binary number
Add one to the answer if the number
using binary addition
Class Activity
Convert to 8-bit One’s Complement and
Two Complement:
1)
2)
3)
4)
5)
-3410
-2710
-1710
-4110
-5310
Binary Coded Decimal
Binary-coded decimal, or BCD, is a method of using binary digits to represent the decimal digits
0 through 9. This involves converting each decimal digit to its four-digit equivalent.
0
1
2
3
4
0000
0001
0010
0011
0100
5
6
7
8
9
0101
0110
0111
1000
1001
Even the signs are assigned a four-bit code:
+ = 1110
– = 1111
Examples would be used to allow students to practice sign
and magnitude, one’s complement, two’s complement and
binary coded decimal.
Home Work
Convert the following to BCD:
1) -56710
2) -48910
4) -28210
5) -17310
3) -36110
Number Systems | 6
Representing Alphanumeric Characters
Every piece of data that is input into a computer, whether it is a number, symbol or some other
character, it would be represented inside the computer by binary codes (since this is the language
the computer understands). Each symbol is represented by a different code.
There are two coding systems that provide a unique binary character code for each character for
use in the computer. They are:
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
ASCII – American Standard Code for Information Interchange
EBCDIC – Extended Binary Code Decimal Interchange Code
ASCII
This is a character encoding system used to define a set of characters which can be displayed by
a computer on a screen, as well as some control characters which have special functions. This
coding system is used on most microcomputers and represents the symbols as a 7 or 8 bit binary
code. The codes are sequential for the alphabet and for numbers. For example, if A is represented
as 1000001, then B would be 10000010. (To make it 8 bit add a 0 to the left).
Converting in ASCII:
If we know the ASCII code for ‘A’ is 1000001, we can find the code for ‘H’. To do this:
512
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256
128
64
32
16
8
4
2
Convert the binary code to decimal.
(1000001 = 64 + 1 = 65)
Determine how far the required letter is from the letter given.
(‘H’ is 7 letters after ‘A’)
Add this number to the decimal value.
(65 + 7 = 72)
Convert this final decimal value to binary.
72 = 1001000 in binary
Therefore the ASCII CODE FOR ‘H’ is 1001000.
Character
0
1
2
3
4
5
ASCII Code
00110000
00110001
00110010
00110011
00110100
00110101
Character
A
B
C
D
E
F
ASCII Code
01000001
01000010
01000011
01000100
01000101
01000110
Character
N
O
P
Q
R
S
ASCII Code
01001110
01001111
01010000
01010001
01010010
01010011
Number Systems | 7
1
6
7
8
9
00110110
00110111
00111000
00111001
G
H
I
J
K
L
M
01000111
01001000
01001001
01001010
01001011
01001100
01001101
T
U
V
W
X
Y
Z
01010100
01010101
01010110
01010111
01011000
01011001
01011010
Abbreviation of Extended Binary-Coded Decimal Interchange Code. Pronounced eb-sih-dik,
EBCDIC is an IBM code for representing characters as numbers. Although it is widely used on
large IBM computers, most other computers, including PCs and Macintoshes, use ASCII codes.
EBCDIC, Extended Binary Coded Decimal Interchange Code, was developed in 1963 by IBM as
an extension to the older Binary Coded Decimal (BCD) standard. BCD encodes the 10 digits of
the standard counting system using 4 binary counters, or bits, where an individual bit can be
either a 1 or a 0.
BCD for Numeric Characters
Digit EBCDIC
0
0000
1
0001
2
0010
3
0011
4
0100
5
0101
6
0110
7
0111
8
1000
9
1001
EBCDIC is eight bits, or one byte, wide. Each byte consists of two nibbles, each four bits wide.
The first four bits define the class of character, while the second nibble defines the specific
character inside that class. For example, setting the first nibble to all-ones, 1111, defines the
character as a number, and the second nibble defines which number is encoded.
In recent years, EBCDIC has been expanded to 16- and 32-bit variants to allow for
representation of large, non-latin character sets. Each EBCDIC variant is known as a codepage,
identified by its Coded Character Set Identifier, or CCSID. EBCDIC codepages have been
created for a number of major writing scripts, including such complex ones as Chinese, Korean
and Japanese.
Number Systems | 8
All IBM mainframe computers, even the most recent models, continue to support EBCDIC so
that legacy applications can be run without modification.
References:
1)
2)
3)
4)
5)
6)
7)
Oxford Information Technology for CXC CSEC by Glenda Gay and Ronald Blades
Introduction to Information Systems by James A. O’Brien
http://chortle.ccsu.edu/AssemblyTutorial/Chapter-08/ass08_13.html
http://www.tpub.com/neets/book13/53s.htm
http://www.computerhope.com/jargon/t/twocompl.htm
www.wikipedia.com
http://www.wisegeek.com
Number Systems | 9